Introduction

This chapter is devoted to developing digital techniques to integrate multidimensional stochastic differential equations. A variety of different spectral densities of the stochastic forcing can be achieved with the algorithms presented.

Very often (for example, when several dimensions are involved, or an external deterministic forcing is present, or fluctuation–dissipation relations do not hold for the physical system modelled by the set of stochastic differential equations) a digital simulation is the only viable way of extracting the detailed information one is looking for. Also, digital techniques can simulate the theoretical model in a sort of ideal environment, where everything is under control and no non-idealities are present. It is actually the author's personal belief that digital simulation in stochastic physics should be thought of as a theoretical tool, like, say, Padé approximations or steepest descents, and it should be considered as the natural complement of modelling.

The chapter is organized as follows: in Section 7.2 a general algorithm for integrating sets of autonomous differential equations in the presence of just one external stochastic forcing is derived. In Section 7.3 the algorithm is generalized to include non-autonomous differential equations and multidimensional external forcings. Sections 7.4 and 7.5 contain applications to, respectively, overdamped and underdamped non-linear oscillators: what is presented in these two sections should be regarded as examples of digital simulation capabilities, and papers cited there should be consulted for more details and references.

While in Volume 1 the treatments are confined exclusively to Fokker–Planck systems, a range of contemporary problems, indicative of the rich diversity of applications of noise driven dynamics, are reviewed in this volume. Though most problems currently treated are classical, recent work on dissipative quantum tunnelling has focused attention on quantum mechanical applications of stochastic dynamics. Chapter 1 builds on the theory of the Anderson–Kubo oscillator, generalized to account for colored noise effects, with applications to molecular spectroscopy. Helpful comparisons with the classical theory of Brownian motion are drawn. Chapter 2 reviews the important and highly topical problem of the crossover from classical Kramers-type thermal activation to the quantum tunnelling of particles out of a metastable state. Having set the stage with a discussion of classical Brownian motion in a periodic potential, the theory is applied to the diffusion of an impurity hopping among interstitial sites in a crystal lattice. An interesting non-Markovian application follows wherein the impurity is driven by a random force weighted around a characteristic frequency. Enhanced activation rates are observed when the noise correlation becomes comparable to the inverse characteristic frequency. This review concludes with a discussion of the formidable problem of quantum tunnelling in the low damping regime. Even classical treatments of this topic are fraught with difficulties, as the following review in Chapter 3 demonstrates. Building on Kramers' original treatment, the escape rate of particles from a metastable state is discussed in the limits of low and moderately high damping.

Historical comments by a participant in the field represent a personal viewpoint. They are not a detached and scholarly contribution to the history of science. The field of this volume, and the (hopefully) closely related field of this discussion, involves a number of ‘schools’, of varying degrees of distinctness, and I cannot represent them all in this brief discussion. The Brussels School associated with Ilya Prigogine is one example. Another group, which is much more loosely knit and less clearly identifiable, can be associated with the work and influence of the late Elliott Montroll. This work comes closer to my viewpoint, represented below, but has had public visibility through recent books and conferences (Montroll and Lebowitz, 1979; Shlesinger and Weiss, 1985; Weiss, 1983, 1986). Both of the aforementioned ‘schools’ have a close relationship to physical chemistry, and we will, later in our discussion, return to some of the pioneering (but long neglected) insights which arose in this area. On the other hand, our field has equally deep, or perhaps deeper, roots in electronics. We will take it upon ourselves to represent that heritage. The approach we will stress describes some degrees of freedom and their dynamics explicitly; the remaining degrees of freedom are regarded as a source of noise and of damping. It is a viewpoint which has wide applicability, but is particularly directly and accurately applicable to electrical circuits.

Introduction

The investigation of the coherence properties of light sources has traditionally involved the study of correlation functions of different orders of the electric field (Born and Wolf, 1975; Mandel and Wolf, 1965). Field and intensity correlation functions have been measured for laser light (for a review, see Lax and Zwanziger, 1973) and have been compared with predictions obtained from quantum laser theory (Haken, 1970, 1981; Lax, 1969; Louisell, 1973; Sargent, Scully and Lamb, 1974). This was, in fact, one of the major confirmations of the correctness of the basic formalism of quantum laser theory. Most of the experiments that probed the coherence properties of lasers were, however, carried out on a very limited variety of lasing media. Interest was also centered on the regime of operation near threshold, where the light from the laser is extremely weak; hence photoelectron counting and correlation techniques were developed for the measurement of coherence properties of the laser light.

