Sprin e roceedings in Physics 1

Springer Proceedings in Physics

Volume 1 Fluctuations and Sensitivity in Nonequilibrium Systems Editors: W. Horsthemke and D. K. Kondepudi

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Fluctuations and Sensitivity in Nonequilibrium Systems Proceedings of an International Conference, University of Texas, Austin, Texas, March 12-16,1984

Editors W. Horsthemke and D. K Kondepudi

With 108 Figures

Springer-Verlag Berlin Heidelberg New York Tokyo 1984

Professor Dr. Werner Horsthemke Department of Physics, Center for Studies in Statistical Mechanics, University ofTexas, Austin, TX 78712, USA

Dilip K. Kondepudi, PhD Center for Studies in Statistical Mechanics, University of Texas, Austin, TX 78712, USA

ISBN-13: 978-3-642-46510-9 e-ISBN-13: 978-3-642-46508-6 DOl: 10.1007/978-3-642-46508-6

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Preface

This volume contains the invited lectures and a selection of the contributed papers and posters of the workshop on "Fluctuations and Sensitivity in Nonequil ibrium Systems", held at the Joe C. Thompson Conference Center, Un i vers ity of Texas at Austin, March 12-16, 1984. The workshop dealt with stochastic phenomena and sensi-tivity in nonequilibrium systems from a macroscopic point of view. Durin9 the last few years it has been realized that the role of fluctuations is far less trivial in systems far from equilibrium than in systems under thermodynamic equilibrium condi-tions. It was found that random fluctuations often are a determining factor for the state adopted by macroscopic systems and cannot be regarded as secondary effects of minor importance. Further, nonequilibrium systems are also very sensitive to small systematic changes in their environment. The main aims of the workshop were: i) to provide scientists with an occasion to acquaint themselves with the state of the art in fluctuation theory and sensitivity analysis; ii) to provide a forum for the presentation of recent advances in theory and experiment; iii) to bring toge-ther theoreticians and experimentalists in order to delineate the major open problems and to formulate strategies to tackle these problems.

The organizing committee of the workshop consisted of W. Horsthemke, O.K. Konde-pudi, G. Dewel, G. Nicolis, I. Prigogine and L. Reichl. The workshop was organized under the auspices of the Center for Studies in Statistical Mechanics at the Uni-versity of Texas at Austin and was funded by the University of Texas at Austin, by the National Science Foundation, grant CHE-8318608, and by the Instituts Interna-tionaux de Physique et Chimie, fondes par E. Solvay.

Austin, Texas May 1984 W. Horsthemke . O.K. Kondepudi

v

Contents

Port I Basic Theory

Irreversibility and Space-Time Structure By 1. Prigogine (With 7 Figures) ............................................ 2 Stochastic Systems: Qualitative Theory and Lyapunov Exponents By L. Arnold ................................................................ 11

First Passage Times for Processes Governed by Master Equations By B.J. Matkowsky, Z. Schuss, C. Knessl, C. Tier, and M. Mangel

Port II Pattern Formation and Selection

Three Caveats for Linear Stability Theory: Rayleigh-Benard Convection

19

By H.S. Greenside (With 4 Figures) .......................................... 38 Pattern Selection and Phase Fluctuations in Chemical Systems By D. Walgraef, P. Borckmans, and G. Dewel (With 2 Figures) Experiments on Patterns and Noise in Hydrodynamic Systems

50

By J.P. Gollub ................ ...... ....... ...... ............ ...... ...... ... 58

Port III Bistable Systems

Optical Bistability: Steady-State and Transient Behavior. By A.T. Rosenberger, L.A. Orozco, and H.J. Kimble (With 7 Figures) . ............ ..... 62 Experimental Studies of the Transitions Between Stationary States in a Bistable Chemical System. By J.C. Roux, H. Saadaoui, P. de Kepper, and J. Boissonade (With 9 Figures) .......................................... 70 Noise-Induced Transitions in Multi-Stable Systems. By E. Ben-Jacob, D.J. Bergman, B.J. Matkowsky, and Z. Schuss (With 10 Figures) ............... 79 Bistable Flows Driven by Colored Noise. By P. Hanggi (With 2 Figures) 95

VII

Part I V Response to Stochastic and Periodic Forcing

Noise-Induced Transitions. By W. Horsthemke ................................. 106

Dynamical Aspects of External Nonwhite Noise. By J.M. Sancho and M. San Miguel ............................................................... 114

Dynamic Systems with Fast Parametric Oscillations By S.M. Meerkov (With 6 Figures) ............................................ 124 Experimental Studies of Noise-Induced Transitions By F. Moss and P.V.E. McClintock (With 12 Figures) 134 Sensitivity of a Hopf Bifurcation to External Multiplicative Noise By R. Lefever and J.W. Turner (With 1 Figure) ............................... 143

Part V Noise and Deterministic Chaos

Noise and Chaos in Selected Quantum Optical Systems By N.B. Abraham (With 1 Figure) ............................................. 152 Distinguishing Low-Dimensional Chaos from Random Noise in a Hydrodynamic Experiment. By A. Brandstater and H.L. Swinney (With 5 Figures) ............. 166 Sensitive Dependence to Parameters, Fat Fractals, and Universal Strange Attractors. By J.D. Farmer (With 2 Figures) ......................... 172 Noise-Induced Transitions in Discrete-Time Systems By R. Kapral, E. Celarier, and S. Fraser (With 8 Figures) 179 Scaling for External Excitations of a Period-Doubling System By A. Arneodo (With 6 Figures) .............................................. 187

Part VI Sensitivity in Nonequilibrium Systems

General Sensitivity Analysis of Differential Equation Systems By H. Rabitz ................................................................ 196

Nonequilibrium Sensitivity. By D.K. Kondepudi (With 4 Figures) 204 Patterns of Nonequilibrium Sensitivity in Biological Systems By O. Decroly and A. Goldbeter (With 7 Figures) ............................. 214 Chemical Reaction Network Sensitivity Analysis. By R. Larter and B.L. Clarke (With 1 Figure) ................................................. 223

Part VII Contributed Papers and Posters

Chaos in the Conservative Duffing System - Renormalization Group Prediction. By L.E. Reichl and W.M. Zheng (With 4 Figures) .................. 228 Solvable Double-Well Potential Models. By W.M. Zheng ........................ 230

VIII

Nonlinear Fluctuation-Dissipation Relations. By B.J. West and K. Lindenberg ............................................................... 233

Numerical Studies of Fluctuations and Hysteresis in the Homogeneous Schlagl Model. By J. Kottalam and K.L.C. Hunt (With 2 Figures) .............. 242 Could Weak Neutral Currents Have Determined Biological Chirality? By G.W. Nelson .............................................................. 245

Nonequilibrium Chemical Instabilities in Continuous Flow Stirred Tank Reactors: The Effect of Stirring. By L. Hannon (With 2 Figures) ............. 249 The Effect of Random and Periodic Fluctuations on Nonlinear Systems By C.R. Doering (With 2 Figures) ............................................ 253 The Period-Doubling Power Spectrum of Conservative Systems By B. Hu and J.-M. Mao ...................................................... 257

A New Computable Criterion for the Non-Existence of Invariant Circles By W. Chou (With 2 Figures) ................................................. 260 Periodic and Nonperiodic Dynamical Behavior Near Homoclinic Systems By P. Gaspard (With 1 Figure) ............................................... 265 Precursors of Period Doubling Instabilities By K. Wiesenfeld (With 1 Figure) ............................................ 268 Flow Patterns in a Circular Couette System By C.D. Andereck and H.L. Swinney ........................................... 271

Index of Contributors ...................................................... 273

IX

Part I Basic Theory

Irreversibility and Space-Time Structure I. Prigogine Center for Studies in Statistical Mechanics, University of Texas, Austin, TX 78712, USA and Faculte des Sciences, Universite Libre de Bruxelles, Campus Plaine, C.P. 231, B-1050 Bruxelles, Belgium

1. Introduction

I am happy to participate in this conference dealing with fluctuations and sensi-tivity. One of the main outcomes of research in macroscopic physics over the last decades is that we live in a pluralistic universe; we deal both with dissipative systems and with conservative systems [1] (I limit myself here to classical dynamical systems). Dissipative and conservative systems have widely different properties. Briefly, dissipative systems are characterized by asymptotic stability. They forget temporary perturbations. The simplest example, well known to everybody, is a pendu-lum with friction. If perturbed it goes back to the equilibrium position. This equilibrium position is a point attractor. However, we know now that at tractors may be more complicated than isolated points. They may be lines in phase space, such as in periodic chemical reactions or even more complicated mathematical objects like fractals. That is why we speak today of strange attractors. The second common element in dissipative systems is the dissymmetry in respect to time. All dissipa-tive systems have a preferential direction of time; they progress towards their attractors for t going to +00 (and not for t going to _00). We could imagine a world in which some biological systems which belong to the class of dissipative systems would age while others become younger. In such a world some dissipative systems would tend to equilibrium for t~ while others do so for t+-oo. But that is not our world, in which, so far as we know on empirical grounds, there is a universal time asymmetry.

Let us now turn briefly to conservative systems. In classical mechanics con-servative systems would be described by Hamilton's equations of motion. From the abstract point of view, such systems are characterized by a phase space and by a measure which is preserved in time. A striking difference is that dynamical systems are never stable in the same sense as dissipative systems. They do not have the property of asymptotic stability which we have mentioned. When we give a larger amplitude oscillation to a frictionless pendulum, it takes up a new frequency and conserves it as long as friction can be neglected. On the contrary, if we run, our heartbeat increases but returns to normal after we take a rest, As a result, in comparison with dissipative systems dynamical systems are basically unstable. There is no way of forgetting perturbations. As a result,the world of conservative dynam~ ical systems is certainly not a world in which delicately balanced processes such as we see in biology would be possible.

