Noise-Sustained Patterns

^o

N 0 IS E SUSTAINED PATTERNS

Fluctuations and Nonlinearities

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World Scientific Lecture Notes in Physics - Vol. 70

markus loecher Siemens Corporate Research, USA

N O I S E SUSTAINED PATTERNS

Fluctuations and Nonlinearities

U G World Scientific

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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NOISE SUSTAINED PATTERNS Fluctuations and Nonlinearities Copyright © 2003 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-02-4676-5

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To my wonderful wife Sita and our miraculous children Aparna and Sachin

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Preface

While writing this book, I continuously tried to adhere to the following three guiding queries: (i) Why would you, the reader, care to know about noise-sustained patterns? (ii) Can I separate important concepts and essential insights from details and formalisms? (iii) Am I merely summarizing existing published work or can this book significantly enhance the depth and speed of learning the presented material? After plenty of soul-searching and many failed attempts at packing several decades' worth of research into this little tutorial, I came to the conclusion that simplicity and clarity would be of utmost importance. I also realized that a basic grasp of stability theory was crucial to a full comprehension and appreciation of the diverse role fluctuations and noise play in nonlinear systems. Having said that, it became clear that the definition and particularly the ubiquitous presence of nonlinearities needed to be pondered upon. Hence, the introduction strives to convince the reader of the beauty and richness caused by the interplay between noise, instabilities and nonlinearities. In that chapter, I deliberately chose a highly verbose and somewhat redundant style in order to make my point. I hope the reader's patience and endurance will be rewarded by the more technical and concise nature of the subsequent chapters. I consider the main goal of this book to be a speedy and up-to-date introduction to the emerging field of stochastic pattern formation. To me, vii

vm

Preface

the major impediments to efficiently study any new field are intimidation and information overkill. Obscuring simple (but powerful) concepts by complicated mathematical formalisms and nomenclature has unfortunately become a common side effect of scientific progress. At the same time, I strongly believe that, given the right guidance and "filtering tool", feelings of inadequacy and incompetence can be minimized. I hope this book can serve such an objective. The organization of the book is as follows: Chapter 1 gives a general introduction to noise-generated phenomena with particular emphasis on dynamics and stability theory. Chapter 2 on mathematical techniques and computational aspects can be omitted for a first read in case the reader merely desired to obtain a phenomenological overview of the field. Purely temporal stochastic processes are dealt with in Chapter 3. This includes classics such as Kramer's theory and buzz topics such as stochastic resonance. The far-reaching following chapter demonstrates the wealth of novel observations in systems with spatial extent. There we focus on stochastic resonance in arrays of coupled nonlinear elements. Chapter 4 is concluded with a section on stochastic pattern formation. The emphasis of Chapter 5 lies on front propagation and the role of noise in open flow systems. Finally, Chapter 6 should be considered as somewhat experimental and exploratory. The first section on the recently introduced minority games cautiously probes the beneficial function of a nonzero "temperature" on the volatility of market dynamics. The author draws analogies to stochastic resonance and suggests open, "hot" topics of research. The example of noise-sustained structures in the context of traffic dynamics should be regarded as a beautiful bridge between the abstract models introduced in Chapter 5 and real world observations. The author sincerely attempted for the conclusion (Chapter 7) to be more than an obligatory summary, outlining open problems and conveying a sense of relevance hierarchy.

Acknowledgements

This endeavor would have remained stuck in any one of a large number of local minima if it h a d n ' t been for the deterministic and relentless encouragement from my wife, Sita. She temporarily placed her career on a somewhat slower track so t h a t sanity remained the most frequent state in our family. She tried to make u p for whatever lack of attention our daughter had to suffer from my preoccupation with this book. I t h a n k our amazing daughter for being such an incredible sport and a constant source of cute noise in our house. I will be forever grateful to my parents without whose support I would not be where I am. I am indebted to Siemens Corporate Research (SCR) for graciously allowing me to invest a large fraction of my energy into this book. The author has benefited from discussions and collaborations with many colleagues over the last many years, too numerous to mention individually. Very special thanks go to Fabio Marchesoni for generously allowing me to include the theory sections 4.1.3 and 5.3, the credit of which goes entirely to him, as well for his avid proof reading. I applaud Mark Spano for setting new proof reading standards. I never expected the detailed critiques, comments and encouragements he provided, all of which were invaluable; t h a n k you, Mark! It is with great pleasure t h a t I recall my days as a graduate student in the physics department at Ohio University. I consider myself extremely lucky to have had the chance to scientifically and personally mature in such a friendly, open and stimulating atmosphere created by the department. My thesis advisor Earl Hunt, who u p t o this day encourages me to pursue interesting science, deserves a good portion of t h a t praise. I ix

x

Acknowledgements

immensely benefited from working with Bill Ditto and Adi Bulsara during my postdoctoral stint at Georgia Tech. I further want to thank Chris Bizon, Andrea Cavagna, Jerry Gollub, John Lindner, Namiko Mitarai, Takashi Nagatani, Stefania Residori, John Sterman, Paul Umbanhowar and Alexei Zaikin as well as the American Physical Society (APS), the American Institute of Physics (AIP) and the Journal of the Physical Society of Japan (JPSJ) for kindly allowing me to reuse previously published figures.

Contents

Preface

vii

Acknowledgements

ix

Chapter 1 Introduction 1.1 Stability theory revisited 1.1.1 Noise and metastability 1.1.2 Linear stability analysis 1.2 Instabilities and nonlinear events in everyday life 1.2.1 Arms race 1.2.2 Distribution of wealth 1.2.3 Front nucleation and propagation 1.2.4 Supply chain dynamics 1.2.5 A hodgepodge of mechanical instabilities 1.3 Postscript

1 1 1 6 11 11 13 14 16 21 24

Chapter 2 Essentials 2.1 Probabilistic and information theoretic measures 2.2 Matrix manipulations 2.2.1 Circulant matrices 2.2.2 Singular value decomposition 2.3 Delay-differential equations 2.4 The fluctuation-dissipation theorem 2.5 The Fokker-Planck equation 2.6 Numerical techniques for the simulation of stochastic equations 2.7 Experimental aspects of generating noise

25 25 27 37 39 40 43 45 47 50

xi

xii

2.8

Contents

Complex integration

Chapter 3 Noise Induced Temporal Phenomena 3.1 Escape from metastable states 3.2 Stochastic resonance in bistable systems 3.2.1 Double-well potential 3.2.2 Two state theory 3.2.3 The diode resonator 3.2.4 The logistic map 3.3 Postscript

52 59 59 61 63 68 71 74 76

Chapter 4 Adding Spatial Dimensions 77 4.1 Spatiotemporal stochastic resonance 77 4.1.1 Array enhancement 78 4.1.2 Global vs. local dynamics 83 4.1.3 Kink nucleation in a 0), infinitesimal deviations from Xf will grow exponentially. A more detailed exploration including general material from linear algebra will follow in Sec. 2.2. The question arises whether the addition of noise to multivariate linear systems (1.12) might lead to surprising and interesting dynamics. As will be shown analytically in Chapter 2, the answer is clearly no as long as the

1,1.

Stability theory

revisited

9

Jacobian matrix A is normal (see definition 2.21). In the one-dimensional case of Eqs. (1.8) and (1.9) there exists only one fixed point for nonpathological A, and its stability attributes cannot be altered by additive noise. It is worth pointing out that this statement is no longer true if multiplicative noise is considered instead. If the noise term in Eq. (1.3) multiplies the state variable x fix

^

= Xx{t)+at)x{t),

(1.13)

the long time evolution of the average value of x(i) {x(t)) = x{0)e~xt

(1.14)

is controlled by the effective exponent X = X + D, where D > 0 is related to the noise variance by (£(£)£(£ + r)) = 2D5(T). From a mean field point of view, the fixed point indeed loses its stability for A > —D. The interpretation becomes clear by rewriting Eq. (1.13) such that the multiplicative noise arises as a consequence of fluctuations in A: x = [\ + £(t)]x(t). Accordingly, as long as the mean squared fluctuations are larger than the mean A, the fixed point Xf = 0 can become unstable even if A is negative. However, the definition of stability in the stochastic model necessarily is very different from the deterministic cases described so far. For an arbitrary stochastic variable 0 , the first few moments, {On(t)),n = 1 , . . . , iV W)~

W-" ,

(1.17)

where the exponent [i is of order 1. Power-laws are the subject of much interest and research, in particular among the physics community [220]. At first glance this might be surprising since power-laws comply with the very intuitive notion t h a t ordinarily small occurrences tend to be common whereas large events are observed rarely. To understand the significance of power-laws we should look at situations when they do not arise and why they are unambiguous signatures of collective or cooperative mechanisms. T h e central limit theorem (CLT) states the most remarkable fact t h a t the distribution of the mean of n independent identically distributed random variables, tends to a normal distribution as n tends to infinity, regardless of the distribution of the individual random variables. T h e CLT is responsible for the ubiquitous role of the normal or Gaussian distribution. Hence, for large-n non-collective processes one would expect the distribution of e.g. the sum x to exhibit exponentially decaying tails ~ exp([x — (x)]2 /2a2), with a2 = {x2) — (x)2. For large enough x any power-law yields substantially larger probabilities t h a n a Gaussian tail, often referred to as "fat tails". As a consequence, the agents' collective behavior leads t o greatly increased chances for large events (in this case large wealths) t h a n if they were acting independently. Another reason for the elegance and impact of power-laws is their universality; their scaling parameters are often independent of the exact details of the underlying process. One of their most celebrated appearances is in the context of phase transitions and critical phenomena [83]. To even allude to this vast and rich domain of past and active research is entirely beyond the scope of this book.

1.2.3

Front

nucleation

and

propagation

There exists another peculiar threshold-feedback mechanism encountered in various pattern-forming systems. If you ever paid attention to the way snow melts on skylights or paved surfaces you might have noticed t h a t the melting is far from uniform but proceeds in patches. T h e reason is t h e different heat absorption coefficient of snow a n d t h e material (glass or asphalt) underneath. As long as the layer of snow covers the ground completely, sunlight absorption is b o t h very low and uniform. However, if through chance (noise) a tiny hole or seed is formed within the snow

1.2.

Instabilities

and nonlinear events in everyday life

15

Fig. 1.6 One-dimensional caricature of the mechanism that gives rise to the "patchy" melting of snow. The solid line denotes the depth of the snow, while the superimposed (pink) dashed line shows the local heat absorption Q(x,t) as defined in equation (1.18). The black dashed line merely serves as the zero reference for Q. Time increases from top to bottom. The nucleation of a small groove (top panel) leads to a peak in the melting rate. The greatly enhanced heat absorption at the interface leads to outward front propagation and hence an expansion of the circular patch.

coating, the local heat absorption will become much higher, causing the snow to melt locally at a greatly higher rate. To a first approximation the dominant melting mechanism from then on will occur along the snowground boundary causing the (circular) patch to expand rapidly. The front separating the snow layer from the dry patch will move at an approximately constant speed. Figure 1.6 demonstrates this concept in one dimension. For the sake of this illustration we made the following crude and qualitative assumptions for the depth of the snow layer z(x,t), the melting rate —z(x,t) and the local heat absorption Q(x,t): Q(x, t) ~ A + exp(-7z 2 (x, t))

z{x,t)

-Q(x,t) 0

if z(x,t) > 0 if z(x,t) < 0

(1.18)

(1.19)

16

1.

Introduction

where x is the spatial coordinate and of course no melting occurs if the ground is dry (z = 0). The spatially uniform constant A (0 < A 0 if and only if all eigenvalues of J satisfy iftA*, < 0. It is unstable if at least one eigenvalue has 5RAfe > 0 and neutrally stable iu{t) is bounded) if SRAfc < 0. If (\j) > 0 for only one j , the solutions to (2.6) are unstable and therefore unbounded. Note that zero real parts correspond to neutrally stable solutions, which do not decay to zero but remain bounded. It is worth pointing out that the above definitions of stability are inherently asymptotic in nature. For long times, t —> oo, we know that the solution will shrink towards zero. Curiously, transient growth is nevertheless possible and requiring strict decay for all times introduces constraints on the matrix J + JT\ To illustrate this phenomenon, consider the coupled differential equations

* nonnormal transient amplification —> nonlinear interaction —> partial recreation of a misfit field —> further transient growth, etc." [91]. Consequently, laminar flow in a pipe can become unstable at Reynolds numbers for which the Jacobian matrix of the linearized equations of motions has exclusively negative eigenvalues. Inspired by the surprising complexity of nonnormal linear operators, we invite the reader to re-examine our apparently premature conjecture in Chapter 1, where we declared the role of additive noise in linear systems as decidedly "uninteresting". If the effect of fluctuations were to constantly rotate the vector u(t) away from the direction of the eigenvectors, we would expect the system never to reach equilibrium. Instead, random perturbations would continuously stimulate the inherent transient instability, reA misfit disturbance refers to an initial condition with a significant component orthogonal to the eigenvectors.

34

2.

JRIL

Quantiles of Standard Normal

Essentials

J

III.

Quanliles of Standard Normal

Fig. 2.2 Histograms and quantile-quantile (QQ) plots of a representative time series from the integration of Eq. (2.28) for zero initial condition, u(0) = 0, and noise strength a2 = 1. The variance of u% is significantly larger than the variance of ui\ (wf) = 1.94, (u|) = 0.26. Note the "S" shape of the u\ QQ-plot, indicating thinner tails than the normal distribution. The Kullback-Leibler distances (definition 2.3) between 141,2 and the corresponding normal distributions verify this qualitative assessment: KL(«i) = 3 X 10~ 3 , KL(ii2) = 1.3 x 10~ 3 . Numerical details: we employed the modified Euler scheme (2.91) with a time step of At = 0.01; the total number of integration steps was 10 6 . For an explanation of QQ-plots see the text. The KL distances were computed over 50 equally spaced bins.

suiting in sustained noise amplification. While we refrain from a thorough investigation of this hypothesis, we do want to pique the reader's interest in further exploration by means of the stochastic version of (2.18) -1 0

aw 6W

(2.28)

with independent noise sources £1,2 (£) of equal strength (£f) = (£f) = a . As expected from Eqs. (2.26, 2.27) and confirmed in Fig. 2.2, the amplitude of the resulting fluctuations is much larger for u\(t) than for u2(t), the dynamics of which is a simple decaying exponential (2.27). According to

2.2.

Matrix

manipulations

35

(2.72) and (2.74), the so-called Ornstein-Uhlenbeck process ii{t) = -1U(t)

+ at),

(2-29)

with (£(£)£(£')) = cr2 •Strictly speaking, only square matrices can be orthonormal. U is orthonormal in the sense that its column vectors are orthonormal: UTU = IM-

40

2.

Essentials

d.2- • • > d,M > 0 are known as the singular values. The columns of U are called the left singular vectors and the columns of V the right singular vector. The columns of UD are the principal components of X. The principal components of a set of data in M.N provide a sequence of best linear approximations to that data. For completeness, we should mention that definition 2.14, strictly speaking, should be referred to as the thin SVD [146]. The more general full SVD, X = UDVr, includes the Null space of XT in the following manner: D = diag(di,..., dk), where k = min (JV, M), and U e RNxN. If N > M, then the diagonal matrix D has the form ( dx

0 \

0 0

D

(2.50)

0

o7 and U is formed by including the N — M additional eigenvectors UM+I,

•• •

,UN-

U

[Mi|

. . . \UM\UM+1\

•••

\UN\

Note that the columns of U are the eigenvectors of XXT its nonzero eigenvalues: XXTUj

2.3

= d~u i ui

(2.51) corresponding to

•

Delay-differential equations

In section 1.2.4 we briefly touched upon the topic of delay-differential equations (DDEs). Equations (1.21) and (1.22) were examples of differential equations with a single delay. The existing literature [57; 149] naturally is a lot more general and deals with both multiple discrete delays, e.g., dx(t) dt

F[x(t-Ti),---,x(t-TN)].

