Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
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∆W
t
1
,t
2
= W (t
2
)−W (t
1
) t
2
> t
1
h∆W
t
1
,t
2
i = 0.
D
(∆W
t
1
,t
2
)
2
E
= 2D (t
2
− t
1
) .
h∆W
t
1
,t
2
∆W
t
2
,t
3
i = 0 ; t
3
> t
2
> t
1
.
∆W
t
1
,t
2
= ∆W
t
2
−t
1
t
i+1
− t
i
→ 0
∂
∂t
p(∆W, t|∆W
0
, t
0
) = D
∂
2
∂∆W
2
p(∆W, t|∆W
0
, t
0
).
W (t) =
n−1
X
k=0
∆W
t
k+1
−t
k
,
t
n
= t t
0
= 0
ξ(t)
W (t) =
Z
t
0
ds ξ(s) ?
ξ(t)
∆W/∆t
∆t → 0
∆W/∆t ξ(t)
ξ(t)
ξ(t)
ξ(t)
c
ξ,ξ
(t
1
−t
2
) = 2D δ(t
1
−t
2
)
ξ(t)
G
ξξ
(ω) G
ξξ
(ω) = 2D
ξ(t)
x(t)
τ
c
τ
r
D
t
0
= a t ξ(t
0
) =
1
√
a
ξ(t)
c
ξ,ξ
(t
0
1
− t
0
2
) = 2 (D/a) δ (t
0
1
−t
0
2
)
ξ(t)
hξ(t) ξ(t + τ )i = δ(τ )
h∆W
2
∆t
i = ∆t
ξ(t) ∆W
∆t
N(t)
[0, t] N(0) = 0 N(t)
N t
P
N
(t) =
(γt)
N
N!
exp (γt ) ,
t ≥ 0 N(t)
γ
γ
hN(t)i = γ t , h( N(t) −hN (t) i) ( N(t
0
)−hN(t
0
) i) i = γ min
t, t
0
.
˙
P
N
(t) = −γP
N
(t) + γP
N−1
(t) ; N ≥ 1 ,
˙
P
0
(t) = −γP
0
(t) ,
N →
N + 1 W (N → N +
1) = γ
P
N
(t + ∆t)
P
N
(t)
N(t)
∆N(t) > 0 N
0
→
N
0
+ ∆N(t) [t
0
, t)
∆N
0
= 0
t = t
0
∆N(t) t > t
0
γ = const.
∆N(t)
P (∆N, t|∆N
0
= 0, t
0
) =
[γ (t − t
0
)]
∆N
∆N!
exp [−γ (t − t
0
)] , ∆N ≥ 0.
N = N
0
t
0
hNi = 1
hti = γ
−1
[0, τ] P
0
(τ)
D(τ) = 1 − P
0
(τ)
N
0
[τ, τ + ∆τ]
dD(τ) = −
dP
0
dτ
dτ = γ exp (−γτ) dτ = w
N
(τ) dτ .
hti =
∞
Z
0
τγ exp(−γτ)dτ =
∞
Z
0
τw
N
(τ) dτ =
1
γ
.
w
N
(τ)
N(t) =
Z
t
0
ds ξ
SN
(s).
ξ
SN
δ
t
i
ξ
SN
(t) =
X
i
δ (t − t
i
) .
t
i
i
N(t)
∆N = 1
ξ
SN
(t) hξ
SN
(t)i = γ > 0
ξ
SN
(t) − γ = η
SN
(τ)
∆N δ t
i
h(ξ
SN
(t
1
) − γ) (ξ
SN
(t
2
) − γ)i = γ δ (t
1
− t
2
) .
RC RCL
˙y(t) = −
1
τ
c
y +
√
2D
τ
c
ξ(t),
ξ(t)
hξ(t) ξ(t + τ) i = δ(τ) y(t)
P (y) =
r
τ
c
2πD
exp
−
τ
c
2D
y
2
,
c
yy
(τ) =
D
τ
c
exp(−
τ
τ
c
),
S
yy
=
2D
1 + ω
2
τ
2
c
.
I
t
∆ ∆
0
τ
∆
∆ → ∆
0
τ
0
∆
∆
0
→ ∆
I
t
d
dt
P (∆, t |I
0
, t
0
) = −
1
τ
∆
P (∆, t |I
0
, t
0
) +
1
τ
∆
0
P
∆
0
, t |I
0
, t
0
,
d
dt
P
∆
0
, t |I
0
, t
0
=
1
τ
∆
P (∆, t |I
0
, t
0
) −
1
τ
∆
P
∆
0
, t |I
0
, t
0
I
0
P
0
(I) = P (i, t → ∞|I
0
, t
0
)
P
0
(∆) =
τ
∆
τ
∆
0
+ τ
∆
, P
0
∆
0
=
τ
∆
0
τ
∆
0
+ τ
∆
.
