Preface Writing this book, I issued from the course that I gave to scientific associates at the Institute of Calculus Mathematics, Russian Academy of Sciences. In the book, I use the functional approach to uniformly formulate general methods of statistical description and analysis of dynamic systems described in terms of different types of equations with fluctuating parameters, such as ordinary differential equations, partial differential equa tions, boundary-value problems, and integral equations. Asymptotic methods of analyzing stochastic dynamic systems — the delta-correlated random process (field) approximation and the diffusion approximation — are also considered. General ideas are illustrated by the examples of coherent phenomena in stochastic dynamic systems, such as clustering of particles and passive tracer in random velocity field and dynamic localization of plane waves in randomly layered media. The book consists of three parts. The first part may be viewed as an introductory text. It takes up a few typical physical problems to discuss their solutions obtained under random perturbations of parameters affecting the system behavior. More detailed formulations of these problems and relevant statistical analysis may be found in other parts of the book. The second part is devoted to the general theory of statistical analysis of dynamic systems with fluctuating parameters described by differential and integral equations. This theory is illustrated by analyzing specific dynamic systems. In addition, this part considers asymptotic methods of dynamic system statistical analysis, such as the delta-correlated random process (field) approximation and the diffusion approximation. The third part deals with analysis of specific physical problems associated with coherent phenomena. These are clustering and diffusion of particles and passive tracer in a random velocity field, dynamic localization of plane waves propagating in layered random media. These phenomena are described by ordinary differential equations and partial differential equations. Each chapter is appended with problems the reader to solve by himself (herself), which will be a good training for independent investigations. The book is intended primarily for scientific workers; however, it may be useful also for senior and postgraduate students specialized in mathematics and physics and dealing with stochastic dynamic systems. Valery I. Klyatskin Introduction Different areas of physics pose statistical problems in ever-greater numbers. Apart from issues traditionally obtained in statistical physics, many applications call for including fluctuation effects into consideration. While fluctuations may stem from different sources (such as thermal noise, instability, and turbulence), methods used to treat them are very similar. In many cases, the statistical nature of fluctuations may be deemed known (either from physical considerations or from problem formulation) and the physical processes may be modeled by differential, integro-differential or integral equations. We will consider a statistical theory of dynamic and wave systems with fluctuating parameters. These systems can be described by ordinary differential equations, partial differential equations, integro-differential equations and integral equations. A popular way to solve such systems is by obtaining a closed system of equations for statistical characteristics of such systems to study their solutions as comprehensively as possible. We note that often wave problems are boundary-value problems. When this is the case, one may resort to the imbedding method to reformulate the equations at hand to initial value problems, thus considerably simplifying the statistical analysis [1], [2]. The purpose of this book is to demonstrate how different physical problems described by stochastic equations may be solved on the base of a general approach. In stochastic problems with fluctuating parameters, the variables are functions. It would be natural therefore to resort to functional methods for their analysis. We will use a functional method devised by Novikov [3] for Gaussian fluctuations of parameters in a turbulence theory and developed by the author of this book [1], [4] [6] for the general case of dynamic systems and fluctuating parameters of arbitrary nature. However, only a few dynamic systems lend themselves to analysis yielding solutions in a general form. It proved to be more efficient to use an asymptotic method where the statistical characteristics of dynamic problem solutions are expanded in powers of a small parameter which is essentially a ratio of the random impact's correlation time to the time of observation or to other characteristic time scale of the problem (in some cases, these may be spatial rather than temporal