(2.52)

2.3.

Delay-differential

41

equations

and continuously distributed delays, which are best described by integrodifferential equations such as dx{t) dt

F[x{t), f

H{t - T)x{r)dT].

(2.53)

J ~ oo

Here, we will only list a few of the main results and take the liberty to point out that the effect of noise in delay-differential equations has not been as extensively explored as its corresponding role in ODEs. For the reader's convenience we have provided a few pointers in the bibliography [92; 93; 144; 209; 210; 211; 212; 256]. We begin by addressing the natural question whether, in the limit of small delays, some of the stability properties from the theory of ordinary differential equations can be recovered. For scalar DDEs the answer is affirmative and twofold. It is a well-known fact that the solution of the elementary equation (]T

— = ax{t) => x(t) = x(0)eat

(2.54)

is asymptotically stable (limt->oo x(t) = 0) if and only if a < 0. For nonzero delays, dx — + ax(t - r) = 0, with a,r e (0, oo) ,

(2.55)

this decision boundary is somewhat more complex, given by the roots of the characteristic equation A + ae-XT = 0 .

(2.56)

A necessary and sufficient condition for all the roots of equation (2.56) to have negative real parts is 0 < ar < ~. However, in order to guarantee asymptotic stability the requirements on the product ar are more stringent, namely 0 < ar < 1. In order to emulate the exponential decay of the solution of (2.54) the upper bound is even smaller: 0 < ar < 1/e. In summary, we have three nested stability regimes: 0 < ar < | =• (V\)x+ae-x.=0Re(\) 0 < ar < 1 =>- lim x(t) = 0

x{t) = f0eXot,

(2.59)

42

2.

Essentials

where Ao is a real negative root of (2.56), and XQ can be considered as a generalized initial condition x0 =

r— ( x(0) + A0 / ex°ux(u)du 1 + A0r V J_T

). )

(2.60)

Obviously, limT_,o aJb = x(0). The generalization to multiple delays, time-dependent parameters and multidimensional DDEs is straightforward. For example, the case of two independent delays is covered by the following theorem — + a\x{t - T\) + a2x(t — r 2 ) = 0, with a i , a 2 , T i , r 2 G (0,oo) (2.61) 0 < a,\Ti + a2T2 < 1 => lim x{t) = 0 t—>oo

In the case of time-dependent coefficients, we still recover similar stability criteria as long as the suprema of a(t) and r(t) obey the following bounds: If 0 < r(t) < a and pt-^cr /•t-t-o-

0 < inf / t>oJt

ft-\-a rt+ir

a(s)ds < sup / t>oJt

o

a(s)ds
0. If for all j T-Me(fo) + \arg(Pj)\)

i S(u) = —J

r°° e-^G(r)dr,

(2.67)

which allows us to express the fluctuations in the time domain: (V(t + r)V(i)) = 2RkT5{r).

(2.68)

The practical and philosophical troubles associated with processes, which formally have zero correlation time (2.68) or — equivalently — a flat power

44

2.

Essentials

spectrum (2.66) will be addressed in the following section 2.6. For now, the important point to make is the linear relationship between the resistance (dissipation) and the noise (fluctuations) expressed by (2.68) and (2.66). Intuitively, this dependence can be understood either by recalling t h a t b o t h processes originate from the same physical mechanism of electron collisions with the lattice or in terms of energy balance, as follows. Voltage fluctuations across a resistor produce heat proportional to its resistance. This heat must precisely balance the energy extracted from the very same fluctuations. T h e derivation of the "simple" § fluctuation-dissipation theorem is short enough to be presented here. Begin with the one-dimensional stochastic evolution equation of a particle of mass m exposed to r a n d o m Fr(t) and deterministic Fd(t) forces as well as viscous damping proportional to its velocity: mv(t)

=-av(t)

+ Fr(t) + Fd(t).

(2.69)

Simple integration yields the general solution: v(t) = e - 1 " (v(0) + J

e~ = e-^+t^

(v2(0)

+ Fr{u2))duldu2

+ I'

T

e7(ui+U2)(^(wi)

).

(2.71)

Note t h a t the angled brackets (•) denote averaging over all possible trajectories. Assuming white noise, (Fr(Ul)

• Fr{u2))

= C5{Ul - u2),

(2.72)

with yet to be determined constant C, we leave it as an exercise to the reader to verify t h a t the double integrals reduce to (v(ti)v(t2))

= e-^tl+t2V(0) + ^

(V^*1-^ -

e-T('

1+t2

)) .

(2.73)

§Note that the general fluctuation-dissipation theorem is contained within the Onsager Relations [217; 77; 226; 271], which have much broader scope.

2.5.

The Fokker- Planck

equation

45

The exponentials describe transients and decay to zero for t\ = t2 = t —> oo. In summary, the final result for the mean square velocity fluctuations is therefore

(v(tf) = £- .

(2.74)

Up to now, we have only arranged terms and solved integrals. In order to ascertain the value of the proportionality constant C we need to provide further insight from statistical physics. We know that in equilibrium the equipartition of energy demands (E) = \m{v2) = \kT. Equating (2.74) with kT/m fixes the constant C = 2^kT/m = 2akT/m2. All in all, (2.72) reduces to the fluctuation-dissipation theorem, which states that in equilibrium the noise power (mean square fluctuations) has to be proportional to the damping constant (F r (ui) • Fr(u2)) = 2akT8(u1 - u2),

(2.75)

Johnson's equation (2.68) is recovered by replacing the viscous damping constant by the resistance a = R. In Chapter 4 we will re-encounter the fluctuation-dissipation theorem when defining the variance of noise in spatially extended systems.

2.5

The Fokker-Planck equation

As its name suggests, this celebrated partial differential equation was introduced by Fokker [62] and Planck [219]. Its original application was solely to describe the time evolution of the probability density of Brownian motion of particles. For educational reasons we defer its derivation for the moment and instead present the general Fokker-Planck (FP) equation for one variable x dW{x,t) dt

( d _(1)/ . d (-|^(1)(*) + |^

( 2 )

(z))^0M).

(2.76)

W(z,t) is the time-dependent probability density function (pdf) of finding x(t) in the interval (z, z + dz). Readers familiar with the heat or diffusion equation will at once recognize the diffusion coefficient D^ (x) and the drift proportional to D (1 '(a;). For the stochastic differential equation (2.69) the

46

2.

Essentials

Fokker-Planck equation would simply be

wpv

eJm»M_

z vmp) +T«:£.„

(2.81)

•

which is equivalent to the FP Eq. (2.76) with D^(x,t) =

f(x,t)+d^g(x,t)

D^(x,t)=g2(x,t).

(2.82)

2.6.

2.6

Numerical techniques for the simulation

of stochastic equations

47

Numerical techniques for the simulation of stochastic equations

Considering the vast amount of literature devoted to stochastic calculus in general [77; 226; 27l] and stochastic differential equations in particular [76; 148], the title of this section might suggest more ambitious goals than intended by the author. Here, we are merely trying to provide the reader with the necessary tools and insights to "solve" simple, first-order stochastic differential equations (SDEs) of the form ^

= f(y,t)

+ g(y,t) • £(t)

(2.83)

where £(£) represents white noise. Note that SDEs, in which the noise term £(£) appears linearly, are called Langevin equations. In passing we mention that any ordinary differential equation can always be reduced to the study of sets of first-order differential equations. For example, the n-th order equation

can be rewritten as a system of n first-order equations by introducing n new variables yi = dlyjdt% dyp dt dy\ dt dyn dt

2/2 (*)

F(yo,yi,...,yn-i,t)

Naturally, the same "trick" can be applied to higher-order Langevin equations to reduce them to the multivariate extension of (2.83). Since £(£) is a stochastic process, there exists an entire family of solutions y^(t) of Eq. (2.83) depending on the particular realization of £(£). "Solving" a Langevin equation can therefore have different connotations in different circumstances and could imply either a description of the complete probability density P(x, t) or frequently just the evolution of the first two mo-

48

2.

Essentials

ments (y(t)n). In this section we focus on the numerical realization of specific instances of the time evolution y(t). We begin with the most elementary technique for discretizing and iterating ordinary first-order differential equations, the Euler algorithm. Obviously, if g(y,t) = g is constant, the analytic solution of (2.83) is simply found by integrating: pt+At

y(t + At) = y{t) + /

pt+At

f(u)du

+g

£(u)du.

(2.84)

In the absence of the stochastic term, £ = 0, the original Euler algorithm would approximate the deterministic integral by its first Taylor term, Jt f(u)du ~ At • f(t), and the resulting iteration scheme y(t + At) = y(t) + Atf(t) would neglect terms of order {At)2. Conventionally, a numerical integration scheme is said to be of order n if its error t e r m decays as 0(Atn+1). At this point, it seems prudent to examine the properties of the stochastic variable £(£) in detail. T h e t e r m white noise alludes to its flat power spectrum and is inspired from optics, where white light contains all (visible) frequencies. T h e very definition of white noise as a zero-mean Gaussian process with autocorrelation mm

- T)> = 6(r)

(2.85)

leads to curious mathematical properties. Akin to the Dirac delta function S(t), white noise is nowhere differentiable and is not a continuous function. A flat power spectrum, which is equivalent to a zero correlation time, results in infinite total power, which is non-physical. Gardiner succinctly notes "White noise does not exist as a physically realizable process and the rather singular behavior it exhibits does not arise in any realizable context," (Chapter 1 in Ref. [77]). It might be most appropriate to consider white noise as a succession of uncorrelated, infinitely short but infinitely large pulses such t h a t their cumulative effect is finite. Fortunately, integrals of white noise are mathematically much more tractable. In fact, the Wiener process W(t) defined by W{t) = f

Z(u)du

(2.86)

J to

is a continuous, non-differentiable Gaussian variable with constant mean

2.6.

Numerical techniques for the simulation

of stochastic equations

49

and linearly growing variance (W(t)) = (W(0)) = Wo 2

{(W(t)-WQ) )=t-t0.

(2.87) (2.88)

Readers familiar with Brownian motion or discrete random walks in one dimension will immediately recognize the celebrated linear time dependence of the asymptotic mean square displacement1'. There indeed exists an intimate connection: in the limit of infinitesimally small steps I —> 0 and time intervals r —> 0 such that the ratio D = 12/T remains finite, the onedimensional random walk and the Wiener process are statistically identical processes. Reexamining the integrated Langevin equation (2.84), we deduce from (2.88) that the stochastic integral WAt{t) = Jt £(u)du is a Wiener (and therefore a Gaussian) process with variance (wAt(t)2)=At.

(2.89)

Of course, in computer simulations we do not have white noise but only random numbers at our disposal. Let us assume that C,n is a sequence of numerically generated, independent, Gaussian random variables of zero mean and variance unity ((Cn) = 0, (CnCm) = &n,m)- We are now in a position to address the dependence of the noise power on the time discretization step At. In order to match variances between the stochastic integral in (2.84) and our normalized Gaussian "noise" £„, we have to introduce the scaling factor y/~Ai wAt(t = nAt) = y/At-Cn-

(2.90)

Summarizing, the modified Euler algorithm to integrate (2.83) yields the difference equation y{t + At) = y(t) + At • f{t) + y/Al • g • (n .

(2.91)

Interestingly, the order of this integration scheme has decreased to 0(\J~A~i). The definition of the order of the error term needs to be clarified since "This famous problem can be summarized as follows: A person (the "random walker") is constrained to a line and, at discrete times t = nr, takes a random step of length I to the left or right with equal probability. In the limit of large n the mean square displacement of the random walker is proportional to time: ((x(t) — ^o) 2 ) — 2Dt with diffusion constant D = l2/r.

50

2.

Essentials

there are subtle differences from the deterministic case. For stochastic difference equations to be of order 0(Atn) implies that the mean-square error" ([y(t + At) - y(t + At) t r u e ] 2 ) is of order 0(Atn). This stepwise error is cumulative, so that the error after elapsed time t = NAt is of the order 0(Atn^1). Unfortunately, the order of the average mean square error of the moments tends to be even lower. For the sake of brevity, we present the results for arbitrary g(y(t)) in (2.83) without derivation. We encourage the reader to perform the necessary Taylor expansions and double integrals as a valuable exercise. The so-called Milshtein method [193; 194; 231] has the following form y{t + At) = y(t) + VAt • g(y(t)) • (n + At

f(t) + \9(y(t))-d-^-C

(2.92) plus a correction term of order 0(At / ). Note that for constant g{y(t)) = g the Milshtein method (2.92) correctly reduces to the Euler algorithm (2.91). It is remarkable that while there exist countless highly sophisticated techniques to numerically integrate deterministic ordinary differential equations [221], increasing the order of the correction term turns out to be much more difficult for SDEs. The reason is that Runge-Kutta-type predictor-corrector ideas and, e.g., the Bulirsch-Stoer method assume a reasonably smooth and predictable functional dependence. This is certainly not the case for differential equations containing discontinuous, erratic noise terms. In general, stochastic integration schemes tend to require much smaller time steps for satisfactory convergence and accuracy. Fox and Mannella recently proposed fast 2nd order schemes for the integration of SDEs in the presence of white [174] and colored [65] noise. The interested reader can find an extension of these algorithms to coupled SDEs or discretized stochastic partial differential equations in Appendix B. 3 2

2.7

Experimental aspects of generating noise

As was seen in the previous section, the "stochastic simulationist" faces many issues and subtleties in the utilization and design of numerical noise sources. The greater part of these difficulties is entirely due to the underlying, inescapable deterministic nature of computational noise. A true "The average y is over different noise realizations.

2.7.

Experimental

aspects of generating

noise

51

random number generating algorithm does not exist; given the same initial condition, or "seed", the same algorithm will reproduce the identical sequence of random numbers. In addition, because of the finite precision of digital computers, any series of numbers generated by a (necessarily) deterministic algorithm eventually will be periodic, albeit with a very large period. For this reason, the more appropriate term "pseudo-random" numbers is applied whenever context demands that level of exactness. In theory, "real" noise sources such as thermal fluctuations or electronic shot noise should be free of the artifacts and troubles associated with computationally simulated noise. Therefore, experimenters should have their respective desired fluctuations readily available by utilizing these microscopic noise supplies. Unfortunately, the need to amplify and adjust the magnitude of these fluctuations inevitably gives rise to an altogether new but equally subtle and challenging set of problems. The basic dilemma is that all amplifiers display a frequency-dependent gain. At the very least, there is a cutoff frequency beyond which the amplification attenuates significantly The bandwidth is normally defined as the frequency range over which the amplifier exhibits an approximately frequency-independent gain. As a rule of thumb, the larger this bandwidth the better — and costlier — the amplifiers tend to be. For the experiments described in subsequent chapters noise sources equivalent to the one sketched in Fig. 2.3 were used. The decision to amplify shot noise through a diode rather than thermal (Johnson) noise across a resistor was made due to the following advantages, which weigh heavily in practice: (a) The amplitude of shot noise is two orders of magnitude larger than Johnson noise. (b) The amplitude of shot noise depends on the current through the diode and can thus easily be adjusted in many noise sources simultaneously. For the interested reader we are now going to summarize the details of the implementation in the remainder of this section. The first amplifier stage in Fig. 2.3 transforms the current through the PN junction 1N4004 rectifier diode into a proportional voltage. In order to remove any DC bias and to provide zero mean noise, a O.l^F capacitor was added to attenuate frequencies below 1600 Hz. The total gain of the ensuing two amplifier stages is about 10000. Note that the power supply provided the same DC voltage

52

2. 3.3k

Essentials 103k

10k

Fig. 2.3 One realization of an experimental noise source. Amplifying the shot noise through a P N junction diode generates voltage fluctuations. The basic circuit was designed by Professor E. R. Hunt [34; 115].

to all 32 noise sources in the spatially extended setups, thereby providing a one-parameter control for all diodes simultaneously. The bandwidth of these noise sources was at least 500 kHz, which was much larger than the usually considered signal frequencies of ^50 Hz. In this context, considering the noise as white, i.e., to neglect its temporal correlations, seems appropriate.

