All macroscopic physical systems are subject to fluctuations or noise. One of the most useful and interesting developments in modern statistical mechanics has been the realization that even complex nonequilibrium systems can often be reduced to equivalent ones of only a few degrees of freedom by the elimination of dynamically nonrelevant variables. Theoretical descriptions of such contracted systems necessarily begin with a set of either continuous or discrete dynamical equations which can then be used to describe noise driven systems with the inclusion of random terms. Studies of these stochastic dynamical equations have expanded rapidly in the past two decades, so that today an exuberant theoretical activity, a few experiments, and a remarkably large number of applications, some with challenging technological implications, are evident.

The purpose of these volumes is twofold. First we hope that their publication will help to stimulate new experimental activity by contrasting the smallness of the number of existing experiments with the many research opportunities raised by the chapters on applications. Secondly, it has been our aim to collect together in one place a complete set of authoritative reviews with contributions representative of all the major practitioners in the field. We recognize that as an inevitable consequence of the intended comprehensiveness, there will be few readers who will wish to digest these volumes in their entirety. We trust, instead, that readers will be stimulated to choose from the many possibilities for new research represented herein, and that they will find all the specialized tools, be they experimental or theoretical, that they are likely to require.

Introduction

The study of non-linear stochastic processes is usually characterized by considerable difficulties, thereby rendering it necessary to have recourse to computers for the theoretical predictions to be tested. This in turn involves interminable times of calculation with a great waste of money.

Analogue simulation by means of electronic devices allows us to bypass these difficulties. Although the precision of the analogue technique is modest compared to that of computer calculations, it must be remarked that in many cases it is more significant to have available, immediate results rather than very accurate numerical precision.

It is important to underline that the analogue experiment does not completely replace computer simulation; rather, a comparison of the two techniques, wherever possible, has proven to be a powerful method for studying non-linear stochastic phenomena.

In order to give a short description of electronic simulation, a few basic elements of the techniques will be given, then we shall discuss some significant applications. Finally, we shall describe how it is possible to obtain important conclusions via a joint use of computer and analogue simulation. This has been the object of a recent investigation on the role of coloured noise in some dynamical systems.

The basis of the analogue simulation via electronic devices

The integration operation

This operation is usually obtained using an RC-circuit; nevertheless, a Miller integrator, assembled in the configuration shown in Figure 8.1(a) is a better choice because of the following advantages:

1. (a) The high input impedance allows us to add many input signals.

2. […]

As will have become clear to readers of the first two volumes of Noise in Nonlinear Dynamical Systems, the theoretical development of the field, already extensive and wide-ranging, is still proceeding at a rapid pace. In fact, although such investigations were initiated as a means towards a better understanding of the observed properties of the physical world, the progress of the theory seems in many cases to have run far ahead of the corresponding experimental studies. Some important questions then arise. Does nature really behave in the manner predicted by the theory? Under what particular physical conditions is any given (approximate) theory to be relied upon? In most cases, these questions have no easy experimental answers, and the aim of this third volume is to survey the limited progress that has been made to date.

Credible experimental tests of stochastic theory have been completed on only a relatively small number of natural systems. These include, particularly, superfluid helium, liquid crystals and lasers, whose diverse properties provide the subject matter of Chapters 1–6.

Heat flow in superfluid helium can result in the formation of a tangle of quantized vortex lines, a particularly simple and well-characterized version of classical turbulence. The process occurs through a continuous instability of the type for which a variety of noise-induced transitions have been predicted. Chapter 1 describes a direct experimental test of these ideas.

The dramatic effect of external noise on liquid crystals is discussed in Chapters 2 and 3, which treat different aspects of the electrohydrodynamic instability (EHD).