scales). This method is essentially a generalization of the theory of Brownian motion. It is termed the delta-correlated random process (field) approximation. For dynamic systems described by ordinary differential stoc;hastic equations with Gaus sian fluctuations of parameters, this method leads to a Markovian problem solving model, and the respective equation for transition probability density has the form of the Fokker- Planck equation. In this book, we will consider in depth the methods of analysis available for this equation and its boundary conditions. We will analyze solutions and validity con ditions by way of integral transformations. In more complicated problems described by partial differential equations, this method leads to a generalized equation of Fokker-Planck type in which variables are the derivatives of the solution's characteristic functional. For Introduction dynamic problems with non-Gaussian fluctuations of parameters, this method also yields Markovian type solutions. Under the circumstances, the probability density of respective dynamic stochastic equations satisfies a closed operator equation. In physical investigations, Fokker-Planck and similar equations are usually set up from rule of thumb considerations, and dynamic equations are invoked only to calculate the coefficients of these equations. This approach is inconsistent, generally speaking. Indeed, the statistical problem is completely defined by dynamic equations and assumptions on the statistics of random impacts. For example, the Fokker-Planck equation must be a logical sequence of the dynamic equations and some assumptions on the character of random impacts. It is clear that not all problems lend themselves for reducing to a Fokker-Planck equation. The functional approach allows one to derive a Fokker Planck equation from the problem's dynamic equation along with its applicability conditions. For a certain class of random processes (Markovian telegrapher's processes, Gaussian Markovian process and the like), the developed functional approach also yields closed equations for the solution probability density with allowance for a finite correlation time of random interactions. For processes with Gaussian fluctuations of parameters, one may construct a better physical approximation than the delta-correlated random process (field) approximation, — the diffusion approximation that allows for finiteness of correlation time radius. In this approximation, the solution is Markovian and its applicability condition has transparent physical meaning, namely, the statistical effects should be small within the correlation time of fluctuating parameters. This book treats these issues in depth from a general standpoint and for some specific physical applications. In recent time, the interest of both theoreticians and experimenters has been attracted to relation of the behavior of average statistical characteristics of a problem solution with the behavior of the solution in certain happenings (realizations). This is especially im portant for geophysical problems related to the atmosphere and ocean where, generally speaking, a respective averaging ensemble is absent and experimenters, as a rule, have to do with individual observations. Seeking solutions to dynamic problems for these specific realizations of medium pa rameters is almost hopeless due to extreme mathematical complexity of these problems. At the same time, researchers are interested in main characteristics of these phenomena without much need to know specific details. Therefore, the idea to use a well developed approach to random processes and fields based on ensemble averages rather than separate observations proved to be very fruitful. By way of example, almost all physical problems of atmosphere and ocean to some extent are treated by statistical analysis. Randomness in medium parameters gives rise to a stochastic behavior of physical fields. Individual samples of scalar two-dimensional fields p(R, t), R = {x^y)^ say, recall a rough mountainous terrain with randomly scattered peaks, troughs, ridges and saddles. Common methods of statistical averaging (computing mean-type averages — (p(R, t)), space-time correlation function ^ (p (R, t)/? (R^, t^)} etc., where (...) implies averaging over an en semble of random parameter samples) smooth the qualitative features of specific samples. Frequently, these statistical characteristics have nothing in common with the behavior of specific samples, and at first glance may even seem to be at variance with them. For exam ple, the statistical averaging over all observations makes the field of average concentration of a passive tracer in a random velocity field ever more smooth, whereas each its realiza tion sample tends to be more irregular in space due to mixture