2.8

Complex integration

In a letter dated Dec. 18, 1811 Gauss wrote to Bessel [78]: "... Was soil man sich nun bei J cpxdx fur x = a + bi denken? Offenbar, wenn man von klaren Begriffen ausgehen will, muss man annehmen, dass x durch unendlich kleine Incremente (jedes von der Form a + f3i) von demjenigen Werthe, fur welchen das Integral 0 sein soil, bis zu z = a + bi iibergeht und dann alle 4>x • dx summirt. So ist der Sinn vollkommen festgesetzt. Nun aber kann der Ubergang auf unendlich viele Arten geschehen: so wie man sich das ganze Reich aller reellen Grossen durch eine unendliche gerade Linie denken kann, so kann man das ganze Reich aller Grossen, reeller und imaginarer Grossen sich durch eine unendliche Ebene sinnlich machen, worin jeder Punkt, durch Abscisse = a, Ordinate = b bestimmt, die Grosse a+bi gleichsam reprasentirt. Der stetige Ubergang von einem Werthe von x zu einem andern a + bi geschieht demnach durch eine Linie und ist mithin auf unendlich

2.8.

Complex

integration

viele Arten moglich. Ich behaupte nun, dass das Integral J 4>xdx nach zweien verschiednen Ubergangen immer einerlei Werth erhalte, wenn innerhalb des zwischen beiden die Ubergange reprasentirenden Linien eingeschlossenen Flachenraumes nirgends (f>x = oo wird. Dies ist ein sehr schoner Lehrsatz, dessen eben nicht schweren Beweis ich bei einer schicklichen Gelegenheit geben werde. Er hangt mit schonen andern Wahrheiten, die Entwicklungen in Reihen betreffend, zusammen. Der Ubergang nach jedem Punkte lasst sich immer ausfiihren, ohne jemals eine solche Stelle wo cf)x = oo wird zu beriihren. Ich verlange daher, dass man solchen Punkten ausweichen soil, wo offenbar der urspriingliche Grundbegriff von J <j>xdx seine Klarheit verliert und leicht auf Widerspriiche fiihrt. Ubrigens ist zugleich hieraus klar, wie eine durch J cf>xdx erzeugte Function fur einerlei Werthe von x mehrere Werthe haben kann, indem man namlich beim Ubergange dahin um einen solchen Punkt wo (fix = oo entweder gar nicht, oder einmal, oder mehrere Male herumgehen kann. Definiert man z. B. logx durch J -dx, von x = 1 anzufangen, so kommt man zu log x entweder ohne den Punkt x = 0 einzuschliessen oder durch ein- oder mehrmaliges Umgehen desselben; jedesmal kommt dann die Constante +2ni oder — 2ni hinzu: so sind die vielfachen Logarithmen von jeder Zahl ganz klar . . . " The English translation would be along the lines of "... What are we supposed to think of J <j)xdx for x = a+bil Abiding by precise terms, we obviously have to assume that x runs from the value where the integral should be 0 up to z = a + bi by using infinitesimal increments (each of the form a + f3i) and summing up all 0.2"

\

\\

•-•

p o.o- 9 0

3D

°

{

_,

CO

2Z

°^

•-•-•-•-— 0

.

.

,

.

3 D/AV

4

.

_ J

6

Fig. 3.4 Amplitude of the spectral peak at v = S7/27T, defined in (3.16), and the signalto-noise ratio SNR as defined in Eq. (3.20) vs. noise power D. Parameters are identical to Fig. 3.3.

SNR = 2{

lim

Au^O

- ( ^

/

S(w)dw W S N ( n ) ,

(3.19)

n-At,

)

rK(D)

(3.20)

plotted versus the noise power D. Figure 3.4 is such a plot and is often called the "fingerprint" or "signature" of stochastic resonance. As a result of using the binary filter (3.12), both measures converge to 0 in the limit of vanishing noise power*. We note in passing that (i) the SNR and x(D) are maximized by noise powers DgfixR = A F / 2 and Dgax, which is the solution of rK(D)2 = Q2{~j- - | ) respectively, which are not identical, and (ii) the time-scaling condition Eq. (3.13) is not exact [66] and does not match either DgfixR or D^ax. *For the "raw", unfiltered time-series, the SNR grows without bounds and the spectral response approaches the input signal amplitude: l i m D ^ 0 {SNR, x} -> {co,A}.

68

3.2.2

3.

Two state

Noise Induced Temporal

Phenomena

theory

McNamara and Wiesenfeld [187] studied the simplest discrete model of a symmetric bistable system that switches between two states, which we denote by ±c respectively, with rate TK{D)- In this section, we investigate the effect of a periodic signal A(t) = AvsmQ,t on the probabilities p±(t) to find the system in state ±c at time t. We assume that in the adiabatic limit, f2 g(t)=

exp (a 0 i)

(3.26)

3.2.

systems

69

{1 + S(t')}

(3.27)

Stochastic resonance in bistable

Therefore,

9{t) Jt0

-ant

W.(t')g{t')dt'

oa0t

where s(t) = a 1 7]sin(0i - $ ) / \ / a o + ^ • Replacing p+{t0) in (3.23) with (5xoC = 8(xo,c) gives us the conditional probability p+(t\xo,to) that the system at time t is in the +c state given that the state at time to w a s ^o (which may be +c or —c): P+(t\x0,t0)

git)

Sxocg(to)+

/

(3.28)

W-{t')g{t')d£

The conditional probability density of the two-state output x(t) is p(x,t\x0,t0)

=p+(t\x0,t0)5(x

- c) + p-(t\x0,t0)6(x

+ c)

= p+{t\x0,t0)5(x

- c) + [1 - p+{t\x0, t0)} S(x + c),

which allows us to compute any desired statistical moment of x(t). conditional expectation value is {x(t)\x0,t0}

= /

(3.29) The

xp{x,t\x0,t0)dx

J — OC

=

(3.30)

c[2p+{t\x0,t0)-l].

To focus on generic behavior, independent of initial conditions, we form the asymptotic expectation value (x(t)) as : (x(t))tvs

=

lim

(x(t)\x0,t0)

to—+ — oo

c [2p»(t) - 1] ,

(3.31)

where p+(i) = p+(t\x0,t0 —> -oo). To calculate p+(£), notice that the initial condition term in (3.28) can be rewritten in terms of a definite integral: 6„

g{k g(t)

<Wexp

[W+{t') + W-(t')]dt'\

.

(3.32)

Since the rates W± may be assumed to be bounded from below by a positive constant, the integral in (3.32) will approach +oo as to —> - c o , and the

70

3.

Noise Induced Temporal

Phenomena

initial condition term (3.32) will decay exponentially, yielding P+(*) = ^

f ^

W- (t')g(t')dt> = \ [1 + 8(t)].

(3.33)

It follows from (3.30) that (x(t)rs

=

lim (x(t)\x0,to)

2 K A

= c-s(t) =

J f

sin(m - $ ) .

(3.34) Note that for xm = c the amplitude prefactor in (3.34) is identical with x{D) from Eq. (3.16). The autocorrelation function can be computed as (x(t + T)x(t)\x0,t0)

=

xyp(x,t + T\y,t)p(y,t\x0,to)dxdy.

(3.35)

In the stationary limit this greatly simplifies to (x(t)x(t + r ) ) a s =

lim {x(t +

r)x(t)\x0,t0)

= c 2 e^ ao|T l [l - s{t)2} + c2s{t + T)s{t).

(3.36)

Note that even in the stationary limit to —> — oo, the autocorrelation function is not time independent. Assuming the typical experimental case in which the starting times t from run to run are random with respect to the signal and modulation phases, it is appropriate to average the correlation function over the initial phases. Alternatively, we can avoid adding another phase variable to (3.21) by cycle-averaging the correlation function over t. Here, one cycle is defined as the period of W± (t). Cycle-averaging over t yields a time-independent correlation function K{T) = Ct/2nJQ27v/Q (x(t)x(t + T))asdt. Utilizing n

f2"'n

2,i.

4r2Kc2A2

2. . , _ 1 s (

' "* = » S # S i '

( 7)

"

and 2

r27r/Q

l

2 r

^ ) ^+ ^ =

D 2 (

4

r2A2 + Q2)cos»r,

(3.38)

3.2.

Stochastic resonance in bistable

systems

71

the asymptotic correlation function becomes K(T) =

1 4r2c2A2 c2e-2r'(x,t)-$(x,t)s+A

2

1d

^(x,t) K

„. . ,- . , ' ' +B(x) cosut+y/eri(x,t),

, (4.1)

where x £ [0,L] and rj(x,t) is white noise, ^-correlated both in x and t. In the absence of drive and noise, besides the obvious spatially homogeneous solutions $ o = 0 (unstable) and $± = i y / m (stable), there is a family of multi-soliton solutions $(fe) [14] that have k zeros in the interval [0, L], which are related to the kink-antikink pairs introduced later in this chapter. It was found that for open ("von Neumann") boundary conditions d$(x,t)/dx\x=otL = 0, the relevant potential barrier separating ± 77

78

4-

Adding Spatial

Dimensions

is not V[$0] - V[$±] b u t in fact AV = V [ $ « ] - V[$±\. A plot of the numerically obtained, kink-shaped solution ^ ^ can be found in [14]. In the weak noise limit, the mean exit times out of the homogeneous states $ ± are therefore T± ex e x p 2 A V / e . A thorough theoretical t r e a t m e n t of the closely related nucleation theory in a $ 4 model was put forward by Marchesoni et al. [180] and is t h e subject of section 4.1.3. A discretized version of the Ginzburg-Landau equation was simulated by Lindner et al. [161; 162]. Stochastic resonance in spatially extended systems has experienced a surge in research activities lately and can roughly be divided into spatially continuous domains [53; 85; 102; 265; 281] and coupled elements [2; 22; 37; 69; 117; 118; 119; 131; 135; 144; 166; 168; 203; 229; 234; 239; 250]. The enhancement of the visual perception of digitized pictures [240], reminiscent of the dithering effect [70], fits neither category. In this section, we present experimental and theoretical results for array enhancement and spatiotemporal synchronization in arrays of nonlinear elements. T h e role of the individual sites will be played by the three systems introduced in Chapter 3. Besides its fundamental appeal, spatiotemporal stochastic resonance promises important insights into the mechanism of information transferal in biological systems. T h e main results and figures are reproduced from Ref. [166].

4.1.1

Array

enhancement

T h e t e r m array enhanced stochastic resonance (AESR) was coined by Lindner et al. [161] to describe spatiotemporal SR in a numerical model of coupled, bistable oscillators. T h e dynamics of an individual element is governed by t h e double well potential (3.9). In analogy t o C h a p t e r 3, we consider the overdamped limit and symmetrically couple the oscillator into an array of N identical elements. Furthermore, we apply a global drive As'muit and local, i.e., spatially u n c o r r e c t e d noise (£,n{t)£,n'(t')) = Snn>S(t - t') to the array. The resulting A^-dimensional ordinary differential equation can be written as xn = -V'{xn)

+ e (xn+1

+ xn--i - 2xn) + Asinuit

+ £n(t),

(4.2)

where the prime denotes the spatial gradient, the dot denotes timedifferentiation, and the coupling is nonnegative, e > 0. Open b o u n d a r y conditions are employed, so t h a t for n = 1 and n = N the coupling t e r m in (4.2) is replaced by e (x2 - z i ) and e (XJV_I - xN) respectively. T h e maxi-

4-1-

Spatiotemporal

stochastic

resonance

79

mum signal-to-noise ratio was significantly improved for selected values of the coupling strength and the number of elements in the chain. It was also shown that optimizing the SNR of an individual oscillator coincided with the onset of maximal spatiotemporal synchronization. Scaling laws for the optimum noise intensities and coupling strengths resulting in a maximum SNR as a function of the number of oscillators, N, were derived in Refs. [119; 162]. AESR in a two-dimensional array was investigated by Sungar et al. [250]. The authors found that due to the extra degrees of freedom the maximum achievable SNR is significantly increased while the optimum coupling strength is reduced. Experimental evidence of AESR in a chain of diffusively coupled nonlinear resonators was first demonstrated by the author and coworkers [166; 168]. The SNRm.AX of the output of one of the middle resonators is observed to increase with the number of elements (here up to 32) for open and periodic boundary conditions. Both the optimal noise intensities as well as the optimal coupling strength grow with the length of the array. The array enhancement is most discernible when the resonators are fairly identical. In the experimental setup, the variation in the depth of the potential wells is slightly less than 10%. In this section, it is verified that global spatiotemporal synchronization coincides with optimized local performance of a single element in the chain. It is furthermore demonstrated that spatial correlation lengths are maximized by a nonzero noise power analogous to the behavior of the SNR. A block diagram of the symmetrically coupled diode resonators is given in Fig. 4.1. The next-neighbor coupling is realized via the coupling resistors Re- In previous work [123] the coupling strength was found to be roughly proportional to 1/Rc- The signals are virtually noise-free within the accuracy of instrument precision; the amount of correlated noise in the array is negligible. (For an interesting discussion on the interplay between internal and external noise in ensembles of nondynamical elements, see [69].) This situation is in close analogy to arrays of phase-locked loops in antennae, which can be assumed to be practically free of correlated noise but are exposed to spatially uncorrelated fluctuations. A different noise source is used for each site, so that there is zero noise correlation between sites. The individual noise sources are far from identical; the standard deviation in the voltage fluctuations between sites is around 10%. In the case of an array there are two additional dimensions in parameter space to be mapped out: the coupling strength as well as the number of

80

4-

Adding Spatial

Dimensions

35.1 kHz

Rc

Fig. 4.1 The circuit diagram sketching the individual noise sources and the 2nd drive added to the main drive. Each diode resonator consists of a diode and an inductor in series. The diffusive coupling is realized via the coupling resistor RQ • The coupling strength is roughly proportional to \/RQ.

elements. In Fig. 4.2, the maximum SNR for a fixed number of diode resonators (eleven) and various coupling strengths are plotted versus input noise intensity. Note that smaller values of RQ result in stiffer coupling. The horizontal bars are a measure of the width of the peaks of the SNR curves. For optimal coupling (in this case 100 kfi) the peak is very broad. Two main tendencies found in [161] are verified here. As the coupling becomes more and more rigid, (i) SNRma^ occurs at higher noise intensities, and (ii) its value initially rises, peaks at some intermediate coupling strength and abates. Intuitively, (i) and (ii) can be understood in the following way [161; 180]. With stiffer coupling, the chain behaves more and more like a rigid rod, so that a greater fraction of oscillators switch phase synchronously. But the intensity of the sum of N uncorrected Gaussian noise sources scales like v^V, thus requiring higher noise intensities at the individual sites as

4.1.