Let us illustrate this remark by considering first the classical example of a pendulum. The representation of the trajectories in phase space is well known (see Fig. 1). Point E corresponds to the equilibrium position. Points HI' H2 are in fact identical and correspond to the unstable situation in which the pendulum stays on its head. Point E is an "elliptic" point. If the system is in the neighborhood of E and we perturb it, it will shift its trajectory to a new periodic motion around E. In contrast, points HI and H2 are "hyperbolic" points corresponding to the cross-ing of a stable and an unstable trajectory called the separatrices. The separatriceE Sl' S2 separate the region of vibration around equilibrium from the region of rota-tion. The elliptic point E is orbitally stable in the sense that the perturbed system remains on a neighboring orbit. On the contrary, the hyperbolic points are unstable. In classical examples such as the pendulum there are generally only a few

2

v Fig. 1. Phase space of a pendulum. Here V is the velocity and e the angle of de-flection; Sl' S2 are the separatrices; E is an elliptic point; HI and H2 are hyper-bolic points--see text

hyperbolic points. It is interesting that, on the contrary, in most systems which are at present at the center of interest in dynamics there are a multitude of both elliptic and hyperbolic points. A very simple example which we shall use for illus-tration corresponds to the baker transformation.

We take a square and flatten it into a rectangle, then we fold half of the rect-angle over the other half to form a square again. This set of operations is shown in Fig. 2 and may be repeated as many times as one likes. Each time the surface of the square is broken up and redistributed. The square corresponds here to the phase space. The baker transformation transforms each point into a well-defined new point. Although the series of points obtained in this way is "deterministic," the system displays in addition irreducible statistical aspects. Let us take, for in-stance, a system described by an initial condition such that a region A of the square is initially filled in a uniform way with representative points. It may be shown that after a sufficient number of repetitions of the transformation, this cell, what-ever its size and localizatiop, will be broken up into pieces (see Fig. 3). The----essential point is that any region, whatever its size, thus always contains differ-ent trajectories diverging at each fragmentation. Although the evolution of a point is reversible and deterministic, the description of a region, however small, is basi-cally statistical.

q=11J (\ q=l. !!!! I

p=1 I 2 p=1

8

I~) q=11J 112 p=1

8-1

Fig. 2. Realization of the baker transformation (B) and of its inverse (B-1). The path of the two spots gives an idea of the transformations

p

q

Fig. 3. Time evolution of an unstable system. Time going on, region A splits into regions A' and A", which in turn will be divided

3

A characteristic feature of the baker transformation is that each point corre~ sponds to the crossing of two orthogonal lines, one vertical, which corresponds to a contracting fiber, the other horizontal, which corresponds to a dilating fiber. Each point therefore corresponds to a hyperbolic point. There is also an abundance of elliptic points; however, they measure zero in the same sense as the measure of rational numbers is vanishing. Still, we would have to expect to find in the baker transformation what is often called orbital randomness. According to the fact that we start with an elliptic or with a hyperbolic point, we would have quite different behav.ior.

There is also another very important concept involved in unstable dynamical sys~ tems. That is the concept of a Lyapounov exponent or a Lyapounov time. In such systems the distance or between two points increases exponentially with time.

TL may be called the "Lyapounov time." (We consider of course here an average Lyapounov time.)

In conclusion, nature presents us with two types of dynamical systems, dissipa~ tive systems and conservative systems. What is the relation between the two? This question has intrigued physicists for more than one hundred years. It is a diffi~ cult question, but I believe we now come closer to the answer.

2.. The Search for Unification

Thermodynamics provides us with a fundamental insight on the difference between dissipative systems and conservative systems. Indeed, the second law of thermo~ dynamics introduces a basic new quantity, the entropy S, which is fundamentally re~ lated to dissipation. Dissipation produces entropy. But what is then the meaning of entropy? Here Boltzmann came up 110 years ago with a most original ideal entropy is basically related to probability:

S = k 19 P

It is because the probability increases that entropy increases,

Let us immediately emphasize that in this perspective the second law would have great practical importance but would be of no fundamental significance. In his ex~ cellent book The Ambidextrous Universe, Martin Gardner writes: "Certain events go only one way not because they can't go the other way but because it is extremely unlikely that they go backward" [2]. By improving our abilities to measure less and less unlikely events, we ~ould reach a situation in which the second law would playas small a role as we want. This is the point of view that is often taken to~ day. However, this point of view is difficult to maintain in the presence of the important constructive role of dissipative systems, which we have emphasized at the beginning of the lecture. In fact, here we are in front of a problem which is quite similar to the famous quarrel about hidden variables, to which physicists have de~ voted so much time over the last years. Is probability the outcome of our ignorance, of our averaging over hidden variables? Or is this probability genuine, expressing some nonlocality in space~time? In the case of quantum theory the answer is at pre~ sent quite clear. The introduction of probability in quantum theory comes from the existence of Planck's constant h. It is interesting that in the problem of irre~ versibility we shall ~lso come to the conclusion that the probability is genuine and that the underlying nonlocality comes from the instability of motion of the dynamical systems to which the second law of thermodynamics can be applied.

However, there seems at first to be a basic difficulty to attempt any unification between thermodynamics and dynamics. In dynamical theory we can introduce a distri~ bution function p which evolves in time according to the law

4

where U is a unitary operator (it is often written as exp iLt, where L is the so-called fiouville operator). The unitary operator Ut satisfies the group relation

> t,s < O.

As a result of the unitary character of the dynamical evolutio~ entropy, which is expressed as a functional of p, remains unchanged in the course of dynamical evolu-tion. This is in striking contrast with what happens with probabilistic processes such as Markov chains. There also we may express the evolution in terms of some operator Wt acting on the initial distribution function

However, this new operator is no longer unitary and satisfies now the semigroup condition

W W t s

W t+s t,s ~ O.

The problem of unification of dynamical systems and dissipative systems is essen-tially the problem of elucidation of the relation between p, corresponding to dynam-ical evolution, and p, corresponding to the evolution of probabilistic processes. We cannot go into details here [3J. Let me simply mention that we have shown that it is possible to go from the dynamical distribution to the probabilistic one in terms of the transformation

p = Ap where A is a suitable operator breaking the time symmetry and introducing a nonlocal description in space-time. In other words, in dissipative systems the fundamental laws are no longer the laws of dynamics alone as we have to include the second law of thermodynamics. As a result, the basic object which is evolving in dissipative systems 1S no longer the initial distribution function p but a transform of this distribution function, the transformation being itself determined by the dynamical laws. Dissipative systems correspond therefore to a new level of description.

A historical analogy may come to one's mind. In the early days of statistical mechanics. the Ehrenfests had emphasized the need to introduce a coarse grained distribution function which would satisfy the second law in contrast with the fine grained distribution p which would satisfy the laws of dynamics. This idea is at the basis of well-known textbooks such as the outstanding book by TOLMAN on statis-tical mechanics [4J. However, the coarse grained distribution was considered to be the outcome of our ignorance of the fine grained distribution and its resulting from some average procedure as applied to the fine grained density. This cannot be the whole story as the arbitrariness of the coarse grained distribution would lead to arbitrariness in the temporal evolution of dissipative systems, which is not born out by experiment. Moreover the coarse graining must include the breaking of sym-metry of the initial equations of motion. Anyway, our method of unification of thermodynamics and dynamics can be viewed as a way of making more precise the in-tuitions of the founders of statistical mechanics.

3. From Dynamics to Thermodynamics

As expressed in the preceding section, in our view the transition from dynamics to thermodynamics corresponds to the transformation from the dynamical fine grained distribution p to the distribution p which satisfies a Markov chain property. The existence of this transformation can be rigorously proved for an important class of dynamical systems which are highly unstable, such as the baker transformation which we have described. It can also be proved, but by perturbation techniques, for a larger class of dynamical systems such as described usually in terms of kinetic

5

equations. Let us emphasize the two elements which are necessary to insure the ex-istence of this transformation.

The first is, as already mentioned, a high degree of instability. It is this instability which makes the concept of trajectory unphysical and leads therefore to the very possibility of applying in an objective sense to probability concepts. Systems which have this high degree of instability may be called intrinsically random. But there is a second element. Only for systems in which the symmetry of time is broken can we hope to formulate the second law of thermodynamics. In other words, it is only when there exist states of motion whose velocity inverse is forbidden that the transformation from dynamics to thermodynamics makes sense. This means that the second law can be expressed as a selection principle selecting only initial condi-tions which are compatible with the approach to equilibrium in our distant future.

Let us now illustrate the idea of the second law as a selection principle. As already mentioned, at every phase point there are now two manifolds (of lower dimen-sionality than the entire phase space): one which progressively contracts under dynamical motion for increasing t, and the other expanding with t. As the result of the baker transformation as represented in Fig. 4, a vertical line will progressively contract to smaller and smaller vertical lines under successive application of the baker transformation ("contracting" fibers), whereas a horizontal line will double with each application of the baker transformation ("dilating" fibers).

-- contracting fiber

-- dilating fiber

~ Baker transformation--contracting fiber and dilating fiber

The contracting and expanding manifolds, when they exist, are evidently time asymmetric objects. The contracting manifold moves, in a sense, as a single unit towards the future. All its points tend toward the same fate in the future, but they have diverging histories as we look back more and more:into the past. Expanding manifolds are just the opposite. Points on it have diverging future behaviors, but progressively converging histories as we look farther and farther back into the past.

It is the existence of such time asymmetric objects which enables one to construct the symmetry-breaking transformation A by assigning non-equivalent roles to expanding and contracting manifolds. In fact, it can be shown that the choice of A which gives rise to entropy increasing evolution ~or t > 0) as the physically realized symmetry-breaking transformation implies the exclusion of (singular) distribution functions concentrated on contracting manifolds from the set of physically realizable states.

What is physically realizable and what is not is, of course, an empirical ques-tion. What our formulation of the second law achieves is to link the second law and associated "arrow of time" with a limitation, on the fundamental level, of pre-paring certain types of initial conditions. It is interesting that in physically interesting models of dynamical systems the type of initial conditions that are ex-cluded by the symmetry-breaking transformation A are precisely those which one never intuitively expects to be unrealizable.

6

Many examples can be given. In scattering theory we may have a plane wave giving rise to an outgoing spherical wave [5]. We could have also an incoming spherical wave which is transformed into a plane wave. Both processes are strictly symmetri-cal from the point of view of the laws of dynamics. However, only one of these two types of phenomena occur in nature. It is very gratifying that the selection princi-ple as included in the second law of thermodynamics has a very simple physical mean-ing: we cannot prescribe a common future for ensembles in unstable dynamical systems.