Prologue

The statement that systems near the onset of instabilities are sensitive to fluctuations may be a haggard cliche, but is true nevertheless. The very notion of ‘instability’ means that a small fluctuation tends to grow with time rather than decay away. This has some distinct disadvantages, as was driven home to me during an experiment performed by Bob Miracky on an electrical circuit. Our goal was to determine the precise parameter value at which a period doubling bifurcation occurred, in order to compare this with analytic results derived from the circuit equation. As an experimental matter, this turned out to be tricky due to some curious interactions of the circuit with unwanted electrical noise.

Figure 7.1 reproduces a typical sequence of power spectra generated by sweeping a control parameter. If one has a good machine, the power spectrum is a very accurate means for determining the onset of period doubling bifurcations: the signature is the birth of a sharp spectral line at one-half the fundamental oscillation frequency ω0, this line initially growing from zero height. Figure 7.1(a) shows the system well below the bifurcation point; however, in addition to the expected lines at frequencies ω = 0 and ω = ω0 there appear two bumps in the broadband noise. Though mysterious, these bumps are not a problem since they are far away from the crucial frequency ω0/2. Unfortunately, as the bifurcation point is neared these bumps move together, eventually colliding at ω0/2 and obscuring the onset of the sharp line heralding the bifurcation.

Introduction

This book is a reflection of the recent interest in the correlation properties of fluctuations driving linear and nonlinear systems. The interest arises because fluctuations in physical systems are never truly white and because of the sensitive dependence of certain system properties on these correlations. In the last few years a number of theoretical techniques have been proposed for the proper treatment of colored noise in stochastic systems. Also, a number of experiments have been designed to explore the effects of color on system response.

In this chapter we concentrate on one class of properties that are particularly responsive to the color of the noise that drives a physical system. The first time T(xp) that an evolving stochastic physical process X(t) reaches a preassigned value xp is called the first passage time and depends sensitively on the correlation properties of the noise. Of course the first passage time T(xp) is itself a statistical quantity and is characterized by a distribution function f(t) as distinct from the distribution P(x, t) for the state variable.

The first passage time from an initial state X(0) = x0 to xp is sensitive to the distribution density P(x, t) for values of x covered by X(t) as it evolves from x0 to xp. The biggest contributions to T(xp) arise from those portions of a trajectory for which P(x, t) is small. These improbable regions of phase space through which the trajectory passes are the most difficult to traverse and therefore contribute most to the first passage time.

All macroscopic physical systems are subject to fluctuations or noise. One of the most useful and interesting developments in modern statistical mechanics has been the realization that even complex non-equilibrium systems can often be reduced to equivalent ones of only a few degrees of freedom by the elimination of dynamically non-relevant variables. Theoretical descriptions of such contracted systems necessarily begin with a set of either continuous or discrete dynamical equations, which can then be used to describe noise driven systems with the inclusion of random terms. Studies of these stochastic dynamical equations have expanded rapidly in the past two decades, so that today an exuberant theoretical activity, a few experiments, and a remarkably large number of applications, some with challenging technological implications, are evident.

The purpose of these volumes is twofold. First we hope that their publication will help to stimulate new experimental activity by contrasting the smallness of the number of existing experiments with the many research opportunities raised by the chapters on applications. Secondly, it has been our aim to collect together in one place a complete set of authoritative reviews with contributions representative of all the major practitioners in the field. We recognize that as an inevitable consequence of the intended comprehensiveness, there will be few readers who will wish to digest these volumes in their entirety. We trust, instead, that readers will be stimulated to choose from the many possibilities for new research represented herein, and that they will find all the specialised tools, be they experimental or theoretical, that they are likely to require.

Introduction

Over the last few years the investigation of the transition to turbulence (Ahlers and Behringer, 1978; Croquette and Pocheau, 1984; Gollub and Benson, 1980; King and Swinney, 1983; Swinney and Gollub, 1981) has mainly focused on two systems: on the Rayleigh–Benard convection, which arises when a thin layer of a simple fluid is heated from below; and on the Taylor instability, for which the simple fluid in the gap between two concentric cylinders is subjected to an external torque by rotating the inner or both cylinders. In both configurations a transition from a quiescent state to spatial turbulence via some intermediate, spatially periodic patterns is observed.

