Klyatskin V.I. Lectures on Dynamics of Stochastic Systems
Подождите немного. Документ загружается.
176 Lectures on Dynamics of Stochastic Systems
Solution Characteristic functional 8[x; v, v
∗
] of the problem solution satisfies
the equation
∂
∂x
8[x; v, v
∗
] =
˙
2
x
x,
k
2
b
M(R
0
)
8[x; v, v
∗
]
+
i
2k
Z
dR
0
v(R
0
)1
R
0
δ
δv(R
0
)
−v
∗
(R
0
)1
R
0
δ
δv
∗
(R
0
)
× 8[x; v, v
∗
] (6.98)
with the Hermitian operator
b
M(R
0
) = v(R
0
)
δ
δv(R
0
)
− v
∗
(R
0
)
δ
δv
∗
(R
0
)
.
Here, functional
˙
2
x
x; ψ(ξ, R
0
)
=
d
dx
ln
*
exp
i
x
Z
0
dξ
Z
dR
0
ε(ξ, R
0
)ψ(ξ, R
0
)
+
is the derivative of the logarithm of the characteristic functional of filed ε(x, R).
Remark Equation (6.98) yields the equations for the moment functions of field
u(x, R),
M
m,n
(x; R
1
, . . . , R
m
; R
0
1
, . . . , R
0
n
)
=
u(x, R
1
) . . . u(x, R
m
)u
∗
(x, R
0
1
) . . . u
∗
(x, R
0
n
)
,
(for m = n, these functions are usually called the coherence functions of order 2n)
∂
∂x
M
m,n
=
i
2k
m
X
p=1
1
R
p
−
n
X
q=1
1
R
0
q
M
m,n
+
˙
2
x
x,
1
k
m
X
p=1
δ(R
0
− R
p
)−
n
X
q=1
δ(R
0
− R
0
q
)
M
m,n
.
(6.99)
General Approaches to Analyzing Stochastic Systems 177
Problem 6.15
Assuming that ε(x, R) is the homogeneous Gaussian random process delta-
correlated in x with the correlation function
B
ε
(x, R) = A(R)δ(x), A(R) =
∞
Z
−∞
dxB
ε
(x, R)
and starting from the solution to the previous problem, derive the equations for
the characteristic functional of wave field u(x, R), which is the solution to linear
parabolic equation (1.89), page 39, and moment functions of this field [50].
Solution
∂
∂x
8[x; v, v
∗
] = −
k
2
8
Z
dR
0
Z
dRA(R
0
− R)
b
M(R
0
)
b
M(R)8[x; v, v
∗
]
+
i
2k
Z
dR
0
v(R
0
)1
R
0
δ
δv(R
0
)
−v
∗
(R
0
)1
R
0
δ
δv
∗
(R
0
)
8[x; v, v
∗
],
∂xM
m,n
=
i
2k
m
X
p=1
1
R
p
−
n
X
q=1
1
R
0
q
M
m,n
−
k
2
8
Q(R
1
, . . . , R
m
; R
0
1
, . . . , R
0
m
)M
m,n
,
where
Q(R
1
, . . . , R
m
; R
0
1
, . . . , R
0
m
) =
=
m
X
i=1
m
X
j=1
A(R
i
− R
j
)−2
m
X
i=1
n
X
j=1
A(R
i
− R
0
j
)+
n
X
i=1
n
X
j=1
A(R
0
i
− R
0
j
).
Problem 6.16
Derive the equation for the one-time characteristic functional of the solution to
stochastic nonlinear integro-differential equation (1.102), page 44.
Solution Characteristic functional of the velocity field
8[t; z(k
0
)] = 8[t; z] =
ϕ[t; z(k
0
)]
,
178 Lectures on Dynamics of Stochastic Systems
where
ϕ[t; z(k
0
)] = exp
i
Z
dk
0
bu(k
0
, t)z(k
0
)
,
satisfies the unclosed variational-derivative equation
∂
∂t
8[t; z] =
Z
dkz
i
(k)
−
1
2
Z
dk
1
Z
dk
2
3
i,αβ
(k
1
, k
2
, k)
δ
2
δz
α
(k
1
)δz
β
(k
2
)
− νk
2
δ
δz
i
(k)
8[t; z]
+
˙
2
t
t;
δ
iδf (κ, τ )
ϕ[t; z]
,
where
˙
2
t
[
t; ψ(κ, τ )
]
=
d
dt
ln
*
exp
i
t
Z
0
dτ
Z
dκf (κ, τ )ψ(κ, τ )
+
is the derivative of the characteristic functional logarithm of external forces
b
f (k, t).