Let us now describe more precisely how to achieve the construction of the A trans-formation.

4. Internal Time

For the class of unstable systems of which the baker transformation is the simplest example, the construction of A may proceed most directly through the consideration of what MISRA and I have called the internal time T [3]. To grasp the intuitive meaning of internal time, think about a drop of ink in a glass of water. The form the drop takes gives us an idea of the interval of time which has elapsed. We may consider the baker transformation and look how ink will be distributed in the square as a result of successive transformations. This succession is represented in Fig. 5. A shaded region may be imagined to be filled with ink, an unshaded region by water. At time zero we have what is called a generating partition, Out of this partition we form a series of either horizontal partitions when we go into the future or verti-cal partitions going into the past. These are the basic partitions. An arbitrary distribution of ink in the square can be written formally as a superposition of the basic partitions. With each basic partition we may associate an "internal" time that is simply the number of baker transformations we have to perform to go from the generating partition to the one under consideration. We therefore see that this type of system admits indeed a kind of internal age.

-m-I}~-=--I o 2

pas! . t .. \jenera!'n\j parhhon future Fig. 5. Starting with the partition at time 0, we repeatedly apply the baker trans-formation. We generate horizontal stripes in this way. Similarly,going into the past we obtain vertical stripes

The internal time T is quite different from the usual mechanical time, since it depends on the global topology of the system. We may even speak of the "timing of space," thus coming close to the ideas recently put forward by geographers, who have introduced the concept of "chronogeography" [6]. When we look at the structure of a town, or of a landscape, we see temporal elements interacting and coexisting. Bra-silia or Pompeii would correspond to a well-defined internal age, somewhat like one of the basic partitions in the baker transformation. On the contrary, modern Rome, whose buildings originated in quite different periods, would correspond to an average time exactly as an arbitrary partition may be decomposed into elements corresponding to different internal times.

It is very important to notice that the internal time corresponds now to an oper-ator whose eigenfunctions are the partitions, in the case of the baker transforma-tion, and whose eigenvalues are the times given by the watch. In this simple ex-ample, the change of the average internal age keeps track with the time of the watch. But for more complex dynamical systems the relation between the average time and watch time becomes more complicated. Obviously the internal time cor-responds to a nonlocal description. If we would know exactly the position of a

7

point in the square corresponding to the baker transformation, we would not know the partition to which it belongs. Inversely, if we know the partition we still do not know the position of the trajectories.

The question ''What is time?" has fascinated man since the dawn of modern thought. Aristotle associated time with motion, but he added, "There must be also a soul which counts." In a sense the soul which counts is replaced here by the internal time which is measured by astronomical time but not identical with it. The watch has no time in our sense. Everyday it goes back into its own past. It is we who have a time, and this time expresses the fact, like all chemical systems, we belong to the category of highly unstable dynamical systems for which an object such as T can be defined.

It is interesting that the prototype of the physical world in the classical thought was planetary motion. Now I believe the prototype becomes that of highly unstable dynamical systems out of which internal time and irreversibility may be generated.

5. States and Laws

We have introduced in the preceding section the transformation function A. We have shown that it is simply a decreasing function of the internal time A(T). This has some very interesting implications. Suppose we expand the fine grained distribution function p in terms of the eigenfunctions Xn of the internal time T. We obtain formally

+00 p = n~-oo cn Xn .

If we now apply the transformation A we obtain similarly

p Ap

The A are the eigenvalues of operator A corresponding to the eigenfunction X. The imporant requirement is that A varies from I for n7-00 to 0 for n++ oo . Herenn is the eigenvalue of the internal ime. But p and p at a given time are made up in general from contributions coming from both past (nO) in terms of the internal time T. However, while in p future and past playa symmetrical role, this is no longer so in p. Here the contribution of the future states is damped as A goes to zero for n~. The present contains the contributions from the past and cgntributions from the nearby future. This is in contrast with dynamical determin-istic systems where the present implies both the past and the future.

Let us represent A as a function of n (see Fig. 6). Past and present are sepa-rated by a kind of tr~nsition layer. It may be shown that this transition layer is

+- past -CD

~n

transition layer

future-+CD

Fig. 6. Transition between past (n-+- oo) and future (n-++oo)

8

of the order of the Lyapounov time we have introduced, 'L' It is interesting to contrast this representation with the traditional representation of time as a straight line (see Fig. 7). The present then corresponds to a single point which separates past from future. The present comes, so to speak, out of nowhere and disappears into nowhere. Moreover, being reduced to a point it is infinitely contiguous to the past and the future. On the contrary, on our representation the past is separated from the future by an interval measured by the Lyapounov time: we may speak of the "duration" of the present.

post present future Fig. 2. Traditional representation of time

It is interesting that many philosophers, BERGSON (7J, WHITEHEAD (8J, have em-phasized the need to attribute to the present such kind of incompressible duration. The rise of the second law as a dynamical principle leads precisely to this conclu-sion.

From the classical point of view, initial conditions were arb'itrary, and only the law which connects the initial conditions to the final outcome had an intrinsic meaning. But this arbitrariness of initial conditions corresponds to highly idealized situations in which indeed we can prepare initial conditions according to our will. When we take complex systems, be it a liquid or, even more so, some social situation, the initial conditions are no longer submitted to our arbitrariness, but are them-selves the outcome of the previous evolution of the system.

This connection can be made more explicit using the conceptual framework devel-oped in this lecture. Let us compare the distribution function p and p expanded in terms of the eigenfunctions of the internal time operator. We have already empha-sized that in p (this section) future and past enter symmetrically. Moreover, this symmetry is propagated by the unitary transformation. The situation changes radical-ly when we consider the formula for the transformed distribution P. As the A de-crease with n++=, the contributions of the partitions belonging to the futurenare "damped." Past and future enter in a dissymmetrical fashion: we have here states with temporal "polarization." Such states can only be the outcome of an evolution which itself is temporally polarized and will remain so in the future.

We see therefore that states and laws are indeed closely connected. There are self-preserving forms of initial conditions. After all, an initial condition corre-sponds to a time we choose arbitrarily; it can have no basic properties which would distinguish it from all other times. We see therefore that there is a close relation between states and laws. Or, in more philosophical terms, a close relation between Being and Becoming. Being is in this way associated with states, and Becoming with the laws transforming the states.

In conclusion, it seems to me quite remarkable that the advances made in non-equilibrium physics and the theory of dynamical systems have not only a direct im-pact on problems of great experimental and even technological relevance, but that they lead to drastic modifications of our concepts of space and time.

1. See I. Prigogine and I. Stengers: Order Out of Chaos (Bantam Books, New York 1984).

2. M. Gardner: The Ambidextrous Universe: Mirror Asymmetry and Time-Reversed Worlds (Charles Scribner's Sons, New York 1979).

3. See I. Prigogine: From Being to Becoming (W. H. Freeman & Co., San Francisco 1980). More recent references may be found in B. Misra and 1. Prigogine: Letts. in Math. Physics 2, 421 (1983).

9

4. R. C. Tolman: The Principles of Statistical Mechanics (Oxford University Press, London 1938).

5. Cl. George, F. Mayn~ and I. Prigogine: to appear, Adv. Chemical Physics (1984).

6. See D. N. Parks and N. J. Thrift: Times, Spaces and Places; A Chronogeographic Perspective (John Wiley & Sons, New York 1980).

7. H. Bergson: Oeuvres, editions du Centenaire (PUF, Paris 1970).

8. A. N. Whitehead: Process and Reality: An Essay in Cosmology (The Free Press, New York 1969).

10

Stochastic Systems: Qualitative Theory and Lyapunov Exponents

L. Arnold Fachbereich Mathematik, 0-2800 Bremen 33, Fed. Rep. of Germany

The main purpose of this paper is to stress the importance of Lyapunov exponents for the study of nonlinear deterministic and stochastic systems. After some introductory examples we present basic results of Lyapunov exponents for stochastic parameter-excited systems. This includes a formula for the biggest Lyapunov exponent (which determines the stability of the system) from which various quantitative conclusions can be drawn. In particular, the stabilizing and destabilizing effect of noise can be studied via perturbation theory.

1. Introduction and Motivation In the following sections we would like to show the importance of the concept of exponential growth rate~ (or Lyapunov exponents) for the qualitative study of linear and nonlinear deterministic and stochastic systems.

1.1 Irregular Behavior of Deterministic Systems During the last 20 years it became clear that even very simple deterministic nonlinear systems can have an extremely complex and irregular behavior. Even worse, it turned out that irregular or chaotic or "stochastic"/"statistical" behavior of deterministic systems is the rule, and regular behavior is a rare exception.

For an explanation of irregular behavior so-called hyperbolicity conditions playa crucial role. They say, in intuitive terms, that near any fixed trajectory the neighboring trajectories behave like at a saddle point. If this is the case at every point of the state space, the trajectories necessarily get mixed up and show "stochastic" behavior.

Hyperbolicity conditions were formulated in terms of Lyapunov exponents (PESIN [15]). Assume that a dynamical system is described by the iterations (Ft)tEZ of a diffeomorphism F: M ~ M on a manifold M or by a flow (Ft)tER of diffeomorphisms, Ft: M ~ M , generated by a vector field X (or, in other words, by a differential equation) (Note: All objects of differential geometry appearing in this paper are assumed to be smooth = C~). Hyperbolicity of (Ft) at x E M means that the tangent space TxM can be decomposed into subspaces, TxM = E1 (x) e E2(x), such that the linearized map TFt(x) : TxM ~ TFt(x)M is a contraction on E1 (x) and an expansion on E2(x). The Ei(x) immersed into M will give stable and unstable manifolds at x

How can one check the splitting of the tangent space? This is done by studying the Lyapunov exponent (= exponential growth) at x E M in the direction v E TxM ,

1 t A(X,V) = lim sup t log IITF (x)vl I , t~

11

I I I I any Riemannian metric. Hyperbolicity at x E M means tnat A(X,V) * 0 for all v E TxM, and E1 (x) = {v: A(X,V) < o}, E2(x) = {v: A(X,v) > o}. For flows, TFt(x)X(x) = X(Ft(x, so one typically expects A(X,X(X = 0 .

