Klyatskin V.I. Lectures on Dynamics of Stochastic Systems

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146 Lectures on Dynamics of Stochastic Systems

Consequently, we can rewrite Eq. (6.13) in the form of the closed operator equation

∂

∂t

P(x, t) =



t; ig(τ )

∂

∂x

F(x)



P(x, t), P(x, 0) = δ(x − x

), (6.14)

whose particular form depends on the behavior of process z(t).

For the two-time probability density

P(x, t; x

, t

) =

δ(x(t) − x)δ(x(t

) − x

)

we obtain similarly the equation (for t > t

)

∂

∂t

P(x, t; x

, t

)



t; ig(τ )



∂

∂x

F(x) + θ(t

−τ )

∂

∂x

F(x

)



P(x, t; x

, t

) (6.15)

with the initial value

P(x, t

; x

, t

) = δ(x − x

)P(x

, t

where function P(x

, t

) satisﬁes Eq. (6.14).

One can see from Eq. (6.15) that multidimensional probability density cannot be

factorized in terms of the transition probability (see Sect. 4.3), so that process x(t) is

not the Markovian process. The particular forms of Eqs. (6.14) and (6.15) are governed

by the statistics of process z(t).

If z(t) is the Gaussian process whose mean value and correlation function are

z(t)

= 0, B(t, t

) =



z(t)z(t

)



then functional 2[t; v(τ )] has the form

2[t; v(τ )] = −

B(t

, t

)v(t

)

and Eq. (6.14) assumes the following form

∂

∂t

P(x, t) = g(t)

dτ B(t, τ )g(τ )

∂

∂x

(x)

∂

∂x

(x)P(x, t) (6.16)

and can be considered as the extended Fokker–Planck equation.

The class of problems formulated in terms of the system of equations

x(t) = z(t)F(x) − λx(t), x(0) = x

, (6.17)

General Approaches to Analyzing Stochastic Systems 147

where F(x) are the homogeneous polynomials of power k, can be reduced to problem

(6.10). Indeed, introducing new functions

x(t) = ex(t)e

−λt

we arrive at problem (6.10) with function g(t) = e

−λ(k−1)t

. In the important special

case with k = 2 and functions F(x) such that xF(x) = 0, the system of equations (6.10)

describes hydrodynamic systems with linear friction (see, e.g., [24]). In this case, the

interaction between the components appears to be random.

If λ = 0, energy conservation holds in hydrodynamic systems for any realization

of process z(t). For t → ∞, there is the steady-state probability distribution P(x),

which is, under the assumption that no additional integrals of motion exist, the uniform

distribution over the sphere x

= E

. If additional integrals of motion exist (as it is the

case for ﬁnite-dimensional approximation of the two-dimensional motion of liquid),

the domain of the steady-state probability distribution will coincide with the phase

space region allowed by the integrals of motion.

Note that, in the special case of the Gaussian process z(t) appeared in the one-

dimensional linear equation of type Eq. (6.17)

x(t) = −λx(t) + z(t)x(t), x(0) = 1,

which determines the simplest logarithmic-normal random process, we obtain, instead

of Eq. (6.16), the extended Fokker–Planck equation



∂

∂t

+ λ

∂

∂x



P(x, t) =

dτ B(t, τ )

∂

∂x

∂

∂x

xP(x, t),

P(x, 0) = δ(x − 1).

(6.18)

Additive Action

Consider now the class of linear equations

x(t) = A(t)x(t) + f(t), x(0) = x

, (6.19)

where A(t) is the deterministic matrix and f(t) is the random vector function whose

characteristic functional 8[t; v(τ )] is known.

For the probability density of the solution to Eq. (6.19), we have

∂

∂t

P(x, t) = −

∂

∂x

(

(t)x

P(x, t)

)





iδf (τ )



(

x(t)−x

)



. (6.20)

148 Lectures on Dynamics of Stochastic Systems

In the problem under consideration, the variational derivative

δx(t)

δf (τ )

also satisﬁes

(for τ < t) the linear equation with the initial value

δf (τ )

(t) = A

(t)

δf (τ )

(t),

δf

(τ )

(t)



t=τ

= δ

. (6.21)

Equation (6.21) has no randomness and governs Green’s function G

(t, τ ) of homo-

geneous system (6.19), which means that

δf

(τ )

(t) = G

(t, τ ).

