By Valery I. Klyatskin
Fluctuating parameters seem in quite a few actual structures and phenomena. they often come both as random forces/sources, or advecting velocities, or media (material) parameters, like refraction index, conductivity, diffusivity, and so forth. the well-known instance of Brownian particle suspended in fluid and subjected to random molecular bombardment laid the root for contemporary stochastic calculus and statistical physics. different very important examples contain turbulent delivery and diffusion of particle-tracers (pollutants), or non-stop densities (''oil slicks''), wave propagation and scattering in randomly inhomogeneous media, for example mild or sound propagating within the turbulent atmosphere.
Such versions certainly render to statistical description, the place the enter parameters and suggestions are expressed via random approaches and fields.
The basic challenge of stochastic dynamics is to spot the basic features of method (its kingdom and evolution), and relate these to the enter parameters of the process and preliminary data.
This increases a bunch of difficult mathematical matters. you could not often clear up such platforms precisely (or nearly) in a closed analytic shape, and their recommendations count in a classy implicit demeanour at the initial-boundary info, forcing and system's (media) parameters . In mathematical phrases such resolution turns into a sophisticated "nonlinear functional" of random fields and processes.
Part I offers mathematical formula for the fundamental actual types of shipping, diffusion, propagation and develops a few analytic tools.
Part II units up and applies the recommendations of variational calculus and stochastic research, like Fokker-Plank equation to these types, to supply special or approximate ideas, or in worst case numeric techniques. The exposition is prompted and proven with a number of examples.
Part III takes up matters for the coherent phenomena in stochastic dynamical structures, defined through usual and partial differential equations, like wave propagation in randomly layered media (localization), turbulent advection of passive tracers (clustering).
Each bankruptcy is appended with difficulties the reader to resolve by way of himself (herself), in order to be a very good education for self reliant investigations.
· This publication is translation from Russian and is done with new valuable result of contemporary research.
· The publication develops mathematical instruments of stochastic research, and applies them to a variety of actual versions of debris, fluids, and waves.
· available to a huge viewers with basic history in mathematical physics, yet no distinct services in stochastic research, wave propagation or turbulence
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Extra info for Dynamics of Stochastic Systems
Example text
Solution dependence on problem type, medium parameters, and initial data Multiplying the first of these equations by Uk (ro (t) , i), summing over index k, adding the result to the second equation, and introducing the vector function Fi (t\ro,to) = (— + U (ro, t 0 ) TT-) n (t|r 0 , to) , V ok) or0 / we obtain that this function satisfies the linear homogeneous equation jdrm{rdlTk)-T)Fk{r\rQ,t0), Fl(t\r0,tQ) = to whose solution is obviously t\ (t|ro,to) = 0. 20) which can be considered as the linear partial differential equation in terms of derivatives with respect to variables ro and to satisfying at to = t the initial condition r ( t | r o , t ) = ro.
With respect to rotations of the reference system, then field / ( x , i) is called the homogeneous isotropic random field. In this case, the correlation function depends on the length |xi — x 2 |: B / ( x i , i i ; x 2 , t 2 ) = (/(x 1 ,t 1 )/(x 2 ,i 2 )) = £/(|xi - x 2 | ; * i , t 2 ) . The Fourier transform of correlation function of the homogeneous random field with respect to spatial variables defines the spatial spectral function (called also the angular spectrum) $/(k,t) = /fixB/(x,t)e l k x , and the Fourier transform of correlation function of the stationary and homogeneous random field / ( x , t) with respect to both spatial and temporal variables defines the space-time spectrum oc * / ( k , w ) = I dx.
1. Random quantities and their characteristics 53 £ (in so doing, operator Q(d/id£) should be introduced into averaging brackets) and set r) = 0. 11) for fl(v), as the series in cumulants Kn Note that, setting /(£) = £n~ in Eq. 12), we obtain the recurrence formula M (Mo= 1 n= 1 2 " = it(k}niMn-k)\KkMn-k ' ' '-) (413) that relates moments and cumulants of random quantity £. To illustrate practicability of the above formulas, we consider two types of random quantities £ as examples. 1. ) = 0, 2 2 B(v) = -^-, M 2 = K2 = a2 = (^ 2 ) , Kn>2 = 0.