To go further one needs a measure of M which is invariant w.r.t. (Ft) Via the study of Lyapunov exponents one can check the ergodicity of (Ft) , give a formula for the entropy etc. This subject is called "smooth ergodic theory". 1.2 Stability Theory of Randomly Perturbed Systems Suppose we have a nonlinear deterministic differential equation in Rd whose right-hand side is perturbed by a stationary stochastic process E (t), ,

z = f(z,E(t, z(o) = zo . The stochastic analogue of a steady state will be a stationary solution z(t) Suppose there exists a stationary solution which is stationarily connected with E.

To study the stability of z(t) one considers the difference x(t) z(t) - z(t). In first approximation, x(t) satisfies

.

x = A(t)x, x(o) ( 1 ) where

A(t) af.

l -(ax:-(z(t),E(t) J

is a matrix-valued stationary stochastic process. Parameter-excited linear systems of type (1) appear in numerous

applications in science and engineering (cf. ARNOLD and KLIEMANN [3]). Under certain conditions, the original solution z(t) of the nonlinear system will be stable with probability 1 if the trivial solution x = 0 of (1) will be stable with probability 1 The latter will be the case if the Lyapunov exponents

1 A(XO) = lim sup t log Ilx(t;xo ) II (2) t-+oo

satisfy for any initial random variable x o

The paradigmatic example is the damped linear oscillator with random restoring force,

y + 2Sy + (1+aE(ty = 0, S,a E R , (3) E(t) isa stationary process on R with mean zero and variance 1. As in the case of periodic excitation (Floquet theory), engineers aim at stability diagrams, i.e. at determining the regions of stability and instability in the S,a plane. Their fear is that drastic changes can happen even for small a * 0 .

On the other hand, can noise have a stabilizing effect? E.g., can an unstable linear system x = Ax be stabilized by parametric nOise, x = (A+B(tx? How about the nonlinear case? How does the bifurcation behavior change in the presence of noise?

All those questions can be dealt with as soon as one has more detailed information about Lyapunov exponents. Those exponents are the stochastic analogue of the real parts of matrix eigenvalues.

1.3 Random Schrodinger Operators An electron in disordered material is described in quantum physics by a Schrodinger operator in R3 ,

12

H = - f::,. + q(x) where q(x), the potential, is a stationary random field (cf. KUNZ and SOUILLARD [12]). One would like to know the spectrum of H , considered as an operator in L2(R3,dx). In particular, is there localization of the wave function, i.e. does H have a complete orthonormal system of eigenfunctions in L2? The answer is not known, but it is being conjectured that there is localization at least for large disorder.

However, for a one-dimensional random medium, the answer is 'yes' even for arbitrarily small noise intensity provided the parameter-excited system -y + q(t)y = Ey, E E R, or, equivalently,

is unstable, i.e. possesses a positive Lyapunov exponent for each E E R(MOLCANOV [13], KOTANI [11]) . This is in sharp contrast to the case of a periodic potential, where there are no eigenvalues. For almost periodic potential q(t) the situation is much more complex.

1.4 Products of Random Matrices The discrete time versions of the previous sections lead to particular cases of the following mathematical problem: Can the limit theorems of probability theory (in particular, the law of large numbers) for sums of independent and identically distributed random variables in Rd be carried over to (non-commutative) products XnXn-1 ... X1 ' where (Xk ) is a stationary sequence with values in a group G?

A breakthrough was accomplished by FURSTENBERG [6] for the case of non-compact semisimple Lie groups. He proved, for example, that for the group G = SI(d,R) (dxd matrices with determinant = 1) there is a non-random constant A ("Furstenberg's constant") for which

(4) n->

with probability 1 for each Xo E Rd-{O} provided the (Xk ) are independent and the closed subgroup generated by the support of the distribution of X1 in G is irreducible. But A is nothing but the Lyapunov exponent of the orbit of xn = Xnxn -1 starting at Xo , and the result (4) says that this exponent is the same for all orbits '" 0 .

This line of research was carried on by GUIVARC'H [7], KIFER [9], ROYER [16], VIRTSER [17] and others.

2. Basic Results on Lyapunov E~~ts In this chapter we would like to restrict ourselves to the case of a multiplicative noise linear system. Let us emphasize once more that the results can be carried over via "deformation" to nonlinear stochastic systems on manifolds. For the case where the stochastic flow (Ft) is generated by a diffusion process (rather than by a disturbed vector field) see CARVERHILL [5]. 2.1 The Multiplicative E~odic Theorem This basic theorem was proved by OSELEDEC [14] in 1968 and reproved many times. We formulate it for systems of type (1).

Theorem (Multiplicative Ergodic Theorem). Let A(t) be a stationary and ergodic dxd matrix valued stochastic process with

13

EIIA(O)II < co, X = A(t)x with and let A(Xo ) 1, there are r

E = mean. Denote by x(t;xo ) the solution of x(O;xo ) = x O ' Xo an Rd-valued random variable,

be its Lyapunov exponent (2). Then, with probability , 1 ~ r ~ d, fixed numbers

A max

(called the Lyapunov exponents of the system) and random linear subspaces

with dim Ei = d i (non-random) and (i) Rd = E1 $ E2 $ $ Er

(ii) A(Xo ) = lim ~ log I Ix(t;xo ) I I t-.oo

Xo E Ei with probability 1 .

r rd. 1 ~

d , such that:

Ai if and only if

(iii) Let V. = E1 $ $ Ei Then Vo = ~ {oJ c::: V1 c ... c Vr = Rd , and

(iv)

A(Xo ) = A. if and ~ In particular, A(Xo ) with probability 1 r rd.A. = trace EA(o). 1 ~ ~

only if A

max

x E V. - Vi _ 1 with probability 1. 0 ~ Rd if and only if x E - V

r-1 0

Of course, for A(t) = A the Lyapunov exponents are the real parts of the eigenvalues of A, and the subspaces Ei are the generalized eigenspaces. For the oscillator (3) we have, by Theorem 1 (iv) , A1 + A2 = - ~. But when is A2(~'o) > 0, i.e., when is the oscillator unstable? To answer this question we need more quantitative information about Lyapunov exponents. The only thing we can immediately read off from Theorem 1 is part (iv), a formula for the center of gravity of the Ai's.

2.2 A Formula for the Biggest Lyapunov Exponent We now restrict ourselves to Markovian noise. More specifically, let f;(t), the "background noise", be a stationary ergodic diffusion process on a connected Riemannian manifold M of dimension m solving the (Stratonovich) stochastic differential equation

r . df; = Xo(E)dt + r x. (E) 0 dW~ ,

i=1 ~ where XO 'X1 ""'Xr are vector fields on M For diffusion processes on manifolds see IKEDA and WATANABE [8J. The generator of E(t) will be

r Q = Xo + r X~

i=1 ~ For example, Brownian motion on M

by d diffusion vector fields,

14

d dE = (I-f;f;') 0 dW = rx. (E) 0 dWi ,

1 ~

Sd-1 C Rd can be described

where I = (oik-EiEk)dxd = (X1 , ... ,Xd ) and W Wiener process in Rd

1 d , (W , ,W ) is a

Let us assume that E(t) is elliptic in the following sense:

dim LA(X1 , ... ,Xr ) (E) = m for all E EM, LA(Z) denoting the Lie algebra generated by the set of vector fields Z

Let A: M ~ dxd matrices with EI IA(E(o I I < 00 the system

x = A(E(t) lx, x(o)

and consider

( 5)

In polar coordinates r = I Ixl I E R+, S = x/I Ixl I E sd-1, (5) is equivalent to the nonlinear system

~=h(s,E(t, i=q(s,E(t, h(s,E)=(A(E)-q(s,E)s, q(s,E)=s'A(E)s. But this entails

Ilx(t;xd f I whence

t Ilxoll exp f q(s(,) ,E(,d" So

o

t t f q(s(,),E(,d, .

o

Consequently, A(X) depends only on the long-term behavior of the pair (s(t) ,E(t,o which is a (degenerate) diffusion process on the manifold Sd-1 x M described by

d(sE) = (h(S,E)\dt + ~ (0 \ 0 dWi Xo(E)} i=1 Xi(E)} .

The ergodic behavior of this process can be studied by nonlinear deterministic control theory (cf. KLIEMANN and ARNOLD [10]). To avoid non-generic situations, we assume

dim LA(h(',E) ,EEM) (s) = d-1 for all s E sd-1 . This is particularly satisfied if dim LA(A(E)x,EEM) = d for all x E Rd - {o}. The latter can be easily checked and is fulfilled in most cases relevant for applications, e.g., for A(E) = non-constant companion form matrices.

Under the provision of the above assumptions we have Theorem 2 (KLIEMANN and ARNOLD [10]).

(i) The process (s(t) ,E(t on Sd-1 x M has only finitely many different extremal invariant probabilities v. The support of v has the form CxM with C having nonvoid interior on Sd-1 .

(ii) The number A = f q(s,E)dv (6)

d-1 S x M is independent of the v chosen.

(iii) For each fixed x *0 0 we have A=Amax with probability 1 .

Comments. (i) The theorem says that physically realizable solutions (i.e., with nonanticipating initial conditions xo ) can "see" only the

15

biggest Lyapunov exponent Amax. (ii) Formula (6) can be used as a starting point for various quantitative studies of A=Amax . The invariant probabilities have, by our assumptions, smooth densities p satisfying the Fokker-Planck equation L*p = 0, L = h + Q being the generator of (s(t) ,E(t)).

Example. The oscillator (3) satisfies all conditions of Theorem 2 if E(t) is a diffusion in a non-degenerate interval (a,b) cR. Moreover, there is a unique smooth invariant v on S1 x (a,b).

3. Applications In this chapter we give some quantitative conclusions based on the general results in section 2.2 concerning the stabilizing and destabilizing effect of noise.

3.1 Perturbation Expansion of Lyapunov Exponents We are interested in the effect of small (or big) noise on a determinisfid system,

x = (A+B(E(t)))x, small (or big, resp.). By means of (6) we can derive a perturbation expansion for A(). Assume,e.g., is small, the eigenvalue A(A) of A with maximal real part is unique (thus real) and simple with eigenvector So E Sd-1 , and let EB(E(o)) = 0 .