Spatiotemporal

38

z

81

150 k 6.8 kn

34

470 k I

a

Pi

resonance

100 k

36 P5

stochastic

32

30

/

1

28 -60

3.9 k " 2.2 m

uncoupled

-56 -52 -48 Output Noise Power (dB)

-44

Fig. 4.2 Maximum SNR of the middle oscillator ( N = l l ) for seven different coupling strengths. With optimum coupling, in this case Re = WOkfl, the SNR of eleven diode resonators surpassed the SNR of a single element by roughly 8 dB. The horizontal bars are a measure of the width of the peaks.

either N or the coupling increase. Figure 4.3 displays the SNR curves for various array sizes, optimized over coupling. The tails of the graphs are seen to fall off markedly less rapidly for higher number of resonators. This effect is also due to the fact that the noise sources are independent, hence partially canceling each other, whereas the signal power is being added coherently. In Fig. 4.4 the enhancement in SNR max , maximized over noise, versus coupling strength for various array sizes is plotted. The reference value of zero thus corresponds to the maximum SNR achieved with a single diode resonator. The increase in SNR max occurs rather markedly for three sites and appears to saturate with higher numbers. For very low as well as for very high coupling, the SNR max of the chain approaches that of a single oscillator, as discussed above. For 32 oscillators the maximum enhancement achieved is 9.5 dB. Only

82

4-

Adding Spatial

Dimensions

40

30 m •^ tr

z

CO

20

10

' 0.4

I 0.8

'

I 1.2

'

| 1.6

!

| 2.0

input noise intensity (arb. units) Fig. 4.3 SNR curves for different numbers of resonators. From top to bottom: N = 32,11, 3 , 1 . The respective coupling strengths are optimized. The curves and their peaks are seen broaden with increasing number of elements.

7 oscillators were sufficient to increase the maximum SNR by 8 dB. The optimal coupling resistor for 32 oscillators was found to be Rc=Vo kfl, compared to Re = I20kfl for 3 resonators. Since the coupling strength scales like 1/Re, this corresponds to an eight times stronger coupling. When applying the noise globally to five resonators, no coupling enhancement is detected, which appears in agreement with conclusions from earlier numerical simulations [119; 162]. The effect of different boundary conditions is most notable at low numbers of oscillators. When employing periodic boundary conditions with just three oscillators, the maximum reachable SNR is slightly lower and shifts towards weaker coupling. Intuitively, the effective coupling is lower for open boundary conditions than for a closed loop, explaining the shift. Already for 7 and 11 oscillators no dependence on the boundary conditions could be detected.

4-1-

Spatiotemporal

83

resonance

Coupling Resistance (k£2) 1000 100 10

10000 11 Un

stochastic

i n 11

i

i

i

1

i i

III 11 1

1

In i i i i

i

I

i

A ^

>

-A

m -«

8V

-

s

6-

hanc

aat

c

W

//

• -

*

t

+

N

r\ 'A-

v

''*

4-

X

J'

*
S

/'

A

'

N

*

V

s

N = 7\

•

•

\

v

•

+

2-

On 1

Fig. 4.4 coupling different enhance

4.1.2

'

1

•

'

i

''

'"•i

10 100 1000 10000 Coupling Strength (dimensionless)

Maximum SNR enhancement, maximized over noise, is plotted against the strength. To aid the eye the symbols are connected with dotted lines. The lines correspond to the different number of resonators. Larger arrays clearly the maximum SNR.

Global vs. local

dynamics

The emphasis so far has been on the performance of a single element in the array. In order to connect the local dynamics to the behavior of the entire array, the occupancy function is computed [161]. It constitutes a measure of the average spatiotemporal synchrony and is defined as the percent of resonators assuming the preferred phase at the extremes of the modulation signal. The extremes of the modulation signal are understood as the two unique phases of the beat frequency which result in maximum asymmetry in the size of the two basins of attractions. The respective values were inferred experimentally by choosing the amplitude of the secondary drive

84

4-

Adding Spatial

Dimensions

to lie barely above threshold and measuring the phase of the beat at the instant of a phase j u m p . By definition, an occupancy value of 100% corresponds to perfect spatial and temporal synchronization, whereas 50% occupancy can be caused by either complete spatial disorder (high noise power) or by a spatially uniform chain of resonators confined to one phase (low noise limit). T h e occupancy function is measured using a device t h a t periodically samples the state of all resonators simultaneously and then provides a sequential analog output of t h e states. This o u t p u t contains information on correlation lengths, and its time average is a measure of the occupancy function. Figure 4.5 clearly demonstrates the strong correlation between the locally measured SNR and the globally determined occupancy. T h e graphs almost mirror each other and peak at the same noise power, thus verifying t h a t the maximum SNR coincides with optimal spatiotemporal synchronization of the array [119; 161; 162; 168; 180]. While the occupancy function is an efficient, combined measure for spatial and temporal correlations, it hides some of the subtler, purely spatial features. In particular, we are interested in length scales, i.e., distributions of domain lengths, where domains are contiguous blocks of in-phase resonators. The three histograms in Fig. 4.5 illustrate the change in dominant length scales as t h e noise is varied. T h e histograms have to be interpreted with caution: the frequencies reflect the occurrence of domain lengths per temporal snapshot, which naturally favors shorter lengths. In order to obtain the probability of detecting a particular length at a r a n d o m site, the frequencies have t o be rescaled with weights proportional to their respective domain lengths. Low but finite noise power overcomes the coupling only sporadically, giving rise to a dominant peak at bin 32 and rather low frequencies for smaller domain lengths (far left histogram). For i n t e r m e d i a t e / o p t i m u m noise strength the peak at bin 32 is even more pronounced, and shorter domain lengths are almost entirely absent. The probability to detect a domain length > M at a r a n d o m site at a r a n d o m time might serve as a convenient quantifying measure for spatial correlation. For M = 32, this probability increases from 60% t o 80% from t h e far left histogram to the middle one. This counterintuitive behavior — adding more noise actually increasing spatial correlations — could be viewed as the spatial analogy to "temporal" stochastic resonance in a single system where a temporal correlation function is maximized by a finite noise power. High noise intensity leads to very

4-1-

Spatiotemporal

stochastic

85

resonance

40 .S-l

/ j 11 f t.Ln fl-J^El

10

|

re

-J-T^mj. Fa—

-J i I lu.^n f^rjji TP

20 30

in tj\ jn

rTTTrrr^£i—miU.

,10 20 30

10 20 30

\Length 100 90 80 ^ 70 ?

20

cs

60 g50 o

^ 10 t/3

!•

-60

1

1

1

1

1

1

-

-50 -40 -30 Output Noise Power (dB)

Fig. 4.5 Occupancy (A) and SNR (O) for 32 resonators versus noise power (periodic boundary conditions). Note that for this plot the intra-well motion was discarded, so that the SNR truly approaches zero for low noise power. The coupling resistor was chosen to be optimal, Re = 15fcf2. The histograms display the distribution of domain length scales, at noise powers 1.35, 1.6 and 2.3 (from left to right). The arrows indicate the corresponding values of occupancy/SNR. Each histogram is comprised of about 200 data points.

short spatial correlations, which is confirmed in the far right histogram. 4.1.3

Kink nucleation

in a 4 model

It was shown by Marchesoni and coworkers [180] that in the limit of strong coupling the discrete system of coupled Duffing oscillators from Ref. [161] approaches the dynamics of the (p4 model. Scaling laws were derived in a

86

Adding Spatial

Dimensions

subsequent paper [162] which confirmed the analogy between the discrete setup and the continuous a (x) and fix and the two internal modes of 4>N with discrete (nearly degenerate) eigenvalues A^ = \/3wo/2 [42]. A standard calculation yields the following analytical expression for T2 [180]:

ra=»i4(' IT ad

y^y^-A*,/**).

\STTUJQ J

\

kl

J

a> a.

0.3/ / / /

/ / /

w

a> *-•

/-25

/ / / /

0.2-

"35

////

>. "o 0.1 >

////

0.00

/-15 /

/ 0-1

/ // ° 5• °7

\s///

/ y / / / / ^ / / / § . M

1

U.U

/

i

'

i

-0.04

-0.02

•

°'°

i

-0.06

force Fig. 4.11 Same as Fig. 4.8 for the coupled map lattice. Shown are the kink speeds for different values of the coupling parameter (labeling the curves) as a function of applied bias/force.

should be regarded as an almost deterministic process. We measured the average number of iterations it takes to nucleate one pair under an applied bias as well as the average time it takes to fully decay into the stable phase afterwards. Figure 4.12 shows these two quantities as a function of noise intensity D along with the sum and the corresponding SNR. The sum will be the most accessible variable to be measured in experiments. It can be considered as the equivalent of the average waiting time for one uncoupled system. With the added spatial dimension it can be nicely dissected into two distinct components corresponding to the two processes described above. The time it takes to nucleate the first kink decreases exponentially with the noise intensity D. As can be seen in Figure 4.12, the subsequent decay time is a slowly decreasing function of the noise, which implies that

98

4-

Adding Spatial

Dimensions

800 12 tn

600 -

c o

CO i— O)

|

8

~ "o

400

40

200

0

"~' I ' 1 ' 1 0.024 0.028 0.032 0.036 noise strength

DC

0 0.040

Fig. 4.12 Average time ( # of iterations) it takes to nucleate the 1st kink (circles),then to fully decay (diamonds), the sum (squares) as well as the corresponding SNR (triangles, dashed line) versus applied noise. The minimum SNR results from the crossover of the two time-scales, i.e. when the nucleation time is equal to the subsequent full decay.

the (anti)kinks propagate faster for increasing noise. This effect is discussed and utilized in more detail in the following chapter. We plot the SNR on the same graph to illustrate a curious phenomenon: the minimum of the SNR is assumed at almost exactly the noise level where the two time-scales cross. The universality of this coincidence is still a topic of ongoing research. In order to stay close to the experiment, we also measured the half time for the coupled maps. Figure 4.13 is the equivalent of Figure 4.9 for the coupled map lattice. The measurement procedure is identical to the experiment: a step force (bias) of —0.01 is applied at time zero to the coupled maps initially in the less favorable phase. We then measure the number of iterations it takes for half the maps to decay into the stable phase.

4.2.

Doubly stochastic

99

resonance

10000^

(/) c o

«

1000-

E | OJ

> re

100 -

20

30 40 50 1/(noise strength)

60

Fig. 4.13 Same as Figure 4.9 for the coupled map lattice. The coupling values (increasing upwards) are e = 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4.(r = 3.2, Sr = 0.01).

4.2

Doubly stochastic resonance

Apart from short allusions in Chapters 1 and 2 we have — so far — more or less ignored the role of multiplicative noise [54; 86; 87; 147; 242]. In the 1980s, the appearance of new maxima in the probability distribution of nonlinear systems was observed when the intensity of the noise is increased. These novel states exist only in the presence of multiplicative noise, i.e., they have no counterpart in a deterministic or purely additive noise setting. The classic reference for such noise induced transitions is the book by Horsthemke and Lefever [106]. More recent work has focused on first and second order phase transitions in spatially coupled overdamped elements [50; 51; 145; 173; 199; 275; 278]. In particular, the interplay between additive and multiplicative noise has been investigated lately. For an updated, comprehensive review we refer the reader to the book by Garcia-

100

4-

Adding Spatial

Dimensions

Ojalvo and Sancho [75]. In this section we study an TV x N two-dimensional lattice of coupled Langevin equations of the form Xi = f(xi)+g(xi)&(t)

+ — J2

(x j ~ xt) + d{t) + A cos (ait+ (/)), (4.20)

jenn(i)

with / and g defined as f(x) = -ar(l + x 2 ) 2 , g{x) = a2 + x2 .

(4.21)

The sum in Eq. (4.20) runs over all nearest neighbors nn{i) of the zth cell, and the "flattened" index i denotes the two-dimensional cell position i = ix + N(iy — 1), with ix,y e [1,./V]. For the remainder of this section we fix the "purely additive component" of the multiplicative noise function to be a = 1. The additive and multiplicative noise terms are mutually uncorrelated (in space and time), zero-mean Gaussian distributed with auto-correlation functions:

(4-22)

&(W)>=*f<M*-*')> m)Q(t')) = ^kj(t-t')-

(4-23)

In the absence of periodic forcing, the system (4.20), which is monostable in its deterministic version, can undergo a phase transition to a noise-induced bistable state [50; 145]. Intriguingly, this noise-induced structure can act as an effective bistable potential that, in combination with the periodic drive and the additive noise, can exhibit collective stochastic resonance! This effect was termed doubly stochastic resonance [277] to emphasize that additive noise causes a resonance-like behavior in the structure, which is sustained by multiplicative noise. The following is a short summary of Ref. [277], which is the first report on doubly stochastic resonance. For A = 0, the model (4.20) can be solved analytically [75] by replacing the nearest-neighbor interaction by a global term in the Fokker-Planck equation corresponding to (4.20). The resulting steady-state probability distribution is given by Pst{x,m)

=

C(m)

=_fn

[x f(y) - D(y - m)d_

-exp 2 /

y^VOzO + a'c22

V^

2 21 s,

2

^V(2/)+CTc

V

•

(4-24)

4-2.

Doubly stochastic

resonance

Fig. 4.14 Transition lines between the ordered and disordered phases plane for different intensities of additive noise: o'i = 0 (curve 1) j? = 1 C (j? = 5 (curve 3). The black dot corresponds to D = 20, CTJ 3, which chosen to demonstrate DSR in the text. Reprinted with permission from

101

in the (D,a^) (curve 2), and are the values [277].

where C(m) is a normalization constant and m is the mean field, defined by m

xPst(x,

m)dx.

(4.25)

The mean field m serves as an order parameter that lets us distinguish between ordered (m ^- 0) and disordered (TO = 0) phases. In the ordered phase the system occupies one of two symmetric possible states corresponding to mean fields ra\ = —TO2 ^ 0. When increasing the multiplicative noise intensity, the system undergoes a phase transition from a disordered to an ordered state, as shown in Figure 4.14. Figure 4.15 demonstrates the delayed transition onset and reduced magnitude of \m\ when noise of growing amplitude is added. When choosing parameters such that the system remains in the disordered state for a range of additive noise powers (indicated by the black dot in Figure 4.14), the authors of [277] study its response to periodic forcing. Numerically simulating (4.20) with a time step At = 2.5 x 10~ 4 ,

102

4-

Adding Spatial

Dimensions

1.0 0.8 0.6

£ 0.4 0.2

0.0 0

2

4

2

6

8

10

Fig. 4.15 The order parameter |m| vs the intensity of multiplicative noise for D = 20 and cr? = 0 (curve 1), cr? = 1 (curve 2), and <J? = 5 (curve 3). Reprinted with permission from [277].

they find significantly enhanced synchronization between the forcing and the instantaneous mean field computed as m(t) = p ^ i ^ W f° r a n optimum value of the additive noise intensity. Time series of the mean field and the corresponding periodic input signal are plotted in Fig. 4.16. Besides the obvious SR-like dependence on additive noise power, we observe decreased amplitudes of hops as a result of the modified effective potential. In fact, the noise-induced bistablity disappears completely for large noise intensities ). The intensity of additive noise is increasing from top to bottom: ai = 0.01, cr? = 1.05 and <ji = 5 respectively. The middle row graph demonstrates optimum input/output synchronization. Parameters for all panes are N2 = 18 X 18, A = 0.1 LU = 0.1, D = 20 and a2 = 3. Reprinted with permission from [277].

process by which a spatially uniform state loses stability to a non-uniform state: a pattern. Spatiotemporal patterns appear spontaneously in a wide range of physical, chemical, and biological systems when they are driven sufficiently far from thermodynamic equilibrium. The classic example is Rayleigh-Benard convection in a fluid layer heated from below. For sufficiently strong heating, fluid motion sets in, typically in the form of convection rolls. We refer the reader to the excellent review by Cross and Hohenberg [41] for a comprehensive introduction. In this section we focus on two basic numerical models that, despite their simplicity, display many

104

4-

Adding Spatial

Dimensions

(»0

(a)

(c)

..,•

' • • . « •

w

1

"

"

*

**

.... " .-* - ^ v.* -ss s.