As a consequence, we have

δf

(τ )

δ(x(t) − x) = −

∂

∂x

(t, τ )δ(x(t) − x),

and Eq. (6.20) appears to be converted into the closed equation

∂

∂t

P(x, t) = −

∂

∂x

(

(t)x

P(x,t)

)



t; iG

(t, τ )

∂

∂x



P(x, t). (6.22)

From Eq. (6.22) follows that any moment of quantity x(t) will satisfy a closed linear

equation that will include only a ﬁnite number of cumulant functions whose order will

not exceed the order of the moment of interest.

For the two-time probability density

P(x, t; x

, t

) =

δ(x(t) − x)δ(x(t

) − x

)

we quite similarly obtain the equation

∂

∂t

P(x, t; x

, t

) = −

∂

∂x

(

(t)x

P(x,t; x

)



t; i

{

(t, τ ) + G

, τ )

}

∂

∂x



P(x, t; x

, t

) (t > t

)

(6.23)

with the initial value

P(x, t

; x

, t

) = δ(x − x

)P(x

, t

where P(x

, t

) is the one-time probability density satisfying Eq. (6.22). From

Eq. (6.23) follows that x(t) is not the Markovian process. The particular form of

Eqs. (6.22) and (6.23) depends on the structure of functional 8[t; v(τ )], i.e., on the

behavior of random function f(t).

General Approaches to Analyzing Stochastic Systems 149

For the Gaussian vector process f (t) whose mean value and correlation function are

f (t)

= 0, B

(t, t

) =



(t)f

)



Eq. (6.22) assumes the form of the extended Fokker–Planck equation

∂

∂t

P(x, t) = −

∂

∂x

(

(t)x

P(x,t)

)

dτ B

(t, τ )G

(t, t)G

(t, τ )

∂

∂x

P(x, t). (6.24)

Consider the dynamics of a particle under random forces in the presence of friction

[49] as an example of such a problem.

Inertial Particle Under Random Forces The simplest example of particle diffusion

under the action of random external force f (t) and linear friction is described by the

linear system of equations (1.75), page 36

r(t) = v(t),

v(t) = −λ

[

v(t) − f (t)

]

r(0) = 0, v(0) = 0.

(6.25)

The stochastic solution to Eqs. (6.25) has the form

v(t) = λ

dτ e

−λ(t−τ )

f(τ ), r(t) =

dτ

1 − e

−λ(t−τ )

f (τ ). (6.26)

In the case of stationary random process f(t) with the correlation tensor



(t)f

)



= B

(t − t

)

and temporal correlation radius τ

determined from the relationship

∞

dτ B

(τ ) = τ

(0),

Eq. (6.25) allows obtaining analytical expressions for correlators between particle

velocity components and coordinates



(t)v

(t)



= λ

dτ B

(τ )

−λτ

− e

−λ(2t−τ )



(t)r

(t)





(t)v

(t)



dτ B

(τ )



1 − e

−λt



1 − e

−λ(t−τ )

(6.27)

150 Lectures on Dynamics of Stochastic Systems

In the steady-state regime, when λt  1 and t/τ

 1, but parameter λτ

can be

arbitrary, particle velocity is the stationary process with the correlation tensor



(t)v

(t)



= λ

∞

dτ B

(τ )e

−λτ

, (6.28)

and correlations



(t)v

(t)



and



(t)r

(t)



are as follows



(t)v

(t)



∞

dτ B

(τ ),



(t)r

(t)



= 2t

∞

dτ B

(τ ). (6.29)

If we additionally assume that λτ

 1, the correlation tensor grades into



(t)v

(t)



= B

(0), (6.30)

which is consistent with (6.25), because v(t) ≡ f (t) in this limit.

If the opposite condition λτ

 1 holds, then



(t)v

(t)



= λ

∞

dτ B

(τ ).

This result corresponds to the delta-correlated approximation of random process f(t).