In this case, the invariant probability is unique with smooth density peE) satisfying L*()p() = 0, where

L() = h(s,A) + Q + h(s,B(E)) = Lo + L 1 is the generator of (s(t) ,E(t)) on Sd-1 x M Put

2 P () = Po + P1 + P2 + ....

Equating coefficients in L*()p() = 0 yields L6Po = 0 , L~P1 = - LtPo' Because q(s,E) = s'As + s'B(E)s, we obtain

A() = f s'ASp + (fs'B(E)sp +fs'ASP1) o 0

+ 2(JS'B(E)SP1+fs'ASP2)+ . Clearly po(s,E) = 6 (s)n(E) (n(E) = invariant density of E(t)),

So thus

Of course, we have to verify that this expansion is asymptotic, meaning, e.g., A() - A(A) - A1 - 2 A2 = 0(2), which can be done in particular cases (cf. ARNOLD, PAPANICOLAOU and WIHSTUTZ [4]). In many cases A1 = 0 so that the sign of A2 decides whether the noise destabilizes (A2>0) or stabilizes (A2 O. What about y + f(E(t))y = O? By using the relation g' (0) = A for

g(p) = lim -t1 log EI Ix(t;x ) liP, pER, t-- 0

(see ARNOLD [1]), MOLCANOV [13] proved that A > 0 which was a

16

E E M A for

breakthrough in the theory of random Schrodinger operators (compare section 1.3). The general situation is as follows:

Theorem 3 (KLIEMANN and ARNOLD [10]). Consider x A(E(tx with trace A{E) - 0 and spectrum(A(E c iR for all ~ EM. Assume the conditions of section 2.2 (i) If there is a basis in which all A(E) ,EEM, are simultaneously

skew-symmetric then A = 0 . (ii) Let S be the semigroup generated by A(E) ,EEM,

S t 1A(E 1 ) tnA(En) {e ... e , all nEN,ti~o,EiEM} c Sl(d,R) .

If there is a matrix $ E S with spectrurn($) not on the unit circle Izl = 1 in ~, then A > 0 .

For d=2 an even stronger result holds. The condition in part (i) is necessary and sufficient for A = O. For y + f(E(ty = 0 all assumptions are satisfied provided f(E(t is not constant, so we always have A > 0 in this case.

Theorem 3 shows that noise typically tends to spread the spectrum, in other words acts as a destabilizing force.

3.3 Stabilization by Noise Given an unstable system x = Ax, i.e., with A(A) > O. Does there exist a parametric noise B(E(t with EB(E(t = 0 such that x = (A+B(E(t)x is stable, i.e., A (A+B) < O? We have the following necessary and sufficient criterion:

Theorem 4 (ARNOLD, CRAUEL and WIHSTUTZ [2]). (i) 1 d trace A ~ A(A+B). (ii) For every E > 0 there exists a B(E(t such that

1 1 d trace A ~ A(A+B) ~ d trace A + E (iii) In particular, x = Ax is stabilizable if and only if trace

A < 0

The proof is done by a perturbation argument of the type in section 3.1 for big E.

Examples. (i) Since the result carries over to nonlinear systems, one can stabilize Eigen's hypercycle (ARNOLD, CRAUEL and WIHSTUTZ [2]). (ii) The oscillator y + 2By + Y = 0 is stable for B > 0, but with A(B) = -B + ~2_1 ~ -1/2B ~ O(B~) With our method we can bring A close to trace A/2 = -B ,which is a drastic improvement of stability behavior if B is big.

3.4 Quantitative Results for the Oscillator For the system (3) A = A(B,a) ~ -B. The perturbation expansion of section 3.1 can be carried out explicitly for small and large a (ARNOLD, PAPANICOLAOU and WIHSTUTZ [4], WIHSTUTZ [18]). Here are sample results. Let (a,b) c R be the state space of E(t).

Theorem 5. (i) A(B,a) is analytic in a for IBI < 1 and a small, and

A(B,a) = - B + A2 (B)a 2 + 0(03 ) for a ~ 0 with A2(B) > 0

17

(ii) For 1131 > 1 A(f3,O) = - 13 + /(32_ 1 - A2 ((3)02 + 0(03 )

with A2 (f3) > 0 for 0 ~ 0

(iii) For 0 ~ ~, 13 E R and a < 0 (in particular, for EE 0) A(f3/0 ) = Yo Elf? + 0(1),

o where EJf= = J Mn(E)dE

a

For 0 ~ ~, 13 E R and a > 0 A(f3,O) = - 13 + AO + 0(1)

where AO>O depends only on E(t). Examples. (i) For Brownian motion in [a,b], a > 0 ,

reflection on the boundaries, AO = 1/12ab . (ii) For E(t) = F(T](t)), T](t) = Brownian motion on

_ 1 (FI (S))2 AO - 8 J ~ ds .

S1

References [1] L. Arnold: SIAM J. Appl. Math. 1984 (to appear)

with

[2] L. Arnold, H. Crauel and V. wihstutz: SIAM J. Control Optim. ~, 451 (1983)

[3] L. Ar.nold, W. Kliemann: "Qualitative Theory of Stochastic Systems", in: Probabilistic Analysis and Related Topics, Vol. 3 (Academic Press, New York 1983)

[4] L. Arnold, G. Papanicolaou and V. Wihstutz: Technical Report, Universitat Bremen (1984)

[5] A. Carverhill: PhD Thesis, University of Warwick (1983) [6] H. Furstenberg: Trans. Amer. Math. Soc. 108, 377 (1963) [7] Y. Guivarc'h: "Quelques proprietes asymptotiques des produits

des matrices aleatoires", in: Lecture Notes in Mathematics 774 (Springer-Verlag, Berlin, Heidelberg, New York 1980)

[8] N. Ikeda, S. Watanabe: Stochastic Differential ~uations and Diffusion Processes (North-Holland, Amsterdam 1981)

[9] Y. Kifer: Z. Wahrscheinlichkeitstheorie verw. Gebiete ~, 83 (1982 )

[10] W. Kliemann, L. Arnold: Technical Report 93, Universitat Bremen (1983)

[11] S. Kotani: Preprint Kyoto University (1983) [12] H. Kunz, B. Souillard: Comm. Math. Phys. 78, 201 (1980) [13] S. A. Mol~anov: Math. USSR Izvestija ~, 69 (1978) [14] V. I. Oseledec: Trans. Moscow Math. Soc. ~, 197 (1968) [15] Y. B. Pesin: Russian Math. Surveys ~, 55 (1977) [16] G. Royer: Ann. Inst. Henri Poincare (Section B) ~, 49 (1980) [17] A. D. Virtser: Theory of Probability and its Applic. 24,

367 (1979) [18] V. Wihstutz: Technical Report 99, Universitat Bremen (1983)

18

First Passage Times for Processes Governed by Master Equations

B.J. Matkowsky, Z. Schuss*, and C. Knessl Department of Engineering Sciences and Applied Mathematics, The Technological Institute, Northwestern University, Evanston, IL 60201, USA C. Tier Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60680, USA M. Mangel Department of Mathematics, University of California, Davis, CA 95616, USA

We calculate the activation rates of metastable states of processes governed by Master Equations, by calculating mean first passage times. We employ methods of singular perturbation theory to derive expressions for these rates, utilizing the full Kramers-Moyal expansions for the forward and backward operators in the Master Equation. In addition we discuss the validity of various diffusion approxi-mations to the Master Equation, showing that such approximations are not valid in general.

1. Introduction

In a series of papers [1-9], we have introduced a method to compute the noise-induced probability density of fluctuations about and mean first passage times from the domains of attraction of deterministically stable solutions of dynamical systems. We considered general state dependent (multiplicative) noise, and con-sidered stable equilibria, stable limit cycles, as well as more general stable solutions of dynamical systems. Our previous work was based on consideration of Langevin equations, or equivalently of the corresponding Kolmogorov forward (Fokker-Planck) and backward equations. In this paper we consider the same ques-tions for processes governed by Master Equations.

Transitions between metastable states of physical systems described by Ifarkov jump processes have been of continuing interest for many years [10-19]. The decay rates or mean lifetimes of these metastable states have attracted much attention because they represent important quantities such as dissociation rates or other activation rates. The calculation of these rates for Markov jump process has been based on the analysis of ~faster Equations. The general discrete time Markov jump process {xn } is described by

(1.1)

where ~n is a sequence of independent random variables, and E~n represents the jump size. The conditional jump density, which we assume to be stationary, is given by

* Permanent address: Department of Applied Mathematics, School of Mathematical Sciences, Tel-Aviv University, 69978 Tel-Aviv, Israel.

19

Pr(~ = zjx = x) = w(z,x) n n

(1. 2)

and the moments are given by

(k = 1,2,"') (1.3)

The small parameter E typically represents the ratio of the mean jump size to the 1

system size. For example,E = nwhere n -is the total number of states [12]. A metastable state for the process {xn } at x = 0 with domain of attraction [-A,B] exists if the drift

~ zw(z,x)dz (1.4)

satisfies ml(O) = 0 and

xml(x) < 0 for x~ (-A,B) A,B > 0 x'" 0 (1.5) Outside the interval [-A,B], the drift carries the process away from the metastable state x = O. Various types of boundaries may be considered. For example, (i) non-characteristic boundary points B,i.e.,ml(B) '" 0, (ii) characteristic bound-ary points, i.e.,ml(B) = O. We will consider problems with one boundary point which is absorbing (characteristic or non-characteristic), and the other which is either absorbing, or reflecting, or is partially absorbing (i.e., sticky). We refer to the latter case as type (iii). The transition density function p(x,y,n) = Pr{x(n) = yjx(O) = x} satisfies the forward Master Equation (ME)

p(x,y,n+l) - p(x,y,n) L;P - i [p (X,y-EZ ,n)w(z ,y-EZ) - P (x,y ,n)w(z ,y) ]dz (1.6)

The Kramers-Moyal expansion [20,21] of (1.6) is given by the forward Kramers-Moyal equation (FKME)

p(x,y,n+l) - p(x,y,n) * L p Y

(1. 7)

Constructing the solutions of (1.6) or (1.7) is, in general, difficult, so that approximate techniques have been developed. The method of approximating the jump process {xn } by a diffusion process has been widely used in the literature [12,20, 21]. In this approximation, the Master Equation (1.6) or the Kramers-Moyal equa-tion (1.7) is approximated by a Fokker-Planck equation. The standard method con-sists in truncating (1.7) after two terms, to obtain

(1.8)

Here the discrete time is replaced by the continuous time variable t En. This

20

procedure is useful for small deviations from the metastable state x = 0, but it has been shown to lead to erroneous results in many cases [12,13,17,22] (see Section 3). In fact, this approximation may lead to decay rates which differ by many orders of magnitude from those obtained from the Master Equation (1.6). Another method, proposed by VAN KAMPEN [12], employs the system size expansion (D expansion) which reduces the Master Equation (1.6) locally to a diffusion process of the Ornstein-Uhlenbeck type

(1. 9)

Here z = y/IE is defined locally near the metastable state and t is the continuous time variable described above. Van Kampen's method is useful only for describing small deviations from the metastable state x = O. Thus, neither approximation (1.8) nor (1.9) allows the determination of global properties such as the proba-bility of large fluctuations or the decay rates of metastable states.