(a)

A «s •' *

(

Fig. 4.17 Upper panel: Snapshots of the field x for different values of correlated additive noise for D = 1.0, ai = 1.8, and cr? = 0. The parameter a increases from left to right: a = 0.1, a = 1.0, and a = 10. Lower panel: Snapshots of the field x in the case of uncorrected additive noise. The parameter which are loosely identified with the experimental dimensionless acceleration and driving frequency, respectively. For fixed kc/ka, as one increases r, the observed bifurcation sequence is a period-1 flat state bifurcating into a period-2 pattern, which then becomes a period-2 flat state, and eventually a period-4 pattern [263]. Among the observed patterns are stripes, squares, hexagons, kinks, and disorder. We refer the reader to Refs. [263; 264] for pictures, bifurcation diagrams, and a thorough stability analysis. In the following we are mainly concerned with the effect of additive and multiplicative fluctuations on the observed patterns. 4.3.1.3

Intermittent pattern switching

We can render the deterministic equations (4.28) and (4.29) into a stochastic model in at least four different, immediately obvious ways, namely: • additive local noise: Cn(xi) = M[£n(xi),r} + ej(n), where (el(n)e'l(n'))=ap(i-i')5(n-n'), • additive global noise: £,'n{xi) = M[£ n (xj),r] + 7](n), where (77(71)77(71')) =alS(n-n'), • multiplicative local noise: Cn(xi) — M[£n(xi),r + Ci(n)], where (O(n)Ci(n')) = x causes information to propagate towards increasing x, therefore introducing a "mean flow" [48]. The reader is referred to the same reference for a nice physical interpretation of Eq. (5.5). The CGLE (5.5) has been shown to be convectively unstable if [47]: ar

v2br ~r < 0 and ar > 0 , Abl

(5.6)

where the subscripts denote the real or imaginary part. The 2nd condition guarantees the instability, while the first one constrains the group velocity to be greater than a critical value. In this regime the CGLE displays features that are reminiscent of real "open-flow" fluid systems. For the following simulations chapter we choose the parameters to be in that regime. The boundary conditions are ^(0, t) = 0 and *S/XX(L, t) = 0 (L stands for the system-length, which is 300 in our case) [47]. We employed a finite difference scheme with fourth order space differencing, dx = 0.3, and 2nd order Runge Kutta for the time advancement, timestep dt = 0.012 [47]. Random numbers uniformly distributed in [—a, a] are added to vpr and ^i at all grid points at each time step. (The magnitude of a was of the order of 10~7.) Figure 5.2 displays the system after transients have settled down, having evolved in time from the initial condition ^(x,t = 0) = 0. The left boundary is fixed, ty(x = 0,t) = 0, and the right boundary is "open", d^(x,t)/dx\x=L = 0 Shown are snapshots of the absolute value and the real part of <J>, as the imaginary part does not yield any quali-

5.1.

Noise-sustained

structures in convectively unstable

200

media

123

300

Fig. 5.2 Numerical simulations of the complex Ginzburg-Landau equation (5.5) with fixed left and open right boundary. Shown is a snapshot after transients settled down. The upper figure displays the modulus of ^ while the real part is shown in the lower figure. The parameters are a = 2, v = 5.2, bT = 1.8, 6; = —1, cr = 0.5, c* = 1, thereby fulfilling the requirements for convective instability (5.6). The microscopic fluctuations grow to macroscopic proportions at x ~ 35.

tatively different insights. Deissler [48] observes three distinctly different regions which are confirmed in Fig. 5.2: the linear region, where the amplitude is small enough for the nonlinearities to be neglected, the transition region, and the fully developed region where nonlinear dynamics dominate. The correlation length £ is observable as the distance from the left boundary where the amplitude is kept at zero to the point in space where the

124

5.

Stochastic

Transport

Phenomena

microscopic noise is just beginning to grow to macroscopic proportions. A rough estimate from Fig. 5.2 gives £ ~ 35, which is also the extent of the linear region. As derived in [48], only a narrow band of frequencies of the broad-band noise is linearly amplified in space. The exponential growth of the waves is clearly seen in the plot of the absolute value of >3/. Even though the real and imaginary parts are fluctuating seemingly randomly in the linear growth region, the absolute value is virtually constant in time for x less than about 80. This is due to the selective amplification of mainly one frequency and the fact that the real and imaginary part are exactly 90° out of phase. The analytic expression for the frequency that is amplified most can be found in [48]. The picture also reveals a secondary (convective) instability. A coherent structure comprised of rather regular traveling waves which break up into turbulent patterns can clearly be noticed. The nature of this secondary instability is investigated in detail in [47]. 5.1.2

Unidirectionally

coupled diode

resonators

The notion of convective instabilities and noise-sustained structures is not limited to spatially continuous systems. In fact, the main requirements, (i) a nonzero propagation velocity and (ii) an inherently linear instability, can be readily met by coupling chaotic or linearly unstable elements into a chain. The diode resonator is a simple electronic circuit, comprised of the series combination of a 30 mH inductance and a General Instruments 852 silicon diode. This circuit has been well studied, in particular in the context of low-dimensional chaotic dynamics. Individually, the circuits follow the period-doubling route to chaos as the drive voltage is increased, and the peak currents through the diode form a nearly one-dimensional first return map [228]. The experimental system, schematically shown in Fig. 5.3, consists of 32 coupled diode resonator circuits driven by a sinusoidal source at a frequency of 70 kHz. The diodes were matched based on their bifurcation sequences such that they all go chaotic at roughly the same drive voltage. For symmetric coupling, the mean group velocity is zero and no convective instability can be observed. Instead, a one-way coupling, proportional to the difference (VJ - V^j~l), where Vj is the voltage across the ith diode, is employed by placing a buffer and resistor between neighboring diodes, as shown in Fig. 5.3. The coupling resistor R, determines the coupling strength. The effect of the coupling is most readily envisioned by consid-

5.1.

Noise-sustained

structures

in convectively unstable media

125

Fig. 5.3 The open flow circuit consisting of 32 diode resonator circuits. A buffer and a resistor R provide the unidirectional coupling.

ering the extreme cases. For R approaching zero, the voltage across the second diode is completely slaved to the first, and so on down the line. Hence all circuits are perfectly synchronized to the first. In the other extreme, with an infinite resistance between circuits, each circuit is completely independent of all of the others. The drive parameter is chosen such that each individual resonator would be chaotic; i.e., its period-1 fixed point is highly unstable. The right boundary is always "open" (coupled only to its left neighbor), but we altered the left boundary to be either fixed or open. The fixed boundary was realized by stabilizing the period-1 fixed point of the first resonator using chaos control techniques. Representative snapshots of the system dynamics for the two settings with parameters chosen to be in the convectively unstable regime are shown in Fig. 5.4. When the first site is allowed to evolve freely in time, a few of the following sites emulate the first chaotic map due to the synchronizing effect of the coupling. This synchrony is eventually destroyed at high enough site numbers since the spatially homogeneous solution is convectively unstable. Controlling the first resonator, one observes spatial period doubling leading to turbulent behavior at higher sites. Analogous phenomena are observed for coupled map lattices [6; 139], which will be addressed in the next section.

126

5.

Stochastic

Transport

Phenomena

T—I—I—|—I—I—I—I—|—I—I—I—I—|—I—I—I—I—|—I—I—i—I—|—I—I—I—I—|—i—r

I

5

10

15

20

,

.

, ,

I .

25

, ,

, I

, ,

30

site Fig. 5.4 The convective instability of the coupled diode resonators manifests itself in (a) breakup of the synchronous state and (b) spatial period doubling. The fixed left boundary condition in (b) was achieved by stabilizing the period-1 fixed point of the first resonator using chaos control techniques. Note that no noise was added to the systems. Inherent, tiny fluctuations are amplified to macroscopic proportion while being convected "downstream".

5.1.

5.1.3

Noise-sustained

Coupled

structures in convectively unstable media

127

maps

Nonlinear partial differential equations such as the CGLE, though already a significant simplification from the original equations of motion for fluid flow, nevertheless do not allow easy theoretical and numerical analysis. Qualitatively similar behavior can be observed in much simpler systems and models. Being discrete in time and space, coupled map lattices (CMLs) are computationally very feasible and yet able to exhibit highly complex patterns. The one-dimensional lattice of unidirectionally coupled logistic maps, first introduced by Kaneko [139], models the experimental system described in the previous section, which is discrete in space and continuous in time, quite well. Here, we are going to derive criteria, that help us to decide the stability type of a given coupled-map lattice [48]. Given the linearized difference equations Xk(t + 1) = axk{t) + e [xk-i{t) + -yxk+1(t)],

(5.7)

where 7 controls the asymmetry of the coupling (unidirectional for 7 = 0 and diffusive or symmetric for 7 = 1). The "spatial" index k covers a finite range k 6 (1,-W). For periodic boundary conditions, Eq. (5.7) can be rewritten as a matrix equation with the circulant N x N matrix Cp = circulant(a, £7, 0, • • •, 0, e). As we know from section 2.2.1 and Eq. (2.43), its eigenvalues are given by Af = a

i

+ ele

^U-i)/N

+ eei2,u-i)[N-i)/Ni

j =

1 )

...)iV.

(

5.

8 )

For open boundary conditions the Jacobian matrix changes into a tridiagonal matrix C° = tridiagonal(e7, a, e), the eigenvalues of which are given by Xf = a + 2^yecosj^,j

= l,...,N.

(5.9)

The only difference between C° and Cp is the removal of the upper righthand corner element e and the lower left-hand corner £7 due to the open boundary conditions. It should be noted that the transitory amplification of small perturbations leading to convective instability is of the same nature as the nonnormal linear transients discussed in section 2.2. Note that while the circulant matrix Cp is normal, for 7 ^ 1 the tridiagonal matrix C° is

128

5.

Stochastic

Transport

Phenomena

not! In general, the deviation from normality for a tridiagonal matrix /ad

au ctd ai

Oil

0

M

\ 0

•• •

0 au Osd

••

0 0 at

at

(5.10) ad)

is proportional to the difference between its squared off-diagonal elements /
1 for at least one j . For truly unidirectional coupling, 7 = 0, it is clear that this corresponds to \a\ < 1 AND m a x j 6 0 , - , M - i \a + eexp (i2nj(N - 1)/N)\ > 1. The latter sum can be simplified significantly, z = \a + eexp [i2irj(N — l)/iV]| 2 = \a + eexp[-i27rj/7V]| 2 = (a + ecosx) 2 + e2 sin2 x, with x = ~2TTJ/N. Z assumes its maximum at x = 0, — IT 43- j = 0, N/2 since 0 = dz/dx = —2e(a + ecosx) sin a; + 2e2 sin x cos x = —2aesinx. Depending on the sign of a, the maximum modulus of the eigenvalues X^ is either |« + e| or \a — e\, which leaves us with the following proposition Proposition 5.1 The unidirectionally coupled map lattice defined in (5.7) with 7 = 0 is convectively unstable if and only if \a\ < 1 and \a\ + jel > 1.

(5.12)

We will return to these results in section 6.2, which investigates spatiotemporal structures in traffic flow.

5.2.

5.2

Noise sustained front

transmission

129

Noise sustained front transmission

While the previous sections focused on the synchronization and noise reduction characteristics of spatiotemporal SR, for the remainder of this chapter we investigate its role in communication by way of coupled, noisy elements. To date, the role of noise in the encoding and transmission of information by neurons remains an open and challenging question [45]. It has been shown by Jung et al. [135] that noise can aid and optimize the spreading of spiral and single wave fronts in a numerical model of an excitable medium. On the experimental side, Kadar et al. [137] recently reported noise-supported traveling waves in a chemical sub-excitable medium, a photosensitive Belousov-Zhabotinsky reaction. Hu et al. [279] demonstrated that, with sufficient coupling, noise can induce undamped signal transmission in a numerical model of a chain of one-way coupled bistable elements. Furthermore, Sendiha-Nadal et al. [237] introduced random spatial fluctuations in an excitable medium and studied their effects on the propagation of auto waves, while Castelpoggi and Wio [25] recently addressed the problem of local vs. global coupling in reaction-diffusion characterizations of "stochastic resonant media". This section summarizes Refs. [165; 167], which demonstrated the constructive role of noise for the propagation of a signal in a metastable medium. The experimental setup, shown in block diagram form in Fig. 5.5, should be familiar by now and is modified in the following manner: The secondary drive, which we refer to as bias, is phase locked to the main drive at exactly half its frequency. As a consequence, there is no beat frequency as in the previous arrangement. Instead, the "input" signal is an induced phase change at one end of the chain, and a detector at the other end provides the "output". We refer to the less stable phase as the metastable state and investigate the effect of local and global noise on the propagation of a front into the more stable domain. The resonators are initially set in the metastable state and can be kicked into the more stable phase with a sufficiently large perturbation. Unlike excitable elements, which after "firing" return to their quiescent state via a refractory period, the resonators in this setup are reset by a global bias after departure from the metastable phase. This arrangement supports single, nonrepetitive wave fronts and provides a "very clean" experimental system for studying the underlying mechanisms of noise sustained wave propagation in subexcitable biological media [67; 132].

130

Fig. 5.5 Thirty two frequency / . An / / 2 on to the first site. provided by resistors

5.

Stochastic

Transport

Phenomena

diode resonators are placed in the period-2 state by the drive at bias serves to make one phase more stable and to force that phase Independent noise sources are applied to each site. Coupling is Rc- The detector outputs the phase of the last site.

The diffusive coupling and bias are chosen such that, in the absence of random fluctuations, the system is unable to sustain any sort of information transmission. We find that the quality and speed of signal propagation can be greatly enhanced by increasing the ambient noise level. Beyond some optimal level, the "channel" is quickly corrupted by noise, resulting in spurious signals and thus poor reliability. Two major forces affect each individual element: local random fluctuations and the influence of its two nearest neighbors. It is clear that in the absence of noise, a local phase jump will lead to a "domino effect" only if the coupling is strong enough. The energetically lower phase then propagates into the metastable phase in the form of a moving kink. The speed of this moving interface depends on both the coupling strength and the amplitude of the applied bias. If the latter two parameters are chosen low enough, kinks in discrete systems will fail to propagate.* Propagation failure of signals due to discreteness of the supporting medium has previously been observed in coupled chemical reactors [153; 154], *Note that in spatially continuous theories, such as a 4 or Sine-Gordon model, the speed of (anti)kinks is truly linear for low forcing amplitudes. A finite cutoff does not exist, and kinks ideally propagate even for extremely low force fields; for more details on discreteness effects in field theories, see, for example, [26; 38; 218].

5.2.

Noise sustained front

131

transmission

0.3 —i

O i_ CD Q. CD

>

0.2

ID

\_ CD Q_ 1/5 CD

CD CD

0.1 —

CL CO

:*:

0.0 0.0

1.0

2.0

3.0

bias (arb. units) Fig. 5.6 Kink speed as a function of applied bias. Kinks will not propagate for bias values less than approximately 1.0 unit.

in proton transport along hydrogen-bonded chains [282], as well as in theoretical and experimental studies of cardiac tissue [35; 142] and in the context of cell differentiation [59]. The kink speed for an intermediate coupling strength [166; 168] as a function of bias amplitude is shown in Fig. 5.6. No kink propagation could be observed for bias values much less than about 1.0 unit. In the experiment, two effects cause the finite cutoff: the discreteness of the system and non-identical elements. The resonators are loosely matched based on their bifurcation sequence. The variation of important parameters such as energy barrier between the two period-2 phases and the levels of the noise generators are within 10%. Since the parameters differ from site to site, kinks get trapped or slowed down at some resonators. The measured kink

132

5.