While these early experiments were performed with the main intent of testing the basic theory, the large variety of lasers now available, and their ever more demanding applications, make it necessary to develop new methods for the measurement and analysis of laser fluctuations. These techniques should be applicable to the measurement of noise in high power lasers; at the same time, in order to delineate the limits of usage, they must be sensitive enough to quantitatively determine the quantum mechanical sources of noise which may lie buried in external noise that is many orders of magnitude larger.

Introduction

The state of chemical equilibrium is endowed with stability: perturbations away from equilibrium always decay to this state. This is a fundamental feature of the statistical mechanics of relaxation processes near equilibrium. The situation is not so simple in the far-from-equilibrium regime. Bifurcations can lead to the appearance of more exotic system states, ranging from simple stationary states akin to the equilibrium state, to periodic, quasiperiodic or even chaotic states. In addition, the asymptotic state may not be unique; given different initial conditions, different final states may be reached. While these phenomena are widespread in nature, chemically reacting fluids, driven to the nonequilibrium regime by external constraints, constitute a ubiquitous and important class of systems of this type (Nicolis and Prigogine, 1977).

The presence of noise can have significant effects on nonlinear systems. It can modify the nature of existing states, create new states, or induce transitions between coexisting states (Horsthemke and Lefever, 1984). Noise may enter the description through internal fluctuations that have their origin in the molecular nature of the system, or through external means due to the inability to control system constraints. In fact, external noise may be intentionally applied to the system in order to achieve a given transition process.

The principal focus of this article is on transitions between bistable states, induced by external noise. The use of external noise sources to promote noiseinduced transitions has several convenient features.

Introduction

In this chapter we discuss a modest refinement of Kramers (1940) result for the noise activated escape out of an underdamped potential well. At very low damping, motion in a potential well (see Figure 3.1) is almost conservative, and the escape up in energy, out of the well, becomes difficult. This was appreciated by Kramers, who derived a result for extremely low damping. For a particle subject to thermal noise and damping proportional to the momentum p with a relaxation rate η, Kramers found an escape rate from the metastable state which is proportional to η. In the low friction limit it is diffusion up along the energy coordinate which limits the escape rate. On the other hand for large friction, the escape out of the potential well is limited by the diffusion along the reaction coordinate. For large friction the escape rate is proportional to η-1. Thus the escape rate as a function of the friction constant peaks at an intermediate value of the damping constant as shown in Figure 3.2. Kramers noted in his paper that it is the smaller of the two rates which applies. At very low damping, as the friction increases, the escape rate must eventually deviate from the linear behavior predicted by Kramers to merge at large η into the heavy damping behavior. It is this departure from the linear behavior for small damping which is the subject of this chapter.

Introduction

Recent work on dye lasers (Fox and Roy, 1987; Jung and Risken, 1984; Lett, Short and Mandel, 1984; Roy, Yu and Zhu, 1985; Short, Mandel and Roy, 1982) and the optical ring laser gyroscope (Vogel et al., 1987a, b) has emphasized the physically important role of colored noise sources. A well-known classical situation in which strongly colored noise has an impact on the physics is the phenomenon of motional narrowing in magnetic resonance (Kubo, 1962). Kubo has shown that a fluctuating magnetic field with very short noise correlation time (almost white noise) does typically not manifestly affect the motion of spins; on the contrary, if the fluctuations are correlated over a long time scale (colored noise) the motion of the spin becomes greatly modified.

Another area where there has been much recent activity addresses escape problems. These are currently in the limelight both from the theoretical viewpoint (Grote and Hynes,1980; Hänggi, 1986; Hänggi and Mojtabai, 1982; Hänggi and Riseborough, 1983; Hänggi, Mroczkowski, Moss and McClintock, 1985) as well as from an experimental point of view (Devoret, Martinis, Esteve and Clarke, 1984; Fleming, Courtney and Balk, 1986; Hänggi et al., 1985; Maneke, Schroeder, Troe and Voss, 1985). In this latter case, a frequency-dependent friction, or noise of finite correlation time, can considerably modify the classical barrier transmission. Except for two-state noise (Hänggi and Talkner, 1985; Masoliver, Lindenberg and West, 1986; Rodriguez and Pesquera, 1986; Van den Broeck and Hänggi, 1984) there exist no exact analytic methods for truly nonlinear systems, being driven by correlated noise.

