Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
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p(x, t)
x
1
x
p
n
t
i
→ t
i
+τ i = 1, 2, . . . , n
t
0
p
2
(x
2
, t + τ|x
1
, t) = p
2
(x
2
, τ|x
1
).
τ = t
2
− t
1
x
0
τ → ∞
p
s
(x) = lim
τ→∞
p
2
(x, τ|x
0
).
hx
ξ
(t)i
p(x, t − t
0
) = p(x, t + T − t
0
)
ϕ
0
= 2πt
0
/T
[0, 2π]
T
hx
ξ
(t)i
x
ξ
(t
0
)=x
0
=
Z
dx x p
2
(x, t|x
0
, t
0
).
t
0
→ −∞
hx
ξ
(t)i
asy
= lim
t
0
→−∞
hx
ξ
(t)i
x
ξ
(t
0
)=x
0
p
2
(x, t|x
0
, t
0
) →
p(x, t)
hxi
asy
= 0
x
ξ
(t
0
) = x
0
hx
ξ
(t) x
ξ
(t+τ) i
x
ξ
(t
0
)=x
0
=
Z
dx
1
dx
2
x
1
x
2
p
2
(x
2
, t+τ|x
1
, t) p
2
(x
1
, t|x
0
, t
0
).
x(t)
Z
dx
2
hx
ξ
(t + τ)i
x
ξ
(t)=x
1
x
1
p
2
(x
1
, t|x
0
, t
0
).
t
0
→ −∞
hx
ξ
(t) x
ξ
(t + τ) i
asy
=
Z
dx
1
dx
2
x
1
x
2
p
2
(x
1
, t, x
2
, t + τ).
τ = 0
hx
2
(t)i τ → ∞
hx
ξ
(t) x
ξ
(t + τ ) i
asy
= hx
ξ
(t + τ)i
asy
hx
ξ
(t)i
asy
.
τ
c
x,x
(τ) =
Z
dx
1
dx
2
x
1
x
2
p
2
(x
1
, τ|x
2
) p(x
2
).
t → t − τ c
x,x
τ c
x,x
(τ) = c
x,x
(−τ)
3
τ
c
=
1
c
x,x
(0)
Z
∞
0
dt |c
x,x
(t)|.
{x(t), y(t)}
p
x,y
(x, t; y, t + τ)
c
x,y
(τ) = c
y,x
(−τ)
|c
x,y
(τ)|
2
≤ hx
2
ihy
2
i
|c
x,x
| ≤ hx
2
i
G
x,x
(ω) =
Z
∞
−∞
dτ c
x,x
(τ) e
−iωτ
= 2
Z
∞
0
dτ c
x,x
(τ) cos(ωτ),
G
x,x
≥ 0
G
x,y
(ω) =
Z
∞
−∞
dτ c
x,y
(τ) exp −iωτ.
G
x,x
(ω = 0) =
Z
∞
−∞
dτ c
x,x
(τ),
c
x,x
(τ = 0) = hx
2
i =
1
2π
Z
∞
−∞
dω G
x,x
(ω).
x(t)
˙x(t) =
dx(t)
dt
= lim
ε→0
x(t + ε) − x(t)
ε
.
lim
ε→0
D
x(t + ε) − x(t)
ε
− ˙x(t)
2
E
= 0.
c
x,x
(t
1
, t
2
)
t
1
t
2
c
x,x
(τ)
˙x
ξ
(t)
h˙x(t
1
) ˙x(t
2
)i =
∂
2
∂t
1
∂t
2
hx(t
1
)x(t
2
)i,
x
ξ
˙x
ξ
G
˙x, ˙x
(ω) = ω
2
G
x,x
(ω).
hx(t
1
) ˙x(t
2
)i =
∂
∂t
2
hx(t
1
)x(t
2
)i
G
x, ˙x
(ω) = iω G
x,x
(ω)
T
n
T = [t
0
, t]
n → ∞ T
I
f
ξ
(t, t
0
) =
Z
t
t
0
ds f(s) x
ξ
(s) = lim
n→∞
q.m.
n−1
X
k=0
f(t
k,n
) x
ξ
(t
k,n
)
t
(n)
k+1
− t
(n)
k
,
t
(n)
0
= t
0
t
(n)
n
= t
t
(n)
k
≤ t
k,n
≤ t
(n)
k+1
q.m.