A third method has recently been proposed, which is an important step in the analysis of large deviations [17]. This method is based on the WKB solution for the stationary density p(y) of (1.6) or (1.7) (cf. KUBO [13]):

p(y) -1 1 A exp {- ~ [~O + E~l + ... ]} (1.10)

where A is a normalization constant. An effective diffusion approximation is constructed by the Fokker-Planck equation

p = -{[ml(y) + E'(L'(y) - L(yNl'(y]p} + E(L(y)p) t y" where (1.11)

1 [ 00 ~+2(Y) k] L(y) = 2 m2 (y) + k~l (k + I)! (~O) (1.12)

It is shown, by means of a birth-death process example, that this diffusion approxi-mation gives the correct decay rate to leading order in E, for boundaries of type (ii). Unfortunately, this approximation does not apply directly to boundary con-ditions of type (iii) as in the Montroll-Shuler model. We also note that in order to obtain higher order terms in the expansion with respect to E, the drift and diffusion coefficients in (1.11) would have to be corrected by terms which are O(E2).

Our purpose is to present an asymptotic theory of large deviations using the full Master Equation. We calculate the decay rates or mean lifetimes of metastable states of both discrete and continuous time jump processes. We relate the decay rate to the first passage time for the process to escape from the domain of attrac-tion of the metastable state. Rather than analyzing the forward Master Equation (1.6) and computing the time-dependent fluxes, we introduce and analyze below the backward Master Equation and the backward Kramers-Moyal equation for the mean first passage time.

21

Let ii(x) be the first (random) time the process {xn } leaves the interval (-A,B), given that Xo = x. The mean first passage time or mean lifetime

B

L 1 p(y,x,n)dy n -A

satisfies .the backward Master Equation [23] (BME)

Ln:: 1 [n(x+z) - n(x)]w(z,x)dz - 1 subject to the condition

n(x) = 0 for x_ (-A,B)

x (-A,B)

*

(1.13)

(1.14)

(1.15)

The operator L in (1.14) is the formal adjoint of L which appears in the forward Master Equation (1.6). As before, we can replace (1.14) by its Kramers-Moyal expansion to obtain the backward Kramers-Moyal equation [20,21] (BKME)

00 k - ~ _(k) Ln = L -, ~ (x)n (x) = - 1

k=l k. 1< (1.16)

We analyze (1.16) by adapting the asymptotic method of MATKOWSKY and SCHUSS [i-9]. Our main results are the following explicit expressions for the mean lifetimes n(O) of metastable state x = O. For a boundary of Type (i)

where the moment generating function ~x(t) is defined by

the "eikonal" function 1jJ(x) is the solution of the first-order equation ~X(1jJI) = 1

and the "amplitude" functionK(x) is given by

K(x)

For type (ii)

11 ...; m2(0)1.~ -1jJ(x)!.'1 I I ;(O)~ -m'(O) L K(x)e V ml (x)m2 (x) 1 x=-A,B

22

(1.17)

(1.18)

(1.19)

(1. 20)

(1.21)

The above discussion was for discrete time Markov processes. Continuous time problems are also cast in the form of Kramers-Moyal equations, which are then solved asymptotically as in the case of discrete time problems.

Formulas (1.17) and (1.21) can also be derived from the approximation (1.11). Thus (1.11) is valid not only for the calculation of the leading terms of the probability of large deviations, and of the decay rates of metastable states, for the birth-death example with type (ii) boundary, as shown in Ref. 17, but for general jump processes of the form (1.1), with boundary conditions of types (i) and (ii). In section 3 we specialize our results to birth-death processes. We also present criteria for the validity of the standard diffusion approximation to the Kramers-Moyal equation (Master ~quation). We observe that the mean life-time is inversely proportional to the first positive eigenvalue of the transition matrix of the Master Equation. Thus our calculation of T yields this eigenvalue, which is exponentially small in E. This eigenvalue was not computed in References 13 and 14, where only the eigenvalues 0(1) and larger were computed. Applications of the analysis in this paper appear in our paper [26]. Finally we present results for higher dimensions. Details and applications will appear in a forth-coming paper [27].

2. Mean First Passage Times

We consider the process {xn } defined in (1.1)-(1.3) above, on the interval [-A,BJ. The mean first passage time n(x) = E(nlxo = x) satisfies the backward Master Equa-tion (B}m) [23] (1.14) and the boundary condition (1.15). Equation (1.14) is equivalent to the backward Kramers-Moyal equation (BD1E) (1.16), which is obtained from (1.14) by expanding about E = O. We construct an asymptotic expansion of the solution of (1.16)-(1.15), and consequently of (1.14)-(1.15), by adapting the method of MATKOWSKY and SCHUSS [1-9]. It is clear from (1.5) that n(x) + 00 as E + O. Thus we assume, for x bounded away from the boundary, that n(x) is of the form

n(x) - C(E)V(X)

where C(E) + 00 as E + 0, and

max vex) = 1 -A

v - v + E:V + E: 2V + ... 012

whose leading term satisfies

o

(2.5)

(2.6)

Hence by (2.2), vO(x) = 1. Since vO(x) does not satisfy the boundary condition (2.4), it is necessary to construct boundary layer corrections [24] near x = -A and x = B. In this section we conside,r boundaries of types (i) and (ii). Other cases can be treated similarly.

For type (i) boundaries, we introduce the stretched variable

(2.7)

into (2.3), and find that the leading term in the boundary layer expansion of the solution V = v(B - E:s) satisfies

L k=l

k (-1) ~(B) (k) k! V (1;) o (2.8)

The boundary and matching conditions are

V(O) = 0 and Vee

with mi(B) > O. We introduce the stretched variable

n = B - x (2.15) IE

into (2.3), and obtain the boundary layer equation

m2(B) 1/2 m1' (B)nV + -2-- V '" O(e: ) (2.16) n nn

and the boundary and matching conditions (2.9). The solution of (2.16) and (2.9) is given by

(- Tu{W') erf\V~) (2.17) The uniform expansion of v(x) is now given by

v (x) - erf --[ B - x c IE

mi (B) J [x + A m2(B) + erf IE Imi(-A) I ] _ 1 m2(-A) (2.18)

where the subscript c denotes the characteristic case (ii). To find the as yet undetermined constant C(e:) in (2.1), we multiply the BKME

(1.14) by the solution u(x) of the stationary foward Kramers-Moya1 equation (KME) [18,19]

k L*u = L (-~) [a (x)u(x)](k) 0

k=l k. k (2.19)

and integrate by parts to obtain B B

~ u(x)Ln(x) -A

00 k k-2 j (.) (k j 1) = L E, L (-1) [a (x)u(x)] J n - - (x)

k=2 k. j=O k _ C(e:)~ (u,V) -A

where we have used n = C(e:)V. Using (1.14), (2.1) and (2.20), we obtain B

- ~ u(x)dx

(2.20)

-A C(e:) - '/I: (u,V) (2.21) We now construct u(x) in the WKB form

_ 1jJ(x) u(x) - K(x)e e: (1 + O(e: (2.22)

Employing (2.22) in (2.19), we find that 1jJ(x) satisfies the "eikona1" equation [13]

25

~ (ezlji ' (x) _ l)w(z,x)dz 0 or (2.23)

where Vnc(x) is given by (2.12). For type (ii) boundaries we employ (2.18) in (2.20) to obtain

ia. (u, V) I x=B - (2.31) Now, using (2.28), (2.31) and (2.21) we obtain the exit time for type (ii) bound-aries as

where Vc(x) is .given by (2.18). Note that if w(z,x) depends on e: as

w(z,x) - wO(z,x) + e:wl(z,x) + ...

(2.32)

the above analysis remains unchanged if w(z,x) is replaced by wO(z,x) and K(x) is replaced by

(2.33)

where K(x) is given by (2.27) with w(z,x) replaced by wO(z,x).

Next, we consider the continuous time jump process x(t) defined by the stochas-tic equation

{ x(t) + b(x(t~t + o(~t)

x(t + ~t) = x(t) + b(x(t8t + e:1;

w.p. 1 - A(X(t ~t + o(~t) e;

(2.34) w.p. A(X(t ~t + o(~t)

e:

where the conditional density of I; is given by

Pr(1; = zlx(t) = x) = w(z,x) (2.35) We assume b(O) = 0 and b'(O) < 0 so that x = 0 is a stable equilibrium of the averaged equation. The moments of I; are denoted by

- kl C k~ ~(x) = E(I; x(t) = x) = } z w(z,x)dz (2.36)

and are assumed to be independent of t. We assume, without loss of generality, that ml(x) = 0, since b(x) and A(x)ml (x) can be combined (cf. (2.38) below). Let , be the first time that the process x(t) hits the boundary of the interval (-A,B). The mean first passage time ;(x) = E(,lx(O) = x) satisfies the BME

b(x)-:t' (x) + At) ~ (r(x + e:z) - -:t(x)]w(z,x)dz = - 1 (2.37) The BKME is given by

n iT (x) = b (x) T' (x) + A (x) L ;- m (x) -:t(n) (x)

e:. n=2 n. n - 1 (2.38)

27

for x (-A,B), T(x) = 0 for xl (-A,B)

The eikona1 equation for *p 0, where E* is the adjoint of L in (2.38), is given by

b(x)~'(x) + \ (eZ~'(x) - l)w(z,x)dz 0

and the transport equation is given by

(b (xl xl l' + L ~ w(', xl< (xl I x + ~i' ,xl," (xl< (Xl} """ (xl d, a F01::::n: t analysis for the discrete time case, we obtain 1IA (x)m2 (0) j, -W( ) It

E(-b'(O X~_A,BK(X)e x E~~(~'(X)~Vnc(X)[l + O(E)] where

1 + b(x)t + A(X) I ~(x)tk/k! k=2

(2.39)

(2.40)

(2.41)

(2.42)

(2.43)

and Vnc(x) is defined by (2.12). The characteristic boundary case can be treated by introducing the obvious modifications in (2.33).