Stochastic

Transport

Phenomena

1

32 Site Number

Fig. 5.7 Staggered snapshots (every 32 drive cycles) of a phase kink moving under the influence of noise across 32 resonators. Local variations in its speed are clearly visible, due to both spatial inhomogeneities and the random forcing. The bias amplitude is 0.55 units and the noise intensity 1.3 units.

velocities should therefore be taken as average values over the entire chain, including local variations. Stronger coupling tends to smooth out differences between elements. We employed a bias value of 0.55 units, which is roughly 1/2 of the corresponding threshold. In this parameter regime, the system is not capable of kink propagation in the absence of random fluctuations. To measure the effect of noise on signal propagation, we force the system to reside in its metastable state; i.e., in the less stable period-2 phase. At the same time we induce the resonator at site 1 to switch phase and subsequently measure the phase of all 32 elements as a function of

5.2.

Noise sustained front

133

transmission

time. Figure 5.7 shows a staggered space-time plot of the entire system with the addition of 1.3 units noise. The induced phase kink exhibits local variations in speed due to the random forcing and the differences in the energy barriers of individual elements. Figure 5.8 displays, for four values of applied noise, the integrated probability of a phase change at site 31 as a function of time after an induced change at site 1. The detector emits a step function which, when averaged over approximately 100 events, produces the smoothly rising curves. In order to measure which part of the output is due to noise nucleated spurious signals, we repeat the same experiment without inducing a kink at the first site. These are shown as the dashed curves. For noise less 1.0 —I

0.5

-;

1.1 units noise

o.o l.o —| 0.5 —

1.5 units noise

o.o —r-

l.o 0.5 1.8 units noise

o.o

1.0 .a m _a o

0.5 2.2 units noise

0.0

' 1000

2000

I 3000

Time (drive periods) Fig. 5.8 Probability of a signal at site 31 with (solid line) and without (dashed curves) induced kink at site 1, for four noise levels increasing from top to bottom. The noise level of 1.1 units is too weak to nucleate any transitions.

5.

134

Stochastic

Transport

Phenomena

1.0

1.5

1.0 —i

0.8 —

>,

0.6 —

-t—'

J5 _Q

o £

0.4 —

0.2 —

0.0

0.5

2.0

2.5

Noise Intensity (arb. units) Fig. 5.9 Maximum of the difference between the probability distributions with and without signal vs. noise intensity for thirty coupled diode resonators. The maximum information transmission is assumed for noise levels 1.0—1.5 units.

than 1.0 units no signal gets through to site 31. At 1.1 units the signal always arrives but travels slowly with a wide distribution of arrival times. At 1.5 units there is a shorter transient time and a narrower distribution of arrival times, but spurious signals begin to appear. By 1.8 units about half the signal is caused by the noise. By 2.2 units the early arrival of spurious transitions - some nucleated near the end of the chain - virtually always preclude the detection of the original input signal. The derivative of an individual curve in Fig. 5.8 gives the probability distribution of arrival events: the probability that the 31st resonator changes phase at a certain time after the system is placed in the metastable state. Two competing effects corrupt the quality and reliability of signal detection in our system: for very low noise levels, the chances of successful signal propagation are

5.2.

Noise sustained front

transmission

135

1500

in T3 O

1000

i_

CD d CD

>

CD

500

0.0

0.5

1.0

1.5

2.0

2.5

Noise Intensity (arb. units) Fig. 5.10 Mode (circles) and width (crosses represent full width at half maximum) of the probability distribution of arrival events vs. noise intensity.

virtually zero, while for very high noise intensities, noise-induced, spurious events contaminate the output. We subtract from the distribution with an induced kink the distribution without induced kinks to obtain the probability (as a function of time) that the signal arriving at site 31 is due to the input signal. The integral of this quantity, while positive, gives the probability of successful signal transmission. This quantity, which is plotted in Fig. 5.9, displays an SR-like sharp rise and subsequent fall behavior. The peak of this curve occurs at noise intensities between 1.0 and 1.5 units, at which values the information flow is optimized by the presence of noise. Important information can be gathered from the distributions that include an induced kink at the first site. The skewed distribution functions shift towards shorter times with increasing noise intensity indicating a higher speed of information transmission. In the absence of noise the kink would

136

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Transport

Phenomena

arrive at site 31 after approximately 1050 drive periods, as indicated by extrapolating the line in Fig. 5.6 down to a bias of 0.55 units. In Fig. 5.10 we plot the mode (probability maximum) as a measure of the arrival times to illustrate the dependency on noise. Also plotted in Fig. 5.10 are the full widths at half maximum value. The distributions become narrower, denoting a smaller spread or variance in arrival times. The author wishes to stress that the spatiotemporal SR effect described in section 4.1 is of a fundamentally different nature. While earlier the system — at least in principle — was translationally invariant, here this spatial symmetry is broken by the application of a constant bias and appropriate initial conditions. The resulting unidirectional information flow allows us to define the boundaries as "input" and "output", regarding the system as a one-way communication channel. Furthermore, the "goal" was synchronization, i.e., concurrent noise-induced hopping of the entire array. Here, the stochastic events are strictly sequential. The most evident consequence is a greatly different scaling behavior: unlike the former examples, in which the coupling entails partial noise cancellation and improved signal-to-noise ratios, the quality of signal transmission in the current setup in fact deteriorates for higher numbers of elements. The sequential nature of the process leads to error accumulation instead of noise suppression. In order to be able to predict parameter values, such as noise intensity, coupling strength, and bias amplitude, which result in satisfactory and reliable signal propagation, we have to look at the underlying mechanism in more detail. There are two relevant time scales which govern the transition rates of the individual elements. The "unperturbed" escape rate Ku = /iexp (—Euf3) (where Eu denotes the threshold energy and (3 the inverse temperature or noise intensity), valid for a resonator with both neighbors in the same, metastable phase. If the left neighbor has performed the transition to the more stable phase, it will modify the energy threshold through the coupling, making the escape more likely: Kp = \i exp (—Ep(3) > Ku (with the perturbed threshold Ep < Eu). The degree to which Eu differs from Ep is a function of both the coupling and the bias amplitude; naturally, for zero bias or vanishing coupling, we have Ep = Eu. If we assume the time for the actual transition to be negligible, it is clear that on the average it will take N • Kpl time units for the kink to traverse TV sites. (This also explains the increase in kink speed with growing noise intensity, since K~l ~ exp (/?).) Over this period of time, the total probability that a resonator "ahead" of the moving front performs an (undesired) transition,

5.2.

Noise sustained front

transmission

137

is equal to Ku-K~l • N(N — l ) / 2 t. Thus, the signal deterioration will scale with the system size like N2. Prom the above considerations it is evident that for a low contamination with spurious signals, the ratio Ku/Kp should be as low as possible; i.e., Ku Ep. For a fixed bias, stronger coupling will increase the difference between Eu and Ev and thus improve on the signal transmission quality. 5.2.1

Propagation

failure in cardiac

tissue

Propagation failure of signals due to discreteness of the supporting medium has previously been observed in theoretical [142] and experimental [35] studies of cardiac tissue. In particular, Keener introduced a modified cable theory, which incorporates the discretizing effects of the so-called gap junctions [142]. Gap junctions, characterized by the (relatively high) intercellular resistance rg, provide the electrical coupling between cardiac cells. Mathematically, the propagation of action potential along cardiac cells is described by various cable theories, which are analogous to wave propagation in one-dimensional conductors (cables). Continuous cable theory either ignores the effects of the gap junctions or replaces the cytoplasmic resistance with an effective resistance; in either case the electrical resistance is assumed to be spatially homogeneous. Here, we focus on the opposite assumption that gap junctional resistance is much more important than cytoplasmic resistance. We thus neglect the dynamics within a cell and assume that the propagation of the action potential is dominated by the delay caused by the gap junctions [141]. Within the context of this discrete cable theory, we can write the current balance as [142] CmS^ at

= - (0„+i - 2„ + „_!) + 5 / m ( 0 n ) rg

(5.13)

where n is the transmembrane potential for the nth cell, S the surface area of cell membrane, and Cm is the membrane capacitance per unit area of membrane. Im specifies the inward ionic currents per unit area of membrane and is generally postulated to be a function of <j>n having three zeros. For t T h e probability that an unperturbed resonator escapes during the time Kp is equal to P = Ku • Kp1. If the kink has propagated i sites, there are N — i — 1 sites left to escape. Thus, after the Oth site is phase flipped, the probability that the remaining N — 1 sites nucleate a spurious kink is equal to Ptotal ~ SiL^j 1 P(N — i — 1) = P • N(N — l ) / 2 . The kink width is assumed to be negligible in this derivation.

5.

138

Stochastic

Transport

Phenomena

0.0 ^

o Coupling Fig. 5.11 Kink speed as a function of coupling strength d = — for the continuous model Eq. (5.14) (dashed line) and the discrete model Eq. (5.13) (solid line). Note that propagation in the discrete model is impossible for rg > r* (d < d* = — ) . For this simulation, CmS = 1 and SIm() = 12 V / 3>(1 - )((/> - 0.5) + 0.5

simplicity, we choose a simple cubic polynomial SIm{4>) = 12V3(1 — (j>) (4> — 0.5) +0.5 [142]. Note that though similar in appearance, Eq. (5.13) is not simply a discretization of its continuous analog km>->

0$ dt

L2 d 2 $ rg

dx2

SIm()

(5.14)

(where L is the size of the cardiac cell), but stands in its own right as a spatially discrete nonlinear wave equation. The most important observation is that propagation can fail in model (5.13) if rg is sufficiently large, but increasing resistances in Eq. (5.14) can never lead to propagation failure. Note that rg depends on the excitability of the tissue. Figure 5.11 shows a plot of the numerically determined speed (solid line) of propagation for model (5.13) as a function of the coupling strength d = l/rg as well as the analytically obtained kink speed c = cc°j^ Vd (dashed line) for Eq. (5.14). (The reader is referred to Ref. [142] for an explanation of Co and Rm-) It is evident that propagation is impossible for rg larger than a certain critical value r*, which turns out to be a monotonically increasing function of excitability [142].

5.2.

Noise sustained front

5

transmission

139

-

0

(d)

Fig. 5.12 Arrival probabilities for the discrete model Eq. (5.13) with (solid line) and without (dashed curves) induced kink at site 1, for five local and global noise levels increasing from top to bottom. The propagation distance spans 40 elements. The time axis is labeled in number of integration steps n. For the employed time step, dt = 0.05, the time range therefore translates from n = 0 , . . . , 6000 to t = 0 , . . . , 300.

140

5.

Stochastic

-f-

Global Nois

•

Local Noise

Transport

Phenomena

Fig. 5.13 Numerically measured average kink velocities as a function of noise strength for both global and local noise. It is insightful to compare these values with Fig. 5.11. The last data point includes a large fraction of spurious kinks which results in an artificially high value for the velocity.

We have performed digital simulations of the stochastic modification of Eq. (5.13):

dt

Pn+l

2<j>n + 0 n _!) + SIm(<j>n)(l + £M(t)) + U(t)

(5.15)

using the Euler-Maruyama algorithm [76; 148] with a time-step of dt = 0.05 and coupling strength e = 0.07 (40 elements). £A(£) and £M(£) are additive and multiplicative Gaussian white noise, bandlimited in practice by the Nyquist frequency IN = j^t- We quantify the noise by its variance a2 = 2D/N , where 2D is the height of the one-sided noise spectrum. Here, we only consider the case of purely additive noise, £M (£) = 0. By following a procedure analogous to the experiment, we obtain the probabilities for successful signal transmission as illustrated in Fig. 5.12. As in

5.2.

Noise sustained

front

transmission

141

the experiment, global noise provides for kink propagation at much lower values of a2 than local noise does. Figure 5.13 compares the velocities of the noise-propelled kink as a function of a2 for both global and local noise. The velocity of the propagating wave front shows an approximately parabolic dependence on the noise power in both cases. For the coupling strength employed, global noise leads to speeds about 15% greater than that observed for local noise. 5.2.2

Information theoretic ROC curves

measures

revisited:

In previous chapters, we frequently employed the signal-to-noise ratio (SNR) as a "performance" measure. However, in practical scenarios, the SNR is often not the best measure of performance. Indeed, a nonlinear signal processor may output a signal which has infinite SNR but is useless because it has no correlation with the input signal. Furthermore, in the absence of periodic input signals, the SNR becomes entirely meaningless. We therefore need alternative measures to assess the quality of an arbitrary signal; information theory provides us with a variety of such measures. The decision whether a kink arriving at the last element corresponds to a signal injected at the first site constitutes a simple binary hypothesis-testing problem [186]. We assign the null hypothesis Ho to "no signal injected" and the opposite for the alternative hypothesis Hi. Denote the corresponding decisions Di as the judgment that hypothesis Hi was in effect. Clearly, there are two possible errors: a so-called Type I error occurs when making the decision that Hi was in effect while the contrary is true. Borrowing notation from radar detection, we refer to this as the probability of false alarm Pf = P(Di\Ho). On the other hand, if Hi was in effect to generate the data and we decide Do, then we have committed a Type II error, which in radar is referred to as a missed detection [186]. We do that with some probability P(Do\Hi), which is related to the probability of detection Pci = P(Di\H\) in an inverse fashion: Pd — 1 — P(Do\Hi). In our experiment, there is no uniquely defined, objectively "best" noise level without first defining a decision strategy. Two suitable approaches are (i) the Neyman-Pears on strategy, in which the probability of detection P^ is maximized while specifying an upper bound for the false alarm probability Pf, and (ii) Bayes' rule which assigns costs to the various outcomes of the decision process, such as correctly detecting a signal or being "deceived"

142

5.

Stochastic

Transport

Phenomena

by a spurious kink. The optimum noise level in the latter case would be the one which minimizes the total average cost. The optimal detection scheme for both criteria (i) and (ii) is commonly expressed in the form of a likelihood ratio test: presented with a measured outcome 77, decide i(H"0u 1 "-

tf PiW/Pofo) < « otherwise ,

(5.16)

where poti(x) are the probability distributions associated with if0,1 respectively. For the front propagation, these (noise dependent) distributions are the derivatives of the rise curves with (ffi) and without (Ho) signal respectively, (see Fig. 5.8.) The choice of the optimum threshold a is found to be a = q/(l — q) in the Bayesian case and is minimized subject to the upper bound constraint on Pf for the Neyman-Pearson strategy [262]. It is intuitively clear that a signal can be detected with least uncertainty when the probability distributions po and p\ are separated as much as possible. In section 2.1 we have already met the Kullback-Leibler distance as a measure of separation between distributions or densities. Much more general information-theoretic distance measures are the Ali-Silvey distances [4; 214], defined by d(po,Pi)

f

—TT p0(x)

Po(x)dx

(5.17)

where f(x) is convex and h(x) is an increasing function. The KullbackLeibler distance (see Def. 2.3) is recovered for f(x) = — logx and h(x) = x. Of particular practical interest is the ds divergence ds(po,Pi)

Po(x)

p0(x)dx.

(5.18)

Remarkably, there exists a simple relation between this distance measure and the minimal probability of error Pg, which is attained when the threshold in (5.16) is set to its optimum value a = q/(l — q) [4; 214]. This one-to-one mapping is given by P£ = \[l-d£(Po,Pi)}.

(5.19)

It follows immediately that minimizing Pg is equivalent to maximizing ds, the separation between the probability distributions. We refer the reader

5.2.