A marked influence of the aspect ratio for both instabilities, as well as of the shape of the container for Rayleigh–Benard convection, is well documented (Ahlers and Behringer, 1978; Croquette and Pocheau, 1984).

No systematic study on the influence of controlled external noise on the transition to turbulence had been done prior to the experiments to be described below and very few qualitative results were known.

The big open question was: Can external noise influence or alter the transition to turbulence qualitatively in pattern-forming nonequilibrium systems, or is it a mere perturbation leaving the main features unchanged?

This was the main motivation triggering our studies on the electrohydrodynamic instability (abbreviated as EHD throughout the rest of this chapter) in nematic liquid crystals.

Introduction

The isolation of a physical system from its environment is a frequent approximation which leads to the idea of a deterministic conservative system. All physical systems are, however, coupled to the outside world, giving rise to the related phenomena of fluctuations and dissipation. In a dissipative dynamical system volumes in phase space contract onto attractors. Fluctuations allow the system to escape from attractors, rendering all attractors metastable. The long time behavior of a noisy dissipative system is thus intermittent, consisting of motion near the various attractors of the system alternating with transitions between attractors. In the limit of small noise, the times spent on the attractors become longer, and the transitions rarer. In this chapter we describe some recent work on stochastic dynamical systems with multiple attractors, with particular emphasis on systems possessing multiple limit cycles.

Although there has been much interest in noisy iterated maps, to our knowledge no one has actually derived a noisy map from a stochastic differential equation. In Section 4.2 we discuss some of the considerations that must go into any such derivation. As an example, we examine how noise affects a driven oscillator in both the phase-locked and unlocked regimes. The details of the noisy dynamics play an important role in determining how noise must affect the resulting map. Section 4.3 contains a formal derivation of a noisy iterated map from a linearized stochastic differential equation.

Introduction

Lasers are part of a large class of non-linear systems which covers a great deal of scientific domains, including physics, chemistry, biology, sociology, etc. All these non-linear systems have common properties and present universal features which become more and more extensive as new studies are performed.

In recent years, the interest in these non-linear systems has considerably increased. In particular, it is now well established that the role of fluctuations in non-equilibrium systems is far more relevant than in systems at thermodynamical equilibrium, and the study of their influence on stationary and transient states has become very important.

Among the different aspects presented by the non-linearity in lasers, two of them will be particularly studied here:

(i) The first one is the optical bistability (OB). An optical system is bistable if it has two output states I for the same value of the excitation A over some range of excitation values (see Gibbs, 1985, for a recent review). A typical response curve of such a system is given in Figure 5.1. Here, the system is defined as bistable between A↓ and A↑, where the effectively occupied state depends on the history of the input parameter (see the caption to Figure 5.1). If OB is a common phenomenon in different fields, the OB transient phenomena have far more general features.

All macroscopic physical systems are subject to fluctuations or noise. One of the most useful and interesting developments in modern statistical mechanics has been the realization that even complex nonequilibrium systems can often be reduced to equivalent ones of only a few degrees of freedom by the elimination of dynamically nonrelevant variables. Theoretical descriptions of such contracted systems necessarily begin with a set of either continuous or discrete dynamical equations which can then be used to describe noise driven systems with the inclusion of random terms. Studies of these stochastic dynamical equations have expanded rapidly in the past two decades, so that today an exuberant theoretical activity, a few experiments, and a remarkably large number of applications, some with challenging technological implications, are evident.

The purpose of these volumes is twofold. First we hope that their publication will help to stimulate new experimental activity by contrasting the smallness of the number of existing experiments with the many research opportunities raised by the chapters on applications. Secondly, it has been our aim to collect together in one place a complete set of authoritative reviews with contributions representative of all the major practitioners in the field. We recognize that as an inevitable consequence of the intended comprehensiveness, there will be few readers who will wish to digest these volumes in their entirety.