Problem 6.17
Assuming that f (x, t) is the homogeneous stationary random field delta-
correlated in time, derive the equation for the one-time characteristic functional
of the solution to stochastic nonlinear integro-differential equation (1.102),
page 44. Consider the case of the Gaussian field f(x, t) [1].
Solution
∂
∂t
8[t; z] =
Z
dkz
i
(k)
−
1
2
Z
dk
1
Z
dk
2
3
i,αβ
(k
1
, k
2
, k)
δ
2
δz
α
(k
1
)δz
β
(k
2
)
− νk
2
δ
δz
i
(k)
8[t; z] +
˙
2
t
[
t; z(k)
]
8[t; z], (6.100)
where
˙
2
t
[
t; ψ(κ, τ )
]
=
d
dt
ln
*
exp
i
t
Z
0
dτ
Z
dκ
b
f(κ, τ )ψ(κ, τ )
+
General Approaches to Analyzing Stochastic Systems 179
is the derivative of the characteristic functional logarithm of external forces
b
f(k, t).
If we assume now that f(x, t) is the Gaussian random field homogeneous and
isotropic in space and stationary in time with the correlation tensor
B
ij
(x
1
− x
2
, t
1
− t
2
) =
f
i
(x
1
, t
1
)f
j
(x
2
, t
2
)
,
then the field
b
f (k, t) will also be the Gaussian stationary random field with the
correlation tensor
b
f
i
(k, t + τ )
b
f
j
(k
0
, t)
=
1
2
F
ij
(k, τ )δ(k + k),
where F
ij
(k, τ ) is the external force spatial spectrum given by the formula
F
ij
(k, τ ) = 2(2π)
3
Z
dxB
ij
(x, τ )e
−ikx
.
In view of the fact that forces are spatially isotropic, we have
F
ij
(k, τ ) = F(k, τ )1
ij
(k).
As long as field
b
f (k, t) is delta-correlated in time, we have that F(k, τ ) =
F(k)δ(τ ), so that functional 2
[
t; ψ(κ, τ )
]
is given by the formula
2
[
t; ψ(κ, τ )
]
= −
1
4
t
Z
0
dτ
Z
dκ F(κ)1
ij
(κ)ψ
i
(κ, τ )ψ
j
(−κ, τ ),
and Eq. (6.100) assumes the closed form
∂
∂t
8[t; z] = −
1
4
Z
dkF(k)1
ij
(k)z
i
(k)z
j
(−k)8[t; z]
−
Z
dkz
i
(k)
1
2
Z
dk
1
Z
dk
2
3
i,αβ
(k
1
, k
2
, k)
δ
2
δz
α
(k
1
)δz
β
(k
2
)
8[t; z]
+ νk
2
δ
δz
i
(k)
Z
dkz
i
(k)
δ
δz
i
(k)
8[t; z].
180 Lectures on Dynamics of Stochastic Systems
Problem 6.18
Average the linear stochastic integral equation
S(r, r
0
) = S
0
(r, r
0
)
+
Z
dr
1
Z
dr
2
Z
dr
3
S
0
(r, r
1
)3(r
1
, r
2
, r
3
)
[
η(r
2
) + f (r
2
)
]
S(r
3
, r
0
),
Solution
G[r, r
0
; η(r)] = S
0
(r, r
0
)
+
Z
dr
1
Z
dr
2
Z
dr
3
S
0
(r, r
1
)3(r
1
, r
2
, r
3
)η(r
2
)G[r
3
, r
0
; η(r)]
+
Z
dr
1
Z
dr
2
Z
dr
3
S
0
(r, r
1
)3(r
1
, r
2
, r
3
)
r
2
δ
iδη(r)
G[r
3
, r
0
; η(r)].
Here,
G[r, r
0
; η(r)] =
S[r, r
0
; f (r) + η(r)]
and functional
r
[
v(r)
]
=
δ
iδv(r)
2
[
v(r)
]
, 2
[
v(r)
]
= ln
exp
i
Z
drf (r)v(r)
.