x
ξ
(t)
T
n
h
I
f
ξ
(t, t
0
)
2
i =
Z
t
t
0
Z
t
t
0
ds
1
ds
2
f(s
1
) f(s
2
)hx
ξ
(s
1
)x
ξ
(s
2
)i.
x
ξ
(t
k,n
)
t
(n)
k+1
− t
(n)
k
∆I
t
k
= I
f=1
ξ
(t
k+1
, t
k
)
I
f=1
ξ
(t, t
0
)
f(t) t
k,n
= t
(n)
k
I
f
ξ
(t, t
0
) =
Z
t
t
0
dI
s
f(s) = lim
n→∞
q.m.
n−1
X
k=0
∆I
t
(n)
k
f(t
(n)
k
).
f ξ
x
ξ
(t)
x
ξ
(t)
x
ξ
(t) x
T
ξ
(t) = x
ξ
(t)
|t| ≤ T x
T
ξ
(t) = 0
x
T
ξ
(ω) =
Z
T/2
−T/2
dt x
ξ
(t) exp(−iωt).
T → ∞
x
ξ
(t)
[ω, ω + ∆ω) ω ≥ 0
∆ω = 2π/T
P
T
ξ
(ω) =
2
T
|x
T
ξ
(ω)|
2
,
T → ∞
P
T
(ω) = hP
T
ξ
(ω)i =
2
T
Z
T/2
−T/2
dt
1
Z
T/2
−T/2
dt
2
hx
ξ
(t
1
) x
ξ
(t
2
)i exp(−iω(t
1
−t
2
)).
P
T
(ω) P
T
ξ
(ω)
T → ∞
P
T
(ω) = 2
Z
T
−T
dτ
1 −
|τ|
T
c
xx
(τ) exp(−iωτ).
∆ω
P
T
(ω)
x
ξ
(t)
P (ω) = lim
T →∞
P
T
(ω) = 2
Z
∞
−∞
dτ c
x,x
(τ) exp(−iωτ).
P (ω) = 2G
x,x
(ω)
G
x,x
(ω)
x
ξ
(t) 2
x(t) = [x
1
(t), . . . , x
n
(t)]
˙x
i
(t) = f
i
(x
1
, . . . , x
n
, t) + g
i
(x
1
, . . . , x
n
, ξ
1
(t) . . . ξ
m
(t), t) .
f(x, t) = [f
i
(x, t), . . . , f
n
(x, t)] g(x, ξ, t) =
[g
i
(x, ξ, t), . . . , g
n
(x, ξ, t)] g
i
(x, ξ = 0, t) = 0
ξ(t) = [ξ
1
(t), . . . , ξ
n
(t)]
hξ
i
(t)i = 0
f(x)
˙x
i
(t) = f
i
(x, t) +
m
X
j=1
g
i,j
(x, t) ξ
j
(t).
j g
i,j
= const
ξ
j
x
i
(t)
x
i
(t) g
i,j
ξ(t)
ξ(t)
p
n
(x
1
, t
1
, . . . , x
n
, t
n
) =
=
1
p
(2π)
n
Det[R(t
i
, t
j
)]
exp
−
1
2
n
X
i,j=1
b
i,j
(x
i
− a(t
i
))(x
j
− a(t
j
))
,
a(t
i
) = hx(t
i
)i
R(t
i
, t
j
)
R
i,j
= h(x(t
i
) − a(t
i
)) (x(t
j
) − a(t
j
))i = hx(t
i
) x(t
j
)i − a(t
i
) a(t
j
),
b
i,j
R
−1
W
i
t
i
D
p
n
(W
1
, t
1
, . . . , W
n
, t
n
) =
n
Y
i=1
1
p
4πD ·(t
i
− t
i−1
)
exp
−
1
4D
(W
i
− W
i−1
)
2
t
i
− t
i−1
,
W
0
= 0 t
0
= 0 W (t)
i ≥ 2
p (W
i
, t
i
|W
i−1
, t
i−1
) =
1
p
4πD (t
i
− t
i−1
)
exp
−
(W
i
− W
i−1
)
2
4D (t
i
t
i−1
)
!
.
p(W
0
, t
0
=
0) = δ(W
0
) t ≥ 0
p
1
(W, t) =
Z
dW
0
p (W, t|W
0
, t
0
) δ(W
0
) =
1
√
4πDt
exp
−
W
2
4Dt
,
hW (t)i = 0
hW (t
1
) W (t
2
)i = 2D min (t
1
, t
2
) ,
t
1
= t
2
= t hW (t)
2
i = 2Dt
D