Over the last few years there has been a quite remarkable increase of activity in the study of nonlinear dynamical systems exposed to external noise. This is especially true in relation to colored noise, that is noise whose correlation time τ is nonzero. The field as it has developed stands on a firm foundation of pioneering, mostly Russian, theoretical work on white noise driven systems dating back to the early 1930s, an exemplar being the landmark 1933 paper by Pontryagin, Andronov and Vitt, of which a first-ever translation into English appears as an appendix to this volume. The early development of the field is reviewed in Chapter 1.

The recent intensification of research effort has in large part been stimulated by two distinct but closely related factors. First, it has become increasingly apparent that most (perhaps all) real physical systems need to be considered in the context of colored noise: they cannot be described adequately within the framework of white noise theory. Secondly, it is the case that, with few exceptions, physically interesting Fokker–Planck systems cannot be solved exactly when driven by colored noise. To meet this challenge, a variety of approximation schemes, carried out within several theoretical frameworks, have been developed, and much effort has been devoted to the testing of their efficacy under different conditions.

Introduction

The apparent generality of the title to this chapter is appropriately reduced by sufficiently precisely defining the terms used. We have in mind the natural extension to quantum mechanical settings of the ideas usually covered in a discussion of stochastic processes in classical mechanical settings. In classical mechanical settings, the fundamental process is Brownian motion, which describes the influence of external fluctuating forces on a small particle. The mathematical apparatus developed for the study of Brownian motion has been generalized and applied to irreversible thermodynamics (Onsager and Machlup, 1953), fluctuating hydrodynamics (Fox and Uhlenbeck, 1970), light scattering from fluids and mixtures (Berne and Pecora, 1976) and chemical reactions (McQuarrie and Keizer, 1981). In each case, the stochastic process involves fluctuations generated by thermal motions. We are not considering the stochastic trajectories (chaos) of classically deterministic motion in non-linear systems (Lichtenberg and Lieberman, 1983), which is an entirely separate context for stochastic processes. Our goal in this chapter is to study the effects of thermal fluctuations in quantum mechanical settings. Therefore, we exclude a discussion of interpreting quantum mechanics as a classical stochastic motion (Nelson, 1966), an idea which should have been put to rest by Grabert, Hänggi and Talkner (1979). We also exclude a discussion of quantum chaos, which is currently a very active area of research (Zaslavsky, 1981).

Just as Brownian motion provides the paradigm for stochastic processes in classical mechanical settings, the Anderson–Kubo oscillator (Anderson, 1954; Kubo, 1954; Kubo, Toda and Nashitsume, 1985) is the paradigm for stochastic processes in quantum mechanical settings.

Introduction and overview

In the year 1851 Foucault demonstrated that the slow rotation of the plane of vibration of a pendulum could be used as evidence of the earth's own rotation. The first optical experiment to detect the earth's rotation was performed by Michelson and Gale (Michelson, 1925a, b) using an unusually large size for an interferometer: 0.4 miles × 0.2 miles. Nowadays high precision measurements of the earth's rotation are performed by using radio telescopes in very long baseline interferometry (Johnson et al., 1979). Moreover, a recent proposal (Small and Chow, 1982) take advantage of the ultra high sensitivity of a ring-laser gyroscope (Aronowitz, 1965, 1971; Chow et al., 1985; Menegozzi and Lamb, 1973; Privalov and Fridrikhov, 1969) of 10 m diameter to monitor changes in earth rate or universal time. The underlying principle of such a device is the optical analog of the Foucault pendulum, the so-called Sagnac effect (Post, 1967; Sagnac, 1913a, b). The frequencies of two counterpropagating waves in a ring interferometer are slightly different when the interferometer is rotated about an axis perpendicular to its plane. Since this frequency difference is proportional to the rotation rate it provides a direct measure of the rotation of the system.

Ring-laser gyroscopes of this size would also allow tests (Schleich and Scully, 1984; Scully, Zubairy and Haugan, 1981) of metric gravitation theories (Misner, Thorne and Wheeler, 1973).