of areas with substantially different concentrations. Introduction Thus, these types of statistical average usually characterize 'global' space-time dimen sions of the area with stochastic processes but tell no details about the process behavior inside the area. For this case, details heavily depend on the velocity field pattern, specifi cally, on whether it is divergent or solenoidal. Thus, the first case will show with the total probability that dusters will be formed, i.e. compact areas of enhanced concentration of tracer surrounded by vast areas of low-concentration tracer. In the circumstances, all sta tistical moments of the distance between the particles will grow with time exponentially; that is, on average, a statistical recession of particles will take place [7]. In a similar way, in case of waves propagating in random media, an exponential spread of the rays will take place on average; but simultaneously, with the total probability, caustics will form at finite distances. One more example to illustrate this point is the dynamic localization of plane waves in layered randomly inhomogeneous media. In this phenomenon, the wave field intensity exponentially decays inward the medium with the probability equal to unity when the wave is incident on the half-space of such a medium, while all statistical moments increase exponentially with distance from the boundary of the medium [1, 8]. These physical processes and phenomena occurring with the probability equal to unity will be referred to as coherent processes and phenomena [9]. This type of statistical co herence may be viewed as some organization of the complex dynamic system, and re trieval of its statistically stable characteristics is similar to the concept of coherence as self-organization of multicomponent systems that evolve from the random interactions of their elements [10). In the general case, it is rather difficult to say whether or not the phenomenon occurs with the probability equal to unity. However, for a number of applica tions amenable to treatment with the simple models of fluctuating parameters, this may be handled by analytical means. In other cases, one may verify this by performing numerical modeling experiments or analyzing experimental findings. The complete statistic (say, the whole body of all n-point space-time moment func tions), would undoubtedly contain all the information about the investigated dynamic system. In practice, however, one may succeed only in studying the simplest statistical characteristics associated mainly with simultaneous and one-point probability distribu tions. It would be reasonable to ask how with these statistics on hand one would look into the quantitative and qualitative behavior of some system happenings? This question is answered by methods of statistical topography. These methods were highlighted by [11], who seems to had coined this term. Statistical topography yields a difi"erent philosophy of statistical analysis of dynamic stochastic systems, which may prove useful for experimenters planning a statistical processing of experimental data. These issues are treated in depths in this book. More details about the material of this book and more exhaustive references can be found in mentioned textbooks [1], [4] [6], recent reviews [2, 9, 12, 13], and recently pub lished textbook [14]. Chapter 1 Examples, basic problems, peculiar features of solutions In this chapter, we consider several dynamic systems described by differential equations of different types and discuss the features in the behaviors of solutions to these equations under random disturbances of parameters. Here, we content ourselves with the problems in the simplest formulation. More complete formulations will be discussed below in the sections dealing with statistical analysis of corresponding systems. 1.1 Ordinary difFerential equations: initial value problems 1.1.1 Particles under the random velocity field In the simplest case, a particle under the random velocity field is described by the system of ordinary differential equations of the first order ^rW=U(r,i), r(<o)=ro, (1.1) where U(r, ^) = uo(r,f) + u(r,^), uo(r, ^) is the deterministic component of the velocity field (mean flow), and u(r, i) is the random component. In the general case, field u(r,t) can have both divergence-free (solenoidal, for which divu(r,t) = 0) and divergent (for which div u(r, t) ^ 0) components. We dwell on stochastic features of the solution to problem (1.1) for a system of particles in the absence of mean flow (uo(r, ^) = 0). Prom Eq. (1.1) formally follows that every particle moves independently of other particles. However, if random field u(r,t) has a finite spatial correlation radius /(;or, particles spaced by a distance shorter than /cor appear in the common zone of infection