Noise sustained front

143

transmission

1.0-r 0.80.6PH

0.40.2-

0.0

0.5

1.0

1.5

2.0 c

2.5

3.0

2

Fig. 5.14 Total probability of error Pe as a function of noise variance o2 for global (solid line) and local noise (dashed line). For a range of optimal noise strengths, Pe virtually vanishes. The qualitative behavior is robust against variations in the measurement time, which here is taken to be 6000 time steps (300 time units). We assume equal priors: q = 1 - q = 0.5.

to Refs. [120; 227] for a generalization of SR in terms of these information theoretic measures. In communication systems one is usually interested in the total probability of error Pe = qP{Dl \H0) + (1 - q)P{DQ\Hl) = qPf + ( l - g ) ( l - Pd), where q and 1 — q are the a priori probabilities of HQ and H\, respectively. If one lacked any prior information, i.e., if one considered both events equally likely, one would assign equal priors q = 1 — q = 0.5. Though in the experimental setup there is no inherent time-scale, i.e. the decision when to reset the chain is rather arbitrary, in digital communication applications we would expect information bits to be sent at a constant rate. Hence, we choose a reasonable time interval at the end of which we measure the probabilities of false alarm Pf and missed detection 1 — Pd as a function of noise power. Then Pf is simply the value of the dashed line in Fig. 5.12 at time t = 300, and Pd is the corresponding value for the solid line. For local and global noise, the total probability of error Pe is displayed in Fig. 5.14; for very low and rather high values of the noise power, Pe is almost unity. However, there exists an optimal noise strength for which both Pf and 1 — Pd nearly vanish, resulting in a sharp minimum of Pe. It appears sensible at this point to introduce a simple and yet powerful

144

5.

Stochastic

Transport

Phenomena

tool which is commonly referred to as the receiver operating characteristics (ROCs). While the exact definitions vary, the reader could think of an ROC curve as a visualization of the inherent tradeoff between Pd and Pf, in general one cannot be lowered without raising the other. This tradeoff is of such a fundamental and pervasive nature that virtually no diagnostic tool can avoid it. It immediately follows that offering just one number * to characterize a diagnostic system, as is frequently done by advertisements or biased reports, is not only meaningless but also misleading. A more subtle consequence of the tradeoff between Pd and Pf is the absence of an objective optimum; without further information such as the NeymanPearson upper limit on one probability or the definition of a cost function, there exists no absolute minimum. ROC curves quantify the commonplace notion that there is "no free lunch", and their unnecessarily technical name might as well be replaced by tradeoff graphs. John Swets [25l] summarizes succinctly that a ROC . . . is the only measure available that is uninfluenced by decision biases and prior probabilities, and it places the performance of diverse systems on a common, easily interpreted scale. In practice, an ROC curve is obtained by plotting either Pd or 1 — Pd versus Pf as a function of the implicit parameter a. Varying a from 0 to oo generates all possible (Pd,Pf) pairs; for most applications a finite sub range is sufficient to map out the graph. Figure 5.15a shows a (made up) set of a "typical" family of ROC curves. The 45° line, where the probabilities of false alarm and detection are equal, serves merely as reference; any diagnostic system can achieve this performance by chance alone. The closer the curves nudge into the upper left corner the better the discrimination of the detector. A common measure for the quality of a detector is the area under an ROC curve, which ranges from 0.5 (worst case) to 1 (perfect discrimination). In order to create an ROC curve corresponding to Fig. 5.12 we first need to compute po and p\. Figure 5.16 displays discretized versions of po and p\ for weak and strong noise respectively. The bimodal nature of the histograms and the unusually good signal separation result in rather atypical ROC curves. Consistent with the behavior of the total error Pe in Fig. 5.14, the signal cannot be detected with any confidence *e.g., "This pregnancy test has a detection rate of 99%".

5.2.

0.0

Noise sustained front

0.5 P

145

transmission

1.0

f

Fig. 5.15 ROC curves showing probability of detection P^ versus probability of false alarm Pf for various levels of (local) noise. Panel a is a made up family of "typical" ROCs. Generally, the discriminative power of a detector increases with the area under its corresponding ROC graph. Panel b corresponds to the probability distributions corresponding to Fig. 5.12. The 45° line corresponds to ± in opposite directions with average speed ±up where UF = 2F/aM0. If the local fluctuations are spatially correlated, say (C(x,t)C(x',t'))

= 2akT6(t - t')[e- | x - a ! 'l / A /2A],

(5.29)

the noise strength D changes into [176; 178]

As speculated in Sec. 5.2, for noise correlation length A smaller than the kink size d, possible spatial inhomogeneities become negligible; i.e., D(\) ~ D for A « d [184]. The global noise regime simulated numerically in Sec. 5.2.1 corresponds to the limit A —> 00 of the source £(x, t) rescaled by the normalization factor A/2A; the Langevin equation (5.28) still applies, but the relevant noise strength is now lirriA^oo 2AD(A) = 4D. This accounts for the observation that global noise sustains kink propagation more effectively than local noise. Note that the enhancement factor of 4, more exactly 4a, is nothing but twice the distance between the DQ potential minima (in dimensionless units). Another important property of global noise is that it cannot trigger the nucleation of a kink-antikink pair and, therefore, minimizes the chances of a "false alarm". For this to occur it would be necessary that a spatial deformation of a stable string configuration (vacuum state) be generated large enough for the external bias to succeed in making it grow indefinitely. Such a 2-body nucleation mechanism would require a local breach of the (f> —> —(f) symmetry of the DQ equation (5.27), which can be best afforded in the presence of uncorrelated in situ fluctuations [179]. The nucleation rate, namely the number of kink-antikink pairs generated per unit of time and unit of length, can be easily computed by combining the nucleation theory of Ref. [179] with the analytical results of Ref. [42] for the DQ theory. For values of the string parameters relevant to Sees. 5.2

5.3.

Theory

151

and 5.2.1, that is for kT and Fd co (so that v ~ 1 — I/7). Note that the spatial extent of the discrete kink solutions ±. n approaches a step function (order-disorder limit); (ii) PN —> w 0 a n d UPN —> %/2CJ0 in t h e highly discrete a n d

continuum limits, respectively. T h e energy barriers of t h e P N potential are thus (almost) quadratic in I. The one-dimensional Langevin equation (5.34) has been studied in great detail by Risken a n d coworkers [226]. In t h e noiseless limit, 77(i) = 0, t h e process X(t) is to b e found either in a locked state with (X) = 0, for 4F/M0 < UJPN , or in a running state with (X) ~ UF, for 4F/M0 > UJPN. This is indeed t h e depinning (or locked-to-running) transition described in Fig. 5.6. At finite t e m p e r a t u r e t h e stationary velocity (X) = u(T) can be cast in t h e form following, 1, the amplitude xn of the nth oscillator obeys xn = -V'{xn)

+ e (xn+i + xn-i

- 2xn) + £n(t),e > 0 .

(5.36)

The potential energy function characterizing each element is given by Eq. (3.9). As before, we choose ki,k2 > 0 to ensure its bistability and we remind the reader that the height and width of the energy barrier are h = k\j\k2 and w = 2^/ki/k2, respectively. With the end of the chain free, we force the first element sinusoidally, ±i = -V'ixi)

+ e(x2 -xi)

+£i(t) + Asinuit,

(5.37)

and study the propagation of this signal along the chain. £„(£) is Gaussian white noise, band-limited in practice by our integration time step dt (which establishes a nonzero correlation time) to a Nyquist frequency /AT = l/(2dt). We quantify the noise by its mean squared amplitude or noise power a2 = 2Dfw, where 2D is the height of the one-sided noise spectrum. We study the case of incoherent or local noise (uncorrelated from oscillator to oscillator) rather than coherent or global noise (identical at each oscillator). The system of stochastic differential equations

154

5.

Stochastic

Transport

Phenomena

Fig. 5.18 Spatiotemporal behavior of a chain of 32 overdamped bistable oscillators, sinusoidally forced at one end and subjected to increasing incoherent noise. Parameters are h = 0.75, w = 2.25, A = 5, w = 0.2, and e = 10. Reproduced with permission from [159].

(1)~(3) is numerically integrated via the Euler-Maruyama algorithm [76; 148], using a time step dt = T/2 1 2 . with the forcing period T = 2ir/ui. We then numerically estimate the spectrum, or power spectral density (PSD), of a long time series of each oscillator. After averaging many such PSDs, we estimate the signal-to-noise ratio (SNR) at, each oscillator as the ratio of the signal power to the noise power, at the frequency of the forcing. The SNR is conventionally expressed in decibels (dB). We estimate the noise power by performing a nonlinear fit to the PSD around, but not including, the forcing frequency. We estimate the signal power by subtracting this noise background from the total power at the forcing frequency. Schematically. SNR = 1 0 log 10

signal power noise power

(5.38)

However, our results are robust with respect to variations in this definition of SNR. Figure 5.18 illustrates the system's spatiotemporal behavior in the presence of increasing noise. Each strip represents the evolution of a chain of 32 oscillators, arrayed vertically and evolving horizontally, for 12

5.4-

Noise enhanced wave

propagation

155

Fig. 5.19 Plots of SNR versus oscillator number n and SNR versus noise variance a2. Intermediate noise results in maximum propagation and oscillators in the chain where noise extends the signal exhibit stochastic resonance. Smooth curves have been fit to the data to aid the eye. Parameters are the same as in Fig. 5.18, except that t = 4. Reproduced with permission from [159].

forcing periods. The position xn{t) is color coded, with blue denoting the left well and red denoting the right well. The first, forced, oscillator is at the top edge of each strip. With little noise, the sinusoidal signal propagates only a short a distance down the chain. Moderate noise extends the propagation, while excessive noise destroys it. Note how a noise variance of a2 = 100 enables the signal to initiate system-wide events, occasionally causing the entire chain to hop from one well to the other. In fact, we chose our initial operating parameters so that, in the absence of forcing, the spatial features are large compared to the chain length. Large segments of the chain spontaneously flop across the bistable potential barrier. Such a partially correlated medium allows noise to support, sustain, and enhance signal propagation. Typically, we parameterize the forcing with amplitude A = 0.5 and angular frequency u = 0.2, and the bistable potential with a barrier height of h = 0.75 and barrier width of w = 2.25 (corresponding to fci « 2.37 and k2 ~ 1.87). However, NEP is robust with respect to variations in these parameters. Fi gure 5.19 illustrates SNR versus oscillator number for increasing noise variance and SNR versus noise variance for increasing oscillator number. For oscillators near the forcing end, the SNR decreases as the noise

5.

156

Stochastic

Transport

Phenomena

-| 30 '35 i

J40 SNR(dB)

02

Fig. 5.20 Smoothed contour plots of SNR versus oscillator number n versus noise variance a2, for increasing coupling. SNR contours are accurate to ± 1 dB. Other parameters same as Fig. 5.18. (Here we employ a larger integration time step of dt = T/210.} Reproduced with permission from [159].

5.4-

Noise enhanced wave

propagation

157

increases. But for oscillators farther away, the SNR goes through a local maximum as the noise increases — the classic signature of stochastic resonance. Oscillators near the forcing do not need help from the noise, as the forcing amplitude there is large (A > h); however, oscillators far from the forcing do need help from the noise as the signal there is attenuated. For a given noise power, SNR decreases with oscillator number downstream along the chain. Defining the propagation length as the number of oscillators (or distance along the chain) for which the SNR exceeds a certain cutoff, say 1 dB (this is roughly the uncertainty in our numerics), we observe that the propagation length is longest for moderate noise. This data can be succinctly combined into a contour plot of SNR versus noise variance versus oscillator number. Figure 5.20 presents a series of such contour plots, for increasing coupling. Grays code SNR, with white indicating large SNR (> 40 dB) and black indicating small SNR (< 1 dB). The bottom-left corners represent the peak SNR of the noiseless first oscillator. The SNR decreases everywhere away from these corners. However, the distinct bulges in the contours, where regions of large SNR extend toward the free end of the chain, are the signatures of NEP. (The 1 dB contour traces out the propagation length as a function of noise.) At sufficiently large coupling, the SNR extensions reach the end of the chain, indicating that noise and coupling have succeeded in sustaining the signal throughout its length. Note how both the extent and the position of these SNR extensions increase with increasing coupling. Figure 5.21 summarizes the scaling of optimal noise and maximum propagation length with coupling. Both appear to scale as the square root of the coupling. This reflects the fact that the correlation length of spatiotemporal features for a local and linearly coupled array scales this way [103]. As the coupling increasingly binds adjacent oscillators together in the same well, the noise variance must increase correspondingly in order to force such correlated oscillators across the bistable potential barrier. It is instructive to compare these results for two-way (bi-directional) bistable chains with the behavior of one-way (unidirectional) bistable chains recently reported by Hu et al. [279]. For a one-way bistable chain driven at one end, if forcing and noise cooperate to flip the first site (from one well to the other), sufficient coupling propagates the flip to the other end with probability unity. When the forcing is just below the threshold for the first site, the requisite noise is small enough not to interfere with the propagation of the flip. A one-way bistable chain readily propagates flips, like a line of tumbling dominoes. Consequently, for sufficiently large coupling,

158

5.

2. We point out that while it is in principle possible for some strategies to be shared by more than one agent, the doubly exponential growth of j(rn) makes this extremely unlikely even for moderately sized m. During the course of the game all deployed strategies are updated according to their potential success. We award a point to a strategy if it would have forecasted the correct bid, ignoring whether it actually was exploited or not. At any given time, each agent selects from among his/her strategies the one with the highest number of points* to pick his/her bid. Apart from the random initial conditions, which comprise the distribution of strategies and the necessarily random first memory string, this minority game is fully deterministic. The macroscopic variable that has received by far the greatest attention is the variance of excess of buyers to sellers, which is defined as

where A(t) is the total bid at time t: A(t) = ^ ~ = 1 a-j(t) and a,j(t) e (0,1) is the individual bid of agent j . The fluctuations around the mean, quantified by the variance, are referred to as volatility in financial markets. The larger *In the unlikely case of a tie, a strategy is randomly chosen among the most successful contenders.

6.

170

Sundry

Topics

a2 is, the larger the total loss of the body of agents.t In a financial context zero volatility would be most desirable as it minimizes risk and generates "wealth" among the largest number of agents. If there were no strategies and all agents played the game randomly, the volatility would simply be of = N/4, according to the law of large numbers. The volatility depends on the parameter m in a nontrivial fashion, as Fig. 6.1 (stars) illustrates: for very small memory sizes the fluctuations of the total bid are significantly larger than if the agents had acted randomly. TThis is, of course, due to the particular payoff function, which does not take into account the size of the minority at all.

15 o Model without memory *—* Model with memory

10

0

0

10

15

m Fig. 6.1 Volatility a as a function of m (s = 2) for the model with and without memmory. The horizontal line is the variance crT = y/N/4 ~ 5 of the random case. The number of agents isN = 101. Error bars are shwon only for the model without memory, while the line just connects the points of the memory model. Reproduced with permission from [27].

6.1.

Minority

game

171

For large m the volatility converges from below towards the random limit, and for intermediate memory sizes, a(m) is actually smaller than err! For the remainder of this section we fix the number of strategies to be s = 2 without losing generality. It has been shown in [32] that the minimum of the curve a(m) persists for larger values of s but gradually grows more and more shallow. Our earlier comment on the likelihood of different agents possessing identical or similar strategies appears to hold true for the behavior depicted in Fig. 6.1. When the strategies are totally uncorrelated, which is more and more likely for increasing m, the game dynamics become more and more random, therefore approaching the variance limit ar. An important breakthrough in the understanding and analysis of the minority game was achieved by Cavagna [27], who demonstrated that the actual memory of the agents is irrelevant! The volatility a(m) of the system is unchanged if we replace the real history by an invented history that in turn can be taken to be a random sequence of bits! In Fig. 6.1, the variance as a function of m is also plotted for this memory-less setup (open circles). The two models surprisingly give the same results and therefore leave the memory size or, more accurately, the dimension of the strategy space D = 2 m as the only relevant variable. As a consequence, there is no need for D being a power of 2 any longer [27] so that from now we conveniently allow D to take on all positive integer values. Cavagna also showed that the dependence of u(D) on s and N is such that all data a collapse onto the same curve if we assume the scaling a2 —> a2/N and m —> 2m/N = D/N. Therefore, in the following, we will always plot a2 /N (or a/\f~N) vs. D/N. Summarizing, it appears that while the information content processed by the agents is extraneous, it is crucial that everyone is exposed to the same "history". In the terminology of Chapter 4, the applied "noise" is global. In the case of local noise, i.e., if each agent received their own information uncorrelated with the others, the dynamics of the game become identical to the random case. For more insights into the controversial topic of whether the actual memory is relevant for the minority game or not, we refer the reader to the literature [155].