Introduction

In nature, states of broken symmetry are found at all levels, from the microscopic to the macroscopic. The state of a physical system is determined by – depending on the level of description – atomic and molecular interactions or by irreversible thermodynamic processes. We speak of a ‘broken symmetry’ if the state does not have the symmetries of the interactions or the processes that determine it; transitions to such states are the symmetry-breaking transitions. From a thermodynamic viewpoint, symmetry-breaking transitions can occur in systems in equilibrium as well as systems far from equilibrium. In equilibrium systems, magnetic transitions in crystalline solids are the best known examples (Briss, 1964; Cracknell, 1975). In nonequilibrium systems, such transitions are known to occur in hydrodynamic, chemical and laser systems (Haken, 1977; Nicolis and Prigogine, 1977). A most striking example of a state of broken symmetry is the ‘state of life’. The chemistry of life has a definite handedness: all proteins are made of L-amino acids and DNA and RNA contain exclusively d-sugars (Miller and Orgel, 1974). Biochemical function depends very delicately on this molecular chirality. d-amino acids are a rarity and they occur only as polypeptides (Bentley, 1969). Though there are many conjectures, the processes that resulted in a transition to such a state of broken chiral symmetry have not yet been identified.

Introduction

One of the important problems of physical kinetics is the investigation of relaxation and fluctuation phenomena in systems which have two or more stable states. Bi- and multistable systems are studied in various fields of physics, and the causes of multistability and the types of stable states are different in different cases. For systems moving in static potential fields (disregarding the interactions that give rise to relaxation and fluctuations in a system) multistability takes place if a potential has several minima. In this case the stable states are the equilibrium states. A number of systems of this type are investigated in solid state physics; in particular, diffusing atoms and impurity centers that reorient within a unit cell (see Narayanamurti and Pôhl, 1970).

Multistability may also arise in systems driven by an external periodic field. The constrained vibrations correspond to the stable states in this case (the attractors with a more complicated structure may also appear here). In particular, nonlinear oscillators of various physical nature refer to such systems (see Landau and Lifshitz, 1976). It is well known that in a certain frequency range the dependence of the amplitude A of the constrained vibrations of a nonlinear oscillator on the resonant external field amplitude h may be S-shaped (cf. curve (c) in Figure 13.1). In the range of the non-singlevalued dependence A(h) the states with the largest and smallest A are stable.

Introduction

In this chapter we investigate the influence of noise in systems where many degrees of freedom are participating. The phenomenon we want to investigate is the possibility to leave and to enter attractor regions in bi- and multi-stable systems. It implies the ability to cross through separatrices (to jump over potential barriers) to alternative attractors of the systems after a certain time.

Since the pioneering work of Kramers this phenomenon has been widely investigated in one- and two-dimensional bistable systems (for reviews see Hanggi, 1986, and Weiss, 1986); several papers also appeared on multistable situations (see, e.g., Büttiker and Landauer, 1982). Stochastic transitions in multistable multidimensional systems are of importance for several physical and nonphysical problems, for example for nucleation induced phase transitions, for the molecular dynamics and the kinetics of chemical reactions, for the storage of information in electronic circuits, for learning procedures in networks of bistable elements, for the search of extrema in combinatorial optimization problems, etc. All those problems possess in general many interacting degrees of freedom, have a large number of stable states, configurations, etc.

Though in all the problems mentioned above stochastic transitions between stable states occur, their dynamics may be quite different. In some situations, for example chemical catalysis, it is desirable that the transition takes place very quickly. In other situations transitions should be avoided; for example, computer memory becomes meaningless if transitions occur on time scales which are comparable with the time of calculations.

Introduction

The Fokker–Planck equation (FPE) deals with those fluctuations of continuous variables which stem from many tiny disturbances, each of which changes the variables in an unpredictable but small way. One therefore has to add noise forces to the deterministic differential equations describing the motion of the variables of the system. This can be done by including these noise forces as additive terms on the right hand sides of the equations of motion (additive noise) or by including them as fluctuating prefactors of some terms (multiplicative noise). Thus, instead of deterministic systems, one has to deal with stochastic differential equations. The properties of the noise forces are important for defining the stochastic differential equations. If the noise forces are δ correlated in time one speaks of white noise, otherwise the term colored noise is used. For white noise the stochastic differential equations are called Langevin equations.

To obtain numeric solutions of these equations one may simulate them on a digital computer or build an analog circuitry simulating the equations electronically. Analytic solutions can be given if the Langevin equations are linear and have additive noise terms. Then, using Green's function of the deterministic system, one can write down the solution of the Langevin equation. If the Langevin equation is nonlinear, it is often impossible to solve it analytically. The best way of treating a nonlinear Langevin equation consists in transforming it to an FPE.