of random field u(r, t), and the behavior of such a system can show new collective features. For steady-state velocity field u(r, ^) = u(r), Eq. (1.1) reduces to |r(t) = u(r), r(0)=ro. (1.2) This equation clearly shows that stationary points r (at which u(f) = 0) remain the fixed points. Depending on whether these points are stable or unstable, they will attract or repel nearby particles. In view of randomness of function u(r), points r are random too. 10 1.1. Ordinary differential equations: initial value problems 11 It is expected that the similar behavior will also be characteristic of the general case of the space-time random field of velocities u{r,t). If some points r remain stable during sufficiently long time, then clusters of particles (i.e., compact regions with enhanced particle concentration, which occur merely in rarefied zones) must arise around these points in separate realizations of random field u(r, t). On the contrary, if the stability of these points alternates with instability sufficiently rapidly and particles have no time for significant rearrangement, no clusters of particles will occur. Simulations [15, 16] show that the behavior of a system of particles essentially de pends on whether the random field of velocities is divergence-free or divergent. By way of example, Fig. 1.1a shows a schematic of evolution of the two-dimensional system of par ticles uniformly distributed in the circle for a particular realization of the divergence-free steady-state field u(r). Here, we use the dimensionless time related to statistical parameters of field u(r). In this case, the area of surface patch within the contour remains intact and particles relatively uniformly fill the region within the deformed contour. The only feature consists in the fractal-type irregularity of the deformed contour. On the contrary, in the case of the divergent velocity field u(r), particles uniformly distributed in the square at the initial instance will form clusters during the temporal evolution. Results simulated for this case are shown in Fig. 1.16. We emphasize that the formation of clusters is purely kinematic effect. This feature of particle dynamics disappears on averaging over an ensemble of realizations of random velocity field . To demonstrate the process of particle clustering, we consider the simplest problem [13], in which the random velocity field u(r,t) has the form u(r,i) = v(t)/(r), (1.3) where v(t) is the random vector process and the deterministic function /(r)=sin2(kr) (1.4) is a function of one variable. Note that this form of function /(r) corresponds to the first term of the expansion in harmonic components and is commonly used in numerical simulations. In this case, Eq. (1.1) can be written in the form -r(t) = v(t) sin 2(kr), r(0) = TQ. In the context of this model, motions of a particle along vector k and in the plane perpen dicular to vector k are independent and can be separated. If we direct the x-axis along vector k, then the equations assume the form —x{t) = Va:{t)sm{2kx), x(0)=a;o, ^R(t) = vn{t)sm{2kx), R(0) = RQ. (1.5) The solution of the first equation in (1.5) is x{t) = - arctan [e'^(*Han(A:xo)] , (1.6) 12 Chapter 1. Examples, basic problems, peculiar features of solutions / ^ "••*• •{.'^••' ".^v , >^i^, >-""r ' "^ '>>,,^ •> * ,/,.-f*^ •.* •^^^"' t = 0 ^ = 0.5 4 •-• r: A '-v~ . • • ,••••-• if /• •• • If . "" - i MV X v.. t = 1 ^ = 2.0 Figure 1.1: Diffusion of a system of particles described by Eqs. (1.2) numerically simulated for (a) solenoidal and (6) divergence-free random velocity field u(r). 1.1. Ordinary differential equations: initial value problems 13 where T{t) = 2kj dTV^ir). (1.7) and we can write the second equation in (1.5) in the form ''-R(i|ro) = sin(2fca;o) ^^^^^ As a result, we have t R(t|ro) = Ro + sin(2/cxo) (dr—-^ 2(J^^l\ir) • 2(. y (l-^) Consequently, if the initial particle position XQ is such that TV kxo = n-, (1.9) where n = 0, ±1,..., then the particle will be the fixed particle and r(t) = VQ. Equalities (1.9) define planes in the general case and points in the one-dimensional case. They correspond to zeros of the velocity field. Stability of these points depends on the sign of function v(t), and this sign changes during the evolution process. As a result, we can expect that particles will be concentrated around these points if Vx{t) ^ 0, which just corresponds to clustering of particles. In the case of the divergence-free velocity field, Vx{t) = 0 and, consequently, T(t) = 0; as a result, we have t x(t\xo) = xo, R(t|ro) = Ro + sin2(A;xo) / O^TVR(T), 0 which means that no clustering occurs. Figure 1.2a shows a fragment of the realization of random process T{t) obtained by numerical integration of Eq. (1.7) for a realization of random process Vx{t); we used this fragment for simulating