If the jump rate is given by A(x)/E2, rather than A(x)/E as above, the BKME takes the form

- 1 -b(x)'t'(x) + 2" A(x)m2(x)'t"(x) + O(E) = - 1 (2.44)

Here we have assumed that m1 (x) = O. We seek an asymptotic solution in the form

't - 'to + E't1 + ... (2.45)

where TO satisfies [23]

- - 1 - -LO'tO = b(x)'tO(x) + 2" A(x)m2(x)'tO(x) TO(x) = 0 for xl (-A,B)

The functions T (x) for n > 1 satisfy n

- 1 xE (-A,B) (2.46)

(2.47)

(2.48)

Note that LO is the backward operator for the standard diffusion approximation [12]. We observe that equation (2.46) is not of singular perturbation type, as is the case if the jump rate is A(x)/E. Thus the standard diffusion approximation is valid in this case.

28

3. A Random Walk with Absorbing Boundaries

As a simple illustrative example we consider the birth-death process defined by

(3.1)

where 8 is a small parameter, and

(3.2)

pr(~n = olxn = x) = 1 - rex) - ~(x)

This corresponds to

w(z,x) = r(x)o(z - 1) + ~(x)o(z + 1) + [1 - rex) - ~(x)]o(z) (3.3)

in (1.2). We assume that ~(x) > rex) for 0 < x < B, and ~(x) < rex) for -A < x < 0, ~(O) = reO) # 0, so that x = 0 is a stable equilibrium point. The corresponding characteristic and non-characteristic cases are ~(B) = reB),

~(-A) = r(-A) , and ~(B) > reB), ~(-A) < r(-A) , respectively. The BME (1.14) becomes

r(x)n(x + 8) + ~(x)n(x - ) - (r(x) + ~(x)n(x= - 1 with boundary conditions

n(x) = 0 for x, (-A,B) The BKME for (3.4) is now

00 k Ln(x) = L ~, [rex) + (_l)k~(x)]n(k)(x)

k=l . - 1

(3.4)

(3.5)

(3.6)

Following the analysis of section 2, we seek the solution of the stationary (for-ward) KME (2.19), which in this case becomes

co k * \' L u(x) = L ,

k=l k. { k 1 (k) [~(x) + (-1) r(x)]u(x) o

in the WKB form J.

-'Ji (x) IE u(x) = K(x)e (1 + 0(

Here the eikonal equation (2.23) for 'Ji(x) reduces to

~x('Ji'(x = r(x)e'Ji'(x) + ~(x)e-'Ji'(x) + (1 - rex) - ~(x Thus,

'Ji(x) x

( log ~(s) ds l res) o

The transport equation (2.26) reduces to

(3.7)

(3.8)

1 (3.9)

(3.10)

29

[r(x)elji'(x) -9,(x)e-lji'(x)]K'(x) + [r'(x)elji'(X) -9,'(x)e-lji'(x) 1 K O. + lji"~X)_ (r(x)elji'(x) + 9,(x)e-lji'(x)~

(3.11) Employing (3.10) in (3.11), we find that

K(x) Cl (3.12)

Ir(x) 9, (x) where Cl is a normalization constant.

In the non-characteristic boundary case, the mean exit time formula (2.31) becomes

2'lT n(x)

sr(O)[9,' (0) - r' (0)]

where

with 9,(B)

B = log reB) a = r(-A) log 9,(-A)

Vnc(x) [1 + O(s)] (3.13) L ..L19,,,;:(=:x)=-=r=( x=)--.LI e -lji (x) Is

x=-A,B Ir(x) (x)

(3.14 )

In the characteristic boundary case, the mean exit time formula (2.33) becomes

n(x) s/r(O)[9,'(O) - r'(O)] L e-1jJ(x)Is_ 119,' (-x) - r' (x) 1 V rex)

x=-A,B

(3.15)

where

V c (x) = erf [ B ~ x r'(B) - 9,'(B)] 2r(B) f [ A + x_I 9,' (-A) - r' (-A)] _ 1 + er IE V 2r(-A) (3.16)

We observe that our results (3.13) and (3.15) agree with the asymptotic expan-sion of the exact solution for types (i) and (ii) boundaries respectively. In addition our result (3.15) agrees with the result in Ref. 17, where the char-acteristic boundary case was treated.

We now compare our results for the mean exit time n(x) with the results obtained by the standard diffusion approximation to the random walk defined by (3.1) and (3.2). The backward operator for the standard diffusion operator is usually obtained from the B~1E (3.6) by truncating the infinite series after two terms, to obtain

s LOu(x) = [rex) - 9,(x)]u' +"2 [rex) + 9,(x)]u" (3.17)

In addition the discrete time is replaced by the continuous time variable t = sn. We note that the backward operator for the diffusion approximation is also obtained from the B~1E by introducing the scaling ~ = x/IE. in which case the

30

Ornstein-Uhlenbeck approximation results. That is, one obtains (3.17) with rex) and (x) linearized about x = O. However, the linear coefficients are then usually replaced by general rex) and (x) so that (3.17) results. The mean exit time T(X) for the diffusion approximation to the process to exit the interval (-A,B), satisfies

T(-A) T(B) = 0

The solution of (3.18) for small E then yields

2'lT T(X) n(x) = -- =

E Er(O)[ 1 (0) - r' (0) 1 2

where x

~(x) 2 ~ (s) - res) ds (x) + res) 0

and 2((B) - reB)) (B - x)

vex) = 1 - e (B) + r (B) E e

vex) \ -$(X)/E I(x) - rex) I L e (x) + rex)

x=-A,B

2(r(-A) - (-A)) (x + A) r(-A) + (-A) E

for case (i). A similar analysis can be carried out for case (ii).

(3.18)

(3.19)

(3.20)

(3.21)

The function ~(x) is the solution of the eikonal equation for the stationary Fokker-Planck (forward) equation

* where LO is the formal adjoint of LO' and

-"'IE p = Ke 'I'

That is, ~ satisfies

r (x) + (x) (~' (x)) 2 + (r (x) _ (x))~ 1 (x) 2

(3.22)

(3.23)

o 0.7.4)

Clearly (3.20) and (3.10) are, in general, not equal. Indeed we observe that 1jJ(x) > ~(x) for x # O. Thus the density of fluctuations predicted by the diffusion approximation has higher tails than the density of the random walk. That is, the probability of large deviations of the diffusion approximation, from equilibrium, is greater than that of the underlying random walk, and consequently the mean first passage time for the diffusion approximation is shorter than that of the random walk. Equation (3.24) can be obtained from (3.9) by expanding the exponentials in powers of 1jJ', and truncating terms higher than quadratic. Thus this diffusion approximation is valid only for x such that 1jJ'(X) is small, which occurs near the equilibrium point x = O. For small deviations from equilibrium, either the standard diffusion approximation or the Ornstein-Uhlenbeck approxi-

31

mation can be used. However, for large deviations from equilibrium, neither can be used. In particular the Fokker-Planck equation (3.22) cannot be used to des-cribe large fluctuations about equilibrium, in this random walk. In fact, the approximating diffusion process cannot be used to calculate the first passage times for deviations of order IE or larger.

4. More General Boundary Behavior

In this section we consider problems with reflecting boundaries and with type (iii) boundaries. For the latter, the method described above may have to be somewhat modified. First we consider a problem with a reflecting boundary. Thus we consider the process (1.1) on the interval (-A,B) with a reflecting boundary at x = -A, and an absorbing boundary at x = B, so that ;(x) is the mean first passage time to reach x = B. Therefore, ;(x) satisfies (1.14), or equivalently (1.16), subject to the boundary conditions

~(-~ 0 ;(x) = 0 x > B (4.1)

The mean exit time ;(x) is determined as in Section 2, except that there is no longer a boundary layer near x = -A, and the contribution from x = -A to the iden-tity (2.20) is asymptotically negligible compared to the contribution of the left hand side. Thus reflecting boundaries do not contribute to the leading term in the asymptotic expansion of the mean exit time.

Next we consider a type (iii) boundary, which is partially absorbing and par-tially reflecting (i.e., sticky). The process, if it reaches such a boundary point, say x = -A, can stay there for a random time, and then jump into the interval with given jump density w(z,-A). Thus w(z,-A) = 0 for z < 0, and ml(O) > O. For many physical examples A = 0, so that the attracting point is also the sticky boundary point. Thus we consider the process (1.1) on the interval (O,B) with a sticky boundary at x = 0, and an absorbing boundary at x = B. Then ;(x) satisfies (1.14) subject to the boundary conditions

\ [;(EZ) - ;(O)]w(z,O)dz - 1

;(x) o x > B The first of these conditions is merely (1.14) evaluated at x = O. We assume that m1 (x) is a regular function of E, for small x, i.e.,

(4.2)

(4.3) We consider two cases. The regular case, in which mll(O) > 0, and the singular case in which mll(O) = O. In addition, in Ref. 27 we also consider the case when ml(x) is not a regular function of E, e.g.,ml(x) contains terms such as

-X/E e

32

We can show that in the regular case, the contribution of the first of the boundary conditions (4.2) is negligible, as in the case of a reflecting boundary. To see this, we construct a boundary layer expansion near x 0, and find that the contribution to (2.20) is again asymptotically negligible. This type of boundary arises in applications such as the Ising-Weiss model of ferromagnetism[26).