Problem 6.19
Assuming that random external force f (r, t) is the homogeneous stationary Gaus-
sian field, derive the equation for the characteristic functional of space-time har-
monics of turbulent velocity field (1.103), page 45 [51].
Solution Characteristic functional of the velocity field
8[z] =
exp
i
Z
d
4
K
0
bu
K
0
z
K
0
and functional
G
ij
K, K
0
; z
=
*
δu
i
(K)
δ
b
f
j
(K
0
)
ϕ[z]
+
General Approaches to Analyzing Stochastic Systems 181
satisfy the closed system of functional equations
(iω + νk
2
)
δ
δz
i
(
K
)
8[z] = −
1
2
F
ij
(K)
Z
d
4
K
1
z
α
(
K
1
)
G
αj
[
K
1
, −K; z
]
−
1
2
Z
d
4
K
1
Z
d
4
K
2
3
αβ
i
(
K
1
, K
2
, K
)
δ
2
8[z]
δz
α
(
K
1
)
δz
β
(
K
2
)
,
(iω + νk
2
)G
ij
K, K
0
; z
+
Z
d
4
K
1
Z
d
4
K
2
3
αβ
i
(
K
1
, K
2
, K
)
δ
δz
α
(K
1
)
G
βj
K
2
, K
0
; z
= δ
ij
δ
4
K − K
0
8[z],
where
b
f
i
(K
1
)
b
f
j
(K
2
)
=
1
2
δ
4
(
K
1
+ K
2
)
F
ij
(K
1
), and F
ij
(K) is the space-time
spectrum of external forces.
Remark Expansion of functionals 8[z] and G
ij
K, K
0
; z
in the functional Taylor
series in z
(
K
)
determines velocity field cumulants and correlators of Green’s
function analog S
ij
K, K
0
= δu
i
(
K
)
/δf
j
K
0
with the velocity field.
This page intentionally left blank
Lecture 7
Stochastic Equations with the
Markovian Fluctuations of Parameters
In the preceding Lecture, we dealt with the statistical description of dynamic systems
in terms of general methods that assumed knowledge of the characteristic functional
of fluctuating parameters. However, this functional is unknown in most cases, and we
are forced to resort either to assumptions on the model of parameter fluctuations, or to
asymptotic approximations.
The methods based on approximating fluctuating parameters with the Markovian
random processes and fields with a finite temporal correlation radius are widely used.
Such approximations can be found, for example, as solutions to the dynamic equations
with delta-correlated fluctuations of parameters. Consider such methods in greater
detail by the examples of the Markovian random processes.
Consider stochastic equations of the form
d
dt
x(t) = f
(
t, x, z(t)
)
, x(0) = x
0
, (7.1)
where f
(
t, x, z(t)
)
is the deterministic function and
z(t) = {z
1
(t), . . . , z
n
(t)}
is the Markovian vector process.
Our task consists in the determination of statistical characteristics of the solution to
Eq. (7.1) from known statistical characteristics of process z(t).
In the general case of arbitrary Markovian process z(t), we cannot judge about
process x(t). We can only assert that the joint process {x(t), z(t)} is the Markovian
process with joint probability density P(x, z, t).
If we could solve the equation for P(x, z, t), then we could integrate the solution
over z to obtain the probability density of the solution to Eq. (7.1), i.e., function P(x, t).
In this case, process x(t) by itself would not be the Markovian process.
There are several types of process z(t) that allow us to obtain equations for density
P(x, t) without solving the equation for P(x, z, t). Among these processes, we men-
tion first of all the telegrapher’s and generalized telegrapher’s processes, Markovian
Lectures on Dynamics of Stochastic Systems. DOI: 10.1016/B978-0-12-384966-3.00007-6
Copyright
c
2011 Elsevier Inc. All rights reserved.
184 Lectures on Dynamics of Stochastic Systems
processes with finite number of states, and Gaussian Markovian processes. Below, we
discuss the telegrapher’s and Gaussian Markovian processes in more detail as exam-
ples of processes widely used in different applications.
7.1 Telegrapher’s Processes
Recall that telegrapher’s random process z(t) (the two-state, or binary process) is
defined by the equality
z(t) = a(−1)
n(0,t)
,
where random quantity a assumes values a = ± a
0
with probabilities 1/2 and n(t
1
, t
2
),
t
1
< t
2
is Poisson’s integer-valued process with mean value n(t
1
, t
2
) = ν|t
1
− t
2
|.