Introduce now the indicator function of the solution to Eq. (6.25)

ϕ(r, v, t) = δ

(

r(t) − r

)

(

v(t) − v

)

which satisﬁes the Liouville equation



∂

∂t

+ v

∂

∂r

− λ

∂

∂v



ϕ(r, v, t) = −λf (t)

∂

∂v

ϕ(r, v, t),

ϕ(r, v; 0) = δ

(

)

(

)

(6.31)

The mean value of the indicator function ϕ(r, v, t) over an ensemble of realizations

of random process f (t) is the joint one-time probability density of particle position and

velocity

P(r, v, t) =

ϕ(r, v; t)

(

r(t) − r

)

(

v(t) − v

)

Averaging Eq. (6.31) over an ensemble of realizations of random process f (t), we

obtain the unclosed equation



∂

∂t

+ v

∂

∂r

− λ

∂

∂v



P(r, v, t) = −λ

∂

∂v

f(t)ϕ(r, v, t)

P(r, v; 0) = δ

(

)

(

)

(6.32)

General Approaches to Analyzing Stochastic Systems 151

This equation contains correlation

f (t)ϕ(r, v, t)

and is equivalent to the equality



∂

∂t

+ v

∂

∂r

− λ

∂

∂v



P(r, v, t) =





iδf (τ )



ϕ(r, v, t)



P(r, v; 0) = δ

(

)

(

)

(6.33)

where functional 2

[

t; v(τ )

]

is related to the characteristic functional of random pro-

cess f(t)

[

t; ψ(τ )

]

exp







dτ ψ(τ )f (τ )







= e

[

t;ψ(τ )

]

by the formula

[

t; ψ(τ )

]

ln 8

[

t; ψ(τ )

]

[

t; ψ(τ )

]

Functional 2[t; ψ(τ )] can be expanded in the functional power series

2[t; ψ(τ )] =

∞

n=1

. . .

(n)

,...,i

, . . . , t

)ψ

) . . . ψ

where functions

(n)

,...,i

, . . . , t

) =

δψ

) . . . δψ

)

2[t; ψ(τ )]



ψ=0

are the n-th order cumulant functions of random process f(t).

Consider the variational derivative

δf

)

ϕ(r, v, t) = −



∂

∂r

δr

(t)

δf

)

∂

∂v

δv

(t)

δf

)



ϕ(r, v, t). (6.34)

In the context of dynamic problem (6.25), the variational derivatives of functions r(t)

and v(t) in Eq. (6.34) can be calculated from Eqs. (6.26) and have the forms

δv

(t)

δf

)

= λδ

−λ(t−t

)

δr

(t)

δf

)

= δ

1 − e

−λ(t−t

)

. (6.35)

Using Eq. (6.35), we can now rewrite Eq. (6.34) in the form

δf (t

)

ϕ(r, v; t) = −



1 − e

−λ(t−t

)

∂

∂r

+ λe

−λ(t−t

)

∂

∂v



ϕ(r, v, t),

152 Lectures on Dynamics of Stochastic Systems

after which Eq. (6.33) assumes the closed form



∂

∂t

+ v

∂

∂r

− λ

∂

∂v



P(r, v, t)



t; i



1 − e

−λ(t−t

)

∂

∂r

+ λe

−λ(t−t

)

∂

∂v



P(r, v, t),

P(r, v; 0) = δ

(

)

(

)

(6.36)

Note that from Eq. (6.36) follows that equations for the n-th order moment func-

tions include cumulant functions of orders not higher than n.

Assume now that f (t) is the Gaussian stationary process with the zero mean value

and correlation tensor

(t − t

) =



(t)f

)



In this case, the characteristic functional of process f(t) is

[

t; ψ(τ )

]

= exp







−

− t

)ψ

)







functional

2[t; ψ(τ )] is given by the formula

[

t; ψ(τ )

]

= −ψ

(t)

(t − t

)ψ

and Eq. (6.36) appears to be an extension of the Fokker–Planck equation



∂

∂t

+ v

∂

∂r

− λ

∂

∂v



P(r, v, t) = λ

dτ B

(τ )e

−λτ

∂

∂v

P(r, v, t)

+ λ

dτ B

(τ )



1 − e

−λτ



∂

∂v

∂r

P(r, v, t),

P(r, v; 0) = δ

(

)

(

)

(6.37)

Equation (6.37) is the exact equation and remains valid for arbitrary times t.

From this equation follows that r(t) and v(t) are the Gaussian functions. For moment

General Approaches to Analyzing Stochastic Systems 153

functions of processes r(t) and v(t), we obtain in the ordinary way the system of

equations



(t)r

(t)



= 2



(t)v

(t)





+ λ





(t)v

(t)





(t)v

(t)



+ λ

dτ B

(τ )



1 − e

−λτ





+ 2λ





(t)v

(t)



= 2λ

dτ B

(τ )e

−λτ

(6.38)

From system (6.38) follows that steady-state values of all one-time correlators for

λt  1 and t/τ

 1 are given by the expressions



(t)v

(t)



= λ

∞

dτ B

(τ )e

−λτ



(t)v

(t)



= D



(t)r

(t)



= 2tD

(6.39)

where

∞

dτ B

(τ ) (6.40)

is the spatial diffusion tensor, which agrees with expressions (6.28) and (6.29).