τ
c
1
τ
c
=
1
τ
∆
+
1
τ
∆
0
,
P (∆, t |∆, t = 0) =
τ
c
2
1
τ
∆
+
1
τ
∆
0
exp
−
t
τ
c
P
∆
0
, t |∆, t = 0
=
τ
c
2
1
τ
∆
0
1 exp
−
t
τ
c
.
I
0
hI
t
i =
∆τ
∆
+ ∆
0
τ
∆
0
τ
∆
+ τ
∆
0
, h(I
t
1
− hI
t
1
i) (I
t
2
hI
t
2
i)i =
D
τ
c
exp
−
|t
1
− t
2
|
τ
c
,
D =
τ
3
c
τ
∆
τ
∆
0
∆ − ∆
0
2
.
∆
0
τ
∆
0
= −∆τ
∆
= const
η
SN
(t) I
t
∆ < 0 τ
∆
|τ
∆
∆|
D = τ
∆
∆
2
.
|τ
∆
∆|
D τ
∆
→ 0 ∆ → −∞
D = const
D (τ
∆
)
(n−2)
γ = ∆τ
∆
p(x, t|x
0
, t
0
)
p(x, t|x
0
, t
0
) x, t
x
0
, t
0
x
t > t
0
x
0
t
0
x
0
t
0
x, t
t − t
0
x
0
x
∂p(x, t|x
0
, t
0
)
∂t
=
∞
X
n=1
(−1)
n
∂
n
∂x
n
K
n
(x, t) p(x, t|x
0
, t
0
),
K
n
(x, t)
K
n
(x, t) =
1
n!
lim
dt→0
1
dt
Z
x
0
− x
n
p(x
0
, t + dt|x, t) dx
0
=
1
n!
lim
dt→0
hdx
n
i
dt
x(t)=x
−
∂p(x, t|x
0
, t
0
)
∂t
0
=
∞
X
n=1
K
n
(x
0
, t
0
)
∂
n
p(x, t|x
0
, t
0
)
∂x
n
0
.
n = 1 n = 2 K
n
= 0
n > 2
∂p(x, t|x
0
, t
0
)
∂t
= L
F
x
p(x, t|x
0
, t
0
)
= −
∂
∂x
K
1
(x, t) p(x, t|x
0
, t
0
) +
∂
2
∂x
2
K
2
(x, t) p(x, t|x
0
, t
0
),
−
∂p(x, t|x
0
, t
0
)
∂t
0
= L
B
x
0
p(x, t|x
0
, t
0
)
= K
1
(x
0
, t
0
)
∂p(x, t|x
0
, t
0
)
∂x
0
+ K
2
(x
0
, t
0
)
∂
2
p(x, t|x
0
, t
0
)
∂x
2
0
.
x
1
, . . . , x
n
p(x
1
, . . . , x
n
, t|x
1,0
, . . . , x
n,0
, t
0
)
∂p
∂t
= −
n
X
i=1
∂
∂x
i
K
i
1
(x
1
, . . . , x
n
, t) p +
n
X
i,j=1
∂
2
∂x
i
∂x
j
K
i,j
2
(x
1
, . . . , x
n
, t) p.
K
i
1
(x
1
, . . . , x
n
, t) = lim
dt→0
hdx
i
i
dt
x
1
(t)=x
1
,...,x
n
(t)=x
n
,
K
i,j
2
(x
1
, . . . , x
n
, t) =
1
2
lim
dt→0
hdx
i
dx
j
i
dt
x
1
(t)=x
1
,...,x
n
(t)=x
n
,
x(t) dx
dt dt
O(dt
2
)
dx
x
dx t
dW
dt
t
x s > t
x(s) dW
dt
t
dW
dt
W (t)
dt → 0
s [t; t + dt]
x(s) s = t
s = t+dt
dx O(dt)
s s = t
s
˙x = f (x, t) +
√
2D g(x, t) ξ(t)
x(t) ξ(t)
hξ(t)i = 0 hξ(t) ξ(t + τ)i = δ(τ)
dt
dx
t
= x(t+dt) − x(t) = f(x(s), s)dt + g(x(s), s)
√
2D dW
dt
,
dW
dt
hdW
2
dt
i = dt t dt