the temporal evolution of coordinates of four particles x(t), X G (0,7r/2) initially located at coordinates xo{i) = || (i = 1, 2, 3, 4) (see Fig. 1.26). Figure 1.26 shows that particles form a cluster in the vicinity of point x = 0 at the dimensionless time t ~ 4 (see [13]). Further, at time t ~ 16 the initial cluster disappears and new one appears in the vicinity of point x = TV 12. At moment t ~ 40, the cluster appears again in the vicinity of point x = 0, and so on. In this process, particles in clusters remember their past history and significantly diverge during intermediate temporal segments (see Fig. 1.2c). Thus, we see that, in this example, the cluster does not move from one region to another; instead, it first collapses and then a new cluster is formed. Here, the lifetime of clusters significantly exceeds the duration of intermediate segments. It seems that this feature is characteristic of the particular model of the velocity field and follows from stationary property of points (1.9). As regards the particle diffusion along the y-direction, no cluster occurs in this direction. Note that such clustering in a system of particles was found to all appearance for the first time in papers [17, 18] as a result of simulating the so-called Eole experiment 14 Chapter 1. Examples, basic problems, peculiar features of solutions T{t) 1.5^X ( T^ b 10 ^ i.d 0 o.d \ '' -10 Luj 1^ -20 0 10 20 30 40 i 0 10 20 30 40 t 13 14 15 16 17t Figure 1.2: (a) Segment of a realization of random process T{t) obtained by numerically integrating Eq. (1.7) for a realization of random process Vx{t)'., (6), (c) x-coordinates sim ulated with this segment for four particles versus time. with the use of simplest equations of atmospheric dynamics. In the context of this global experiment, 500 constant-density balloons were launched in Argentina in 1970-1971; these balloons traveled at a height of about 12 km and spread along the whole of the southern hemisphere. Figure 1.3 shows the balloon distribution over the southern hemisphere for day 105 from the beginning of this process simulation [17]; this distribution clearly shows that balloons are concentrated in groups, which just corresponds to clustering . Now, we dwell on another stochastic aspect related to dynamic equations of type (1.1); namely, we consider the phenomenon of transfer caused by random fluctuations. Consider the one-dimensional nonlinear equation d , ^ :(l-x2)+/(i), x(0) = a;o; A > 0, (1.10) where f(t) is the random function of time. In the absence of randomness (/(t) = 0), the solution of Eq. (1.10) has two stable steady-state states x = ±1 and one instable state X = 0. Depending on the initial condition, solution of Eq. (1.10) arrives at one of the stable states. However, in the presence of small random disturbances /(i), dynamic system (1.10) will first approach the vicinity of one of the stable states and then, after the lapse of certain time, it will be transferred into the vicinity of another stable state. It is clear that the similar behavior can occur in more complicated situations. The system of equations (1.1) describes also the behavior of a particle under the field of random external forces f(r,^). In the simplest case, the behavior of a particle in the presence of linear friction is described by the system of the first-order differential equations jv{t) = v(«), | vW = -Av(0+f(r,f), r(0) = ro, v(0) VQ. (1.11) 1.1. Ordinary differential equations: initial value problems 15 Figure 1.3: Balloon distribution in the atmosphere for day 105 from the beginning of process simulation. The behavior of a particle under the deterministic potential field in the presence of linear friction and random forces is described by the system of equations d , , , dU(r,t) ^, , r{t) = v{t), v(t) dt r(0) ro, v(0) = vo, (1.12) which is the simplest example of Hamiltonian systems. In statistical problems, equations of type (1.11), (1.12) are widely used for describing Brownian motion of particles. 1.1.2 Systems with blow-up singularities The simplest stochastic system showing singular behavior in time is described by the following equation commonly used in the statistical theory of waves x{t) = -Ax^(t) + /(t), x(0) = xo, A > 0, (1.13) where f{t) is the random function of time. In the absence of randomness (/(t) = 0), the solution to Eq.(1.13) has the form 1 x{t) to A (t - to)' Axo" For xo > 0, we have to < 0, and solution x{t) monotonically tends to zero with increasing the time. On the contrary, for XQ < 0, solution x{t) reaches a value of — oo within a finite time to = —I/XXQ, which means that the solution becomes singular and shows the blow-up behavior. In this case, random force /(t) has insignificant effect on the behavior of the system. The eff^ect becomes significant only for positive parameter XQ.