Remark

Temporal correlation tensor and temporal correlation radius of process v(t).

We can additionally calculate the temporal correlation radius of velocity v(t), i.e.,

the scale of correlator



(t)v

)



. Using equalities (6.35), we obtain for t

< t the

equation



+ λ





(t)v

)



= λ

(t − t

−λ(t

−t

)

= λ

λ(t−t

)

t−t

dτ B

(τ )e

−λτ

(6.41)

with the initial value



(t)v

)



t=t



)



. (6.42)

154 Lectures on Dynamics of Stochastic Systems

In the steady-state regime, i.e., for λt  1 and λt

 1, but at ﬁxed difference

(t − t

), we obtain the equation with initial value (τ = t − t

)



dτ

+ λ





(t + τ )v

(t)



= λ

λτ

∞

dτ

(τ

−λτ



(t + τ )v

(t)



τ =0



(t)v

(t)



(6.43)

One can easily write the solution to Eq. (6.43); however, our interest here concerns

only the temporal correlation radius τ

of random process v(t). To obtain this quantity,

we integrate Eq. (6.43) with respect to parameter τ over the interval (0, ∞). The

result is

∞

dτ



(t + τ )v

(t)





(t)v

(t)



+ λ

∞

dτ

(τ

)



1 − e

−λτ



and we, using Eq. (6.39), arrive at the expression

(t)

= D

= τ

(0), (6.44)

i.e.,

(0)



(t)



(0)

∞

dτ B

(τ )e

−λτ







, for λτ

 1,

1/λ, for λτ

 1.

(6.45)

Integrating Eq. (6.37) over r, we obtain the closed equation for the probability

density of particle velocity



∂

∂t

− λ

∂

∂v



P(v, t) = λ

dτ B

(τ )e

−λτ

∂

∂v

P(r, v, t),

P(r, v; 0) = δ

(

)

The solution to this equation corresponds to the Gaussian process v(t) with correlation

tensor (6.27), which follows from the fact that the second equation of system (6.25)

is closed. It can be shown that, if the steady-state probability density exists under the

condition λt  1, then this probability density satisﬁes the equation

−

∂

∂v

vP(v, t) = λ

∞

dτ B

(τ )e

−λτ

∂

∂v

P(v, t),

General Approaches to Analyzing Stochastic Systems 155

and the rate of establishing this distribution depends on parameter λ.

The equation for the probability density of particle coordinate P(r, t) cannot be

derived immediately from Eq. (6.37). Indeed, integrating Eq. (6.37) over v, we obtain

the equality

∂

∂t

P(r, t) = −

∂

∂r

vP(r, v, t)dv, P(r, 0) = δ

(

)

. (6.46)

For function

P(r, v, t)dv, we have the equality



∂

∂t

+ λ



P(r, v, t)dv = −

∂

∂r

vP(r, v, t)dv

− λ

dτ B

(τ )



1 − e

−λτ



∂

∂r

P(r, t), (6.47)

and so on, i.e., this approach results in an inﬁnite system of equations.

Random function r(t) satisﬁes the ﬁrst equation of system (6.25) and, if we would

know the complete statistics of function v(t) (i.e., the multi-time statistics), we could

calculate all statistical characteristics of function r(t). Unfortunately, Eq. (6.37) des-

cribes only one-time statistical quantities, and only the inﬁnite system of equations

similar to Eqs. (6.46), (6.47), and so on appears equivalent to the multi-time statistics

of function v(t). Indeed, function r(t) can be represented in the form

r(t) =

v(t

so that the spatial diffusion coefﬁcient in the steady-state regime assumes, in view of

Eq. (6.44), the form

(t)

∞

dτ

v(t + τ )v(t)

= τ

(t)

= D

= τ

(0), (6.48)

from which follows that it depends on the temporal correlation radius τ

and the vari-

ance of random function v(t).

However, in the case of this simplest problem, we know both variances and all

correlations of functions v(t) and r(t) (see Eqs. (6.27)) and, consequently, can draw

the equation for the probability density of particle coordinate P(r, t). This equation is

the diffusion equation

∂

∂t

P(r, t) = D

(t)

∂

∂r

P(r, t), P(r, 0) = δ

(

)