Telegrapher’s process z(t) is stationary in time and its correlation function
z(t)z(t
0
)
= a
2
0
e
−2ν|t−t
0
|
has the temporal correlation radius τ
0
= 1/
(
2ν
)
.
For splitting the correlation between telegrapher’s process z(t) and arbitrary func-
tional R[t; z(τ )], where τ ≤ t, we obtained the relationship (5.25), (page 129)
h
z(t)R[t; z(τ )]
i
= a
2
0
t
Z
0
dt
1
e
−2ν(t−t
1
)
δ
δz(t
1
)
e
R[t, t
1
; z(τ )]
, (7.2)
where functional
e
R[t, t
1
; z(τ )] is given by the formula
e
R[t, t
1
; z(τ )] = R[t; z(τ )θ(t
1
− τ + 0)] (t
1
< t). (7.3)
Formula (7.2) is appropriate for analyzing stochastic equations linear in process
z(t). Functional R[t; z(τ )] is the solution to a system of differential equations of the
first order in time with initial values at t = 0. Functional
e
R[t, t
1
; z(τ )] will also satisfy
the same system of equations with product z(t)θ(t
1
− t) instead of z(t). Consequently,
we obtain that functional
e
R[t, t
1
; z(τ )] = R[t; 0] for all times t > t
1
; moreover, it
satisfies the same system of equations for absent fluctuations (i.e., at z(t) = 0) with
the initial value
e
R[t
1
, t
1
; z(τ )] = R[t
1
; z(τ )].
Another formula convenient in the context of stochastic equations linear in random
telegrapher’s process z(t) concerns the differentiation of a correlator of this process
with arbitrary functional R[t; z(τ )](τ ≤ t) (5.26)
d
dt
h
z(t)R[t; z(τ )]
i
= −2ν
h
z(t)R[t; z(τ )]
i
+
z(t)
d
dt
R[t; z(τ )]
. (7.4)
Stochastic Equations with the Markovian Fluctuations of Parameters 185
Formula (7.4) determines the rule of factoring the differentiation operation out of
averaging brackets
z(t)
d
n
dt
n
R[t; z(τ )]
=
d
dt
+ 2ν
n
h
z(t)R[t; z(τ )]
i
. (7.5)
We consider some special examples to show the usability of these formulas. It is
evident that both methods give the same result. However, the method based on the
differentiation formula appears more practicable.
The first example concerns the system of linear operator equations
d
dt
x(t) =
b
A(t)x(t) + z(t)
b
B(t)x(t), x(0) = x
0
, (7.6)
where
b
A(t) and
b
B(t) are certain differential operators (they may include differential
operators with respect to auxiliary variables). If operators
b
A(t) and
b
B(t) are matrices,
then Eqs. (7.6) describe the linear dynamic system.
Average Eqs. (7.6) over an ensemble of random functions z(t). The result will be
the vector equation
d
dt
h
x(t)
i
=
b
A(t)
h
x(t)
i
+
b
B(t)ψ
ψ
ψ(t),
h
x(0)
i
= x
0
, (7.7)
where we introduced new functions
ψ
ψ
ψ(t) =
h
z(t)x(t)
i
.
We can use formula Eqs. (7.4) for these functions; as a result, we obtain the equality
d
dt
ψ
ψ
ψ(t) = −2νψ
ψ
ψ(t) +
z(t)
d
dt
x(t)
. (7.8)
Substituting now derivative dx/dt (7.6) in Eq. (7.8), we obtain the equation for the
vector function ψ
ψ
ψ(t)
d
dt
+ 2ν
ψ
ψ
ψ(t) =
b
A(t)ψ
ψ
ψ(t) +
b
B(t)
D
z
2
(t)x(t)
E
. (7.9)
Because z
2
(t) ≡ a
2
0
for telegrapher’s process, we obtain finally the closed system of
linear equations for vectors
h
x(t)
i
and ψ
ψ
ψ(t)
d
dt
h
x(t)
i
=
b
A(t)
h
x(t)
i
+
b
B(t)ψ
ψ
ψ(t),
h
x(0)
i
= x
0
,
d
dt
+ 2ν
ψ
ψ
ψ(t) =
b
A(t)ψ
ψ
ψ(t) + a
2
0
b
B(t)
h
x(t)
i
, ψ
ψ
ψ(0) = (0).
(7.10)