6.1.1

The thermal

minority

game

As pointed out in [28], the original version of the minority game can be modified to be both analytically more tractable as well as more realistic. We will first introduce the continuous version of the MG and then proceed

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to its stochastic version, which is the key rationale for devoting a section of this book to a game theoretical model. Cavagna et al. [28] introduced the concept of a "temperature" to the game dynamics. This section is essentially a summary of their intriguing work, which we highly recommend for further reading. A departure from the binary nature of the MG is motivated by its following shortcomings: (i) Integer variables, such as the strategies and bids, are less suited for time series analysis than floating point numbers. (ii) Geometrical arguments developed for the hypercube of strategies [32; 52; 126; 280] become more natural if the strategy space is continuous. (iii) To model financial markets more realistically, both the size of the individual bids as well as the gains or losses ought to be continuous. The following continuous formulation of the MG meets all of these criticisms and demands. The space of strategies V is assumed to be the surface of a hypersphere of dimension D. A strategy R is therefore a (real) vector in MD subject to the constraint ||i?|| = y/~D. The global information ff(t) presented to the agents is also a vector in R D ' but random and of unit length. We require rf(t) to be uniformly distributed on the unit hypersphere and uncorrelated in time {fj{t)fj{t + T)) = S(T). The (now continuous) bid* corresponding to strategy R is defined as a(R) = R • f)(t)

(6.2)

The sign of a(R) determines whether to buy (positive) or to sell (negative), and its magnitude gives the respective amount. As before, the total bid is simply the sum of all individual bids corresponding to the agent's strategy R* (t) with the highest number of points at that time JV

N

AW = 5>(i?*(t)) = £/?;(*) • m 3= 1

(6.3)

3= 1

All that is left for us to specify is the continuous version of the time evolution of the points P{R, t) for strategy R. In the discrete case, we simply awarded one point to each winning strategy; i.e., P(R,t + 1) = *Note that in Ref. [28] the symbols for the individual bid are b(R) and bi(t) respectively. We prefer to stay close to the original notation, a,i(t), from the earlier MG papers.

6.1.

Minority

173

game

2.0 -i

1.5-

a/N

1/2

random value

1.00.5-

0.0

-I—!~T

4 5 6 7

0.1 D/N Fig. 6.2 The scaled variance a/VN as a function of the reduced dimension D/N for s = 2 and AT = 100. The variance achieves its minimum at a critical dimension dc f» 0.5. The dashed horizontal line serves as a reference for the random case. The first to = 10000 steps were discarded as transients, and we averaged over 100 samples and r = 10000 time steps respectively.

P(R, i) + sign[- -a(R)A(t)\. would be

The natural extension to the continuous case

P(R, t + 1) = P(R, t) -

a{R)A(t)/N,

(6.4)

so that the points P are independent of N and proportional to both the individual and total bids. As can be seen in Fig. 6.2, the main features of the binary MG model are reproduced. There is a worse-than-random phase for < 0.2 and a better-than-random phase for d > 0.2, where d = D/N is the reduced dimension of the strategy space. The volatility a/y/N assumes its minimum value at dc = (D/N)c « 0.5, which we refer to as the critical dimension. The continuous MG constitutes an improvement in analytical tractability, but from a statistical physics point of view a stochastic parameter, which plays the equivalent role of a temperature, appears highly desirable. As an elegant alternative to the deterministic "best-strategy rule", Cavagna et al. [28] introduced the thermal minority game (TMG). The TMG utilizes an inverse temperature (3 = 1/T by computing the prob-

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abilities -l

^•(i)

=$>*(*)

e^^t\i

= l,...,N,j = l,...,s

(6.5)

^fc=i

for each agent i of choosing his/her strategy j . In the zero temperature limit, /? —> oo, the deterministic MG is recovered, while for decreasing j3 the point differences between the strategies grow less and less decisive. For p = 0 strategies are selected completely at random. To fully explore the dynamics of the TMG, we would have to continuously vary both the temperature and the reduced dimension d. Instead, we only present the dependence of a on T for fixed d, trusting that qualitatively similar trends hold for a wide range of dimensions. Fitting well into the general theme of this book, we find that increasing stochasticity can actually benefit the community of agents by reducing the volatility! From Fig. 6.2 we pick a value of D/N which, for T = 0, results in a worse-than-random volatility a > ar, e.g., D/N = 0.1. Figure 6.3 illustrates the major reduction of a(T)

2.0-1

best strategy value (MG)

1.5-

a/N 1.0

0.5 4 10

10"

10

10'

10

10°

Fig. 6.3 The scaled variance a/-/N as a function of the temperature T at D/N = 0.1 for s = 2 and N = 100. Both t0 and r from Eq. (6.1) are taken to be 10, 000. Note that the increased variance for temperatures T > 100 is an artifact of finite simulation times, as pointed out in [30],

6.1.

Minority

game

175

for a large range of temperatures. Even if d lies in the better-than-random phase, finite temperatures still improve the behavior of the system. We are obliged to mention that the apparent minimum and subsequently increasing variance for temperatures above w 100 is an artifact and entirely due to finite simulation times as pointed out by Challet et al. [29; 30]. It is shown that the integration time r required to reach the steady state is of order - NT. Therefore, for AT = 100 and r = 10,000, the results for T > 100 in Fig. 6.3 do not represent the steady state. The "true" volatility curve, shown in Fig. 1 of [29], monotonically decays with increasing temperature. We did not correct Fig. 6.3 for reasons of completeness and also because its principal message is not weakened: any nonzero temperature larger than a critical value Tc always leads to improvement as long as d is less than the critical value dc\ Intuitively, we can understand the favorable response of the system to "noise" in the following manner: if every agent always chose his/her optimal strategy, then the system would be more "unstable" or more critical, i.e., every once in a while there would be rather large

2.0

-0.5-1.005 C

-1.5

Individual gains Variance

en 03

0/N

•D

| c

-2.0 1.0 -2.5 -3.0^

0.001

1—i i i n r q

0.01

1—i i m i i |

0.1

1—i—ri 11rq

1

1—i i i 11 IT[

10

1—i i i i MM

J-0.5

100

Fig. 6.4 Plot of the average individual gains of all 100 agents as a function of the temperature (D/N = 0.1). In addition, the volatility is plotted. The opposite trend is clearly visible.

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losses or gains. The introduction of a temperature or random component "smears out" the criticality somewhat, so that there are fewer extreme events, leading to a reduced variance. Recently, the MG has been reexamined through the eyes of various learning algorithms. We refer the reader to Refs. [266] and [5] for the performance of neural networks and the Q learning algorithm on the MG, respectively. Before concluding this section we would like to stimulate the reader's quest for further research. Among the macroscopic variables, the variance of the total bid is the first interesting moment ((A(t) N/2)n), since the mean A(t) vanishes. Another interesting observable is the average individual gain gt{T) = {-cn{R)A{t)/N) of the agents. It is noteworthy that the temperature dependence of the individual gains is almost the exact opposite to that of the volatility. In Fig. 6.4 we have overlaid the average gains gt(T),i = 1 , . . . , 100 and a/y/N. Interestingly, the individual gains are all negative, which can be understood as follows N

-(ai{t)

• A(t))

J=

"D

N

=-(ai(t) $>,-(*)> = -(a?(i)> - £> 3

random value

c/N"

0-

1.0

-10-

0.5

c o

ill

•••*•• Variance —t— Correlation between bids

D/N = 0.1 -20x10"

T

I

0.001

0.01

TTTT

0.1

r—m

0.0

m

l 10

Fig. 6.5 The cross correlation between the individual bids YliLj(ai(t) as a function of temperature for D/N = 0.1.

100

• &j{t))/{N

— 1)

6.2.

Traffic

dynamics

177

The first term in this sum is simply the negative of the autocorrelation of the individual bid a^t), and the second term is the average of the cross correlations between the bids! In Fig. 6.5 we therefore plot instead of the individual bids, the quantity J2i^j(ai(t) " a j W ) / ( ^ ~~ 1)> which is just a randomly selected individual gain, divided by N - 1. It is evident that the cross correlation between the bids crosses zero at the same temperature as the variance approaches the random value. We close this section leaving these intermediate results as food for thought.

6.2

Traffic dynamics

Traffic research is an interdisciplinary and rapidly evolving field. It has attracted the interest of the engineering sciences as well as that of theoretical physics, mathematics, and cybernetics. Since traffic flow is governed by high-dimensional, complex and decidedly nonlinear dynamics reminiscent of fluid or granular flow, there is a great variety in the structure and simplicity of the models that attempt to extract key features. One of the first attempts to capture the nonlinear effects inherent in realistic traffic flows was put forward by Newell [206]. More insights into the solutions of the Newell model were provided by Whitham [267]. A great deal of important macroscopic traffic patterns were reproduced in a series of computationally highly efficient cellular automaton (CA) models first proposed by Nagel and Schreckenberg [101; 202; 232; 235]. In 1995, Bando et al. [ll] pioneered a microscopic, dynamical model that capitalizes on the concept of a legal or optimal velocity VQ . In this section, we can neither afford nor strive to cover even a fraction of the existing literature on traffic modeling. We refer the interested reader, who would like to delve deeper into this fascinating and rich discipline to a number of excellent monographs and conference proceedings [100; 156; 185; 236; 272]. In addition, an entertaining and yet accurate layman-accessible explanation of the cause of density waves and jams in everyday traffic flow can be found online [112]. As a good introduction to the active community of traffic researchers, we recommend visiting the Web site run by M. Treiber [113]. Since this book deals entirely with stochastic effects, we would like to focus on the consequences of noise on pattern formation in car-following models. In fact, this section builds directly upon the results and insights

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developed in section 5.1. There, the assumption of strongly asymmetric coupling was put forward in a rather ad hoc manner. However, in the context of traffic dynamics, the notion of unidirectional information flow emerges very naturally: from the view of the individual driver, the actions of the preceding vehicle are vastly more significant than those of the following car. For the following discussion, we can safely ignore the minuscule coupling of a vehicle to rear events. Recently, Nagatani [201] as well as Mitarai and Nakanishi [109; 195; 196] have given numerical and analytic evidence for the existence of noise-sustained structures in the time-continuous and difference equation versions of the optimal velocity model. 6.2.1

Time-continuous

model

For later convenience, we summarize the optimal velocity (OV) model, more details of which can be obtained from Refs. [ll; 206]. All dynamical traffic models are based on the common assumption that each driver of a vehicle responds to external stimuli by de/accelerating, which is the only direct control available. We have to distinguish between two diverging conjectures for the driver's dominant strategy to avoid accidents. Follow-the-leader theories assume that each vehicle maintains a velocity dependent safe distance from the preceding vehicle. OV models on the other hand postulate the objective of a legal or optimal velocity, which depends solely on the distance to the preceding vehicle. Furthermore, a linear feedback loop is assumed such that the change in velocity is proportional to the deviation from the desired velocity vn = an(VQ — vn). The proportionality constant an quantifies the responsiveness of the driver and is referred to as the sensitivity. To keep the model simple, one usually ignores the lengths of the vehicles as well as the individual differences in sensitivity. Denoting the position of car n at time t by xn (t), the dynamical equation of the system can therefore be written as xn = a (V0(Axn)

- xn),

with Axn = xn+1 - xn .

(6.7)

As mentioned above, the "goal" velocity of vehicle n depends solely on the distance Axn to the preceding vehicle, which is also referred to as the headway. A subtle aspect of this model worth pointing out is the competing flow of information. While the vehicles move "to the right", i.e., downstream toward larger x, the unidirectional coupling acts "toward the left."

6.2.

Traffic

179

dynamics

Depending on which velocity is larger, information and perturbations propagate either upstream only or in both directions. A fundamental difference from the equations studied in section 5.1 is that the system of differential equations (6.7) is second order, which will necessitate different techniques for a stability analysis (see below). The exact functional form of Vo(z) is rather arbitrary but must meet the following properties: (i) (ii) (hi) (iv)

V0(z) > 0 dVo{z)/dz > 0 V0{z -> oo) -> Vmax < oo Vb(z-0)-0

These requirements quantify our expectations that for small headways the velocity needs to be reduced to avoid a collision and that for longer headways the vehicle can move faster, although it is always limited by some maximum velocity Vmax. Now any monotonic, smooth function that is bounded from below and above must have a general "S" shape. Here, we adopt the optimal velocity function used by Bando et al. [ll] Vo(Ax) = -~-

[tanh (Ax - hc) + tanh (hc)},

(6.8)

where hc is a safety distance referred to as the critical headway. Figure 6.6 shows this velocity function for a typical set of parameters. Note that Vo{Ax) possesses an inflection point at Ax = hc which, as we will see below, leads to bistability. Eq. (6.7) has the stationary solution xsn(t) = b-n + Vo{b)-t

(6.9)

which represents the perfectly periodic arrangement of vehicles traveling at constant velocity Vo{b), spaced b length units apart. In order to find the conditions for absolute linear instability, we linearize the equations of motion and assume periodic boundary conditions b = L/N. xn{t) = xsn{t) + £ n (t) and xN+i

= xx

=» 6, = a (7 • A£n - in) + 0(C),

(6.10)

(6.H)

with 7 = VQQ}). A common method of solving linear, second order differential equations with periodic boundary conditions such as (6.10) is to

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o

o

O.

O

d o o 10

Ax

Fig. 6.6 Optimal velocity as function of the headway Ax as defined in (6.8) for hc = 5 a n d Vrnax = 2.

expand £„ (£) into a Fourier series N-l

£„(*) = X! «^(".*).

with

& M ) = elAj"+wt

(6.12)

J=0

Due to the presumed movement of the vehicles on a finite circle of length L = N • b, the wave numbers are constrained to Xj = 2irj/N. It is easily seen t h a t substituting £j(n,t) into (6.11) yields the dispersion relation 2

aj (e

(6.13)

1) — au> = A(w, Xj)

2UJ = - a ± J a? + 4a-y(eiXi - 1) = 2w,(A).

(6.14)

Linear stability requires the real part of UJJ(X) to be negative which — we leave it to the reader as an exercise — is true if and only if a > 2-y =

(6.15)

2V^(b).

T h e so-defined stability b o u n d a r y a = 2 V Q ( 6 ) = Vmaxsech2 in Fig. 6.7 as the solid line.

(hc-b)

is plotted

6.2.

Traffic

dynamics

181

O

c\i

Linearly Stable

'Convectively Unstable

m

Absolutely Unstable d

—

—

v

V

2 b

0

Fig. 6.7 The boundaries in the (a, b) parameter space separating the three stability regimes. The solid line demarcates the linear stability curve a = 2VQ (b) for Vmax = 2 = hc and the dashed line is a numerical approximation of Eq. (6.17).

To derive the conditions for convective instability we have to consider the modified dispersion relation [109; 158; 195; 196] (\ \

^

Vo{b)

a.

, = 5.0, v, = 1.7,