If mIl (0) = 0 as in the singular case, the contribution from x = 0 to (2.20) is no longer negligible, and requires careful analysis. This is illustrated by the Montroll-Shuler model of dissociation of a gaseous diatomic molecule immersed in an inert gas. In appropriately scaled variables, the problem is described as a birth-death process, on the interval (0,1) with

rex) ~(x) KX (4.4)

where K, 8, and are given constants. The details of this analysis may be found in our paper [26). The boundary conditions are given by

~(l) = 0 ~(O) = neE) + e8 /K

We observe that

K(X + )e-8 - KX

so that

(4.5)

(4.6)

(4.7)

Thus mll(O) dition at x

o and we are in the singular case. We refer to the boundary con-o as a singular boundary condition in the sense that all the

derivatives of ~(x) at x = 0 contribute to the identity (2.20). Therefore we modify our method by constructing a boundary layer solution of the backward Master Equation, rather than of the corresponding Kramers-Moyal equation. We then match this boundary layer solution to the outer solution of the Kramers-Moyal equation, valid in the interior of (0,1). Following this procedure leads to the result

8/E ~(x) - ~ e -8 2 (1 - e-(1-x)8/) K (1 - e )

(4.8)

which agrees with the result of Montroll-Shuler.

5. A Two-Dimensional Process

We consider the process !n defined on a two-dimensional region D by

!n+l = !n + ~ (5.1)

where the conditional jump density of the process ~ is given by Pr(~ = zlx = x) = w(~,~)

-n --n -(5.2)

33

and its moments are denoted by

=\\ (5.3) We assume that the moments are independent of n and of E, and that the averaged dynamical system

(5.4)

has a unique stable equilibrium point in D, located at ~ = 0 say. For simplicity we consider the boundary aD to be noncharacteristic, i.e.,

on aD,. where (v 1 ,v2 ) is the unit outer normal to aD. Characteristic boundaries can be similarly treated. To comDute the mean first passage time n(~) from a point x in D to:lD of the process (5.1) - (5.3), we follow the method in section 2. The result is found to be

1/2 ~ - I -1 1 ~/E \ n - 2'IT H e L (5.6)

where ~ is the absolute minimum on :lD, of the eikona1 function ~, and s are the points on aD where the minimum is achieved. The function~ is a solution of the eikonal equation

o (5.7)

The Hessian matrix H of ~ at x = 0 may be determined from the solution X of the matrix Ricatti equation

MX + XMT = - A

by H = -1 X . Here the matrices M and A are

amlO (Q) :lm10

aXI :lx2 M =

:lm01 (.2)

:lm01 dX2

and

C'o(Q) '11 (Q)) A

m11 (Q) m02 (Q)

Finally the function R(s) is given by

[ (av1 ) i~j ~i _ a R(s) = EK(s) L y. + x

i+j>l a(~ y - av2) ~ - a x 34

given by

i v~~;] mij ., ., l.J.

(5.8)

(5.9)

(5.10)

(5.11)

where s denotes arc length on aD, the function K(~) is a solution of the transport equation

I J "" " . ' [ V ~ [w(~,~)K(~)) (5.12) the function K(s) denotes K(~) on aD, and the constant a is the positive root of the equation

\' mij (s) i j i+j L .,., v l (s)v2 (s)a i+j~l l..J.

o (5.13)

The details of the analysis leading to (5.6) are given in our paper [27], where applications are also considered.

6. Acknowledgements

This work was partially supported by AFOSR Grant 83-0086, DOE Grant DE-AC02-78-ERO-4650, and NSF Grants MCS-83-00562 and MCS-8l-2l659.

7. References

l. 2. 3. 4. 5.

B. Z. B. B. B.

J. Matkowsky and Schuss and B. J. J. Matkowsky and J. Matkowsky and J. Matkowsky, Z.

835 (1982).

Z. Schuss, Matkowsky, Z. Schuss, Z. Schuss, Schuss and

SIAM J. Appl. Math. 33(12), 365 (1977) . SIAl! J. Appl. Math. 35(3), 604 (1979). SIAM J. Appl. Math. 40(2), 242 (1981). SIAM J. Appl. Hath. 42(4), 822 (1982). E. Ben-Jacob, SIAM J. Appl. Math. ~(4) ,

6. B. J. Matkowsky, Z. Schuss and C. Tier, SIAM J. Appl. Math. 43(4), 673 (1983). 7. E. Ben-Jacob, D. Bergman, B. J. Matkowsky and Z. Schuss, rhy~ Rev. A26,

2805 (1982). 8. B. J. Matkowsky and Z. Schuss, Phys. Letters 95A(5), 673 (1983). 9. Z. Schuss, "Theory and Applications of Stochastic Differential EqU1itions,"

Wiley, New York (1980). 10. R. B. Griffiths, C. Weng and J. S. Langer, Phys. Rev. A 149(1), 301 (1966). 11. E. W. Montroll and K. E. Shuler, "The Application of theTheory of Stochastic

Processes to Chemical Kinetics", in Advances in Chemical Physics 1, I. Prigogine, ed., Interscience, New York (1958), p. 361. -

12. N. G. Van Kampen, "Stochastic Processes in Physics and Chemistry", North Holland (1981).

13. R. Kubo, K. Matsuo and K. Kitahara, J. Stat. Phys. 1(1), 51 (1973). 14. Th. W. Ruijgrok and J. A. Tjon, Physica 65, 539 (1973). 15. L. H. Liyanage, C. M. Gulati and J. M. Hill, "A Bibliography on Applications

of Random Walks in Theoretical Chemistry and Physics", in Advances in Molecular Relaxation and Interaction Processes, Elsevier, Amsterdam, pp. 53-72 (1982).

16. B. Carmeli and A. Nitzan, J. Chem. Phys. ~, 5321 (1982). 17. P. Hanggi, H. Grabert, H. Talkner and H. Thomas, Phys. Rev. A29, 371 (1984).

See also P. Hanggi and H. Thomas, Phys. Reports 88, 207-cI982) and H. Grabert, P. Hanggi and I. Oppenheim, Physica l17A, 300 (1983).

18. R. Landauer, to appear in Self-Organizing Systems: The Emergence of Order, F. E. Yates, D. O. Walter and G. B. Yates, eds., Plenum Press, New York (1984).

19. J. Troe, J. Chem. Phys. 66, 4745 (1977) . 20. R. A. Kramers, Physica 2., 284 (1940). 2l. J. E. Moyal, J. Roy. Stat is. Soc. (B) 11, 150 (1949). 22. H. J. Kushner, Stochastics 6, 117 (1982).

35

23. E. B. Dynkin, "Markov Processes", two volumes (translated from Russian) Springer - Berlin (1963).

24. R. E. O'Malley, Jr., "Introduction to Singular Perturbations", Academic Press, New York (1974).

25. C. M. Bender and S. A. Orszag, "Advanced Mathematical Hethods for Scientists and Engineers, McGraw Hill, New York, Ch. 6 (1978).

26. B. J. Matkowsky, Z. Schuss, C. Knessl, C. Tier and M. Hangel, to appear, Phys. Rev. A (1984).

27. B. J. Matkowsky, Z. Schuss, C. Knessl and C. Tier, to appear, SIAM J. Appl. Math. (1984).

36

Part II Pattern Formation and Selection

Three Caveats for Linear Stability Theory: Rayleigh-Benard Convection

H .S. Greenside Princeton Plasma Physics Laboratory, P.O. Box 451, Princeton, NJ 08544, USA

Recent theories and experiments challenge the applicability of linear stability theory near the onset of buoyancy-dri ven (Rayleigh-B~nard) convection. This stabili ty theory, based on small perturbations of infinite parallel rolls, is found to miss several important features of the convective flow. The reason is that the lateral boundaries have a profound influence on the possible wavenumbers and flow patterns even for the largest cells studied. Also, the nonlinear growth of incoherent unstable modes distorts the rolls, leading to a spatially disordered and sometimes temporally nonperiodic flow. Finally, the relation of the skewed varicose instability to the onset of turbulence (nonperiodic time dependence) is examined. Linear stabili ty theory may not suffice to predict the onset of time dependence in large cells close to threshold.

1 Introduction

Instability is an essential feature of driven dissipative systems as the driving is increased. Examples of increasing complexity are a buckling beam, a convecting fluid, and a fusion plasma, where the driving is an increasing force, a temperature gradient, and an ohmic current respectively. When the driving and dissipation are no longer in balance, small fluctuations can grow. What these fluctuations grow into raise several fundamental issues that lie at the forefront of current efforts to understand nonlinear nonequilibrium systems.

These issues are somewhat different depending on whether fluctuations lead to a stationary state or to a time-dependent but statistically steady state. When an unstable system evolves into a stationary state, one would like to know on what time and length scales this evolution occurs. Often the physical system is large in some sense and one would expect a significant degeneracy of possible stationary states. Calculations and experiments show that this intuition is incorrect. Boundary conditions and nonlinear terms sometimes allow only a small

nl~ber of states to be attained, so that a linear analysis can be misleading.

Similarly, when fluctuations grow and a system becomes time dependent, one would like to understand the relation between the spatial structure and the temporal evolution. That there is a relation is evident since there are often long-wavelength instabilities that are sensitive to the cell geometry and to the spatial patterns. These instabilities may not be intrinsically time dependent; the imaginary part of the fastest growing eigenvalue could be zero. A temporally nonperiodic evolution could then arise only from the interaction of incoherent spatial modes. This interaction is most interesting when the growing fluctuations no longer have small amplitude so that a nonlinear analysis is needed. A linear stability theory may then be unsuccessful in predicting the onset of time dependence. It is not yet known how to generalize the linear theory to handle this difficult nonlinear regime.

In the following, we the onset of turbulence has li ttle to do wi th

will discuss these issues of pattern formation and of in the context of convecting fluids [1,2). This choice the practical and widespread applications of thermal

convection to engineering, meteorology, astrophysics, and geology. Instead,

38

Rayleigh-Benard convection comes closest to sat~sfying t