Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
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f g
dx
t
x(s)
[t, t + dt]
ξ(t)
dW
t
q ∈ [0, 1]
s = t + qdt
f g
dx
t
= f
t
dt + q
∂f
∂x
t
dx
t
dt + g
t
√
2D dW
dt
+ q
∂g
∂x
t
dx
t
√
2D dW
dt
,
t
dx
t
dW
dt
dx
t
dx
t
= f
t
dt + g
t
√
2D dW
dt
+ q
∂g
∂x
t
g
t
2 D dW
2
dt
+
q
∂f
∂x
t
g
t
dt + f
t
g
t
+ q
∂g
∂x
t
f
t
√
2D dW
dt
dt.
2DdW
2
dt
dx
t
dt → 0
2D dt
2DdW
2
dt
2Ddt
dt → 0
O(dt)
dx
t
= f(x, t) dt + 2D dt q
∂g(x, t)
∂x
g(x, t) +
√
2Dg(x, t) dW
dt
.
x(t
0
)
t
0
< t
dW
dt
dW
dt
x(t) x
0
x
0
dt → 0
dW
dt
dx
0 t
x(t) = x(0) +
Z
t
0
ds
f(x, s) + 2Dq
∂g(x, s)
∂x
g(x, s)
+
√
2D
Z
t
0
dW
s
g(x, s).
W
t
x(t)
x(t)
K
1
(x, t) =
dhxi
dt
= f (x, t) + 2 D q
∂g(x, t)
∂x
g(x, t)
K
2
(x, t) =
1
2
dhx
2
i
dt
= D g
2
(x, t).
p
2
=
p
2
(x, t|x
0
, t
0
)
∂
∂t
p
2
= −
∂
∂x
f(x, t) + 2 D q
∂g(x, t)
∂x
g(x, t)
p
2
+ D
∂
2
∂x
2
g
2
(x, t) p
2
.
˙x
i
= f
i
(x
1
, . . . , x
n
) +
m
X
j=1
g
i,j
(x
1
, . . . , x
n
) ξ
j
(t),
m
hξ
i
(t) ξ
j
(t + τ)i = 2 D
i,j
δ(τ).
dx
i
K
i
1
= f
i
+ 2 q
X
j,k,l
D
j,l
∂g
i,j
∂x
k
g
k,l
,
K
i,j
2
=
X
k,l
D
k,l
g
i,k
g
j,l
.
q
dt → 0
q = 1/2
q
q
q = 0
q = 1
q = 0
dx
t
dW
t
t x(t)
q = 1/2
I
t
I
t
= ±∆
1/γ
˙x = f(x) + g(x) I
t
.
I
t
= const.
K
1
(x, ±∆) = f (x) ± g(x) ∆ , K
2
= 0 .
I
t
p
∆
(x)
p
−∆
(x)
∂
∂t
p
±∆
(x) = −
∂
∂x
( f(x) ± ∆ g(x)) p
±∆
(x) + γ p
∓∆
(x) − γ p
±∆
(x).
p
±∆
p(x, t) = p
∆
(x, t) +
p
−∆
(x, t) Q(x, t) = p
∆
(x, t) − p
−∆
(x, t)
∂
∂t
p(x, t) = −
∂
∂x
( f p(x, t) + ∆ g(x) Q(x, t)).
Q p
Q(x, t) = −∆
Z
t
−∞
ds exp
−
2γ +
∂
∂x
f
(t − s)
∂
∂x
g(x) p(x, s).
Q(−∞) = 0
∂
∂x
g(x) p(x, s)
δ
∂
∂t
p(x, t) = −
∂
∂x
f(x) p(x, t) + D
∂
∂x
g(x)
∂
∂x
g(x) p(x, t)
,
D = ∆
2
/(2γ)
q
q = 1/2
∂p/∂t = 0
p
s
(x) = N
g(x)
∆
2
g
2
(x) − f
2
(x)
exp
−2γ
Z
x
dx
0
f
f
2
− ∆
2
g
2
.
p
s
(x)
∂p
2
/∂t = 0
j(x, t) = f (x) p(x, t|x
0
, t
0
) − D g(x)
∂
∂x
(g(x) p(x, t|x
0
, t
0
)) ,
g = 1
j
s
= f (x) p
s
− D
∂
∂x
p
s
(x) = const.
j
s
6= 0
j
s
= 0
x → ∞
p
s
(x) = N exp
−
1
D
U(x)
.
U(x) =
−
R
x
dx
0
f(x
0
) N
N
−1
=
Z
X
dx
0
exp
−
1
D
U(x
0
)
,
X x(t)
x
e
dp
s
/dx =
0
f(x
0
) =
0 f
0
(x
0
) < 0
p
s
f
0
(x
0
) > 0
p
s
(x
1
, . . . , x
n
) = N exp
−
1
D
U(x
1
, . . . , x
n
)
,
˙x
i
= −
∂
∂x
i
U(x
1
, . . . , x
n
) +
√
2D ξ
i
(t),
q = 1/2
j
s
p
s
(x) = N
1
|g(x)|
exp
1
D
Z
x
dx
0
f(x
0
)
g
2
(x
0
)
.
x
e
p
s
(x)
g(x)
h(x
e
, D) = f (x
e
) − D g(x
e
)
∂g(x
e
)
∂x
e
= 0.
D
h
0
(x
e
) < 0 h
0
(x
e
) > 0
h
0
(x
e
) = 0
P f
0
(x
0
)
x
e
P
P
D
D
c
∆a = a − a
c
˙x = f(x, a) D
c
∝ ∆a
D
hxi
s
= x
m
K
1
(x
m
) = f (x
m
) + D g(x
m
)
∂g(x
m
)
∂x
m
= 0,
˙x = v , ˙v = −γ(x, v) v −
∂
∂x
U(x) +
√
2Dξ(t),
x v
U(x) γ(x, v) γ = const.
U
p
2
= p
2
(x, v, t|x
0
, v
0
, t
0
)
∂
∂t
p
2
+ v
∂
∂x
p
2
−
∂U
∂x
∂
∂v
p
2
=
∂
∂v
γ(x, v) v p
2
+ D
∂
2
∂v
2
p
2
.
p
2
p
2
→ p
s
(x, v) ∂p
s
/∂t = 0
Φ
0
= ω
0
t ω
0
A ϕ Φ = Φ
0
+ ϕ
U(x) = ω
2
0
x
2
/2
x = A cos(ω
0
t + ϕ) , v = −ω
0
A sin(ω
0
t + ϕ).
A ϕ
˙
A =
sin(ω
0
t + ϕ)
ω
0
γ(x, v) v + y
A
, ˙ϕ =
cos(ω
0
t + ϕ)
Aω
0
γ(x, v) v + y
ϕ
,
y
A
(t) y
ϕ
(t)
y
A
= −
sin(ω
0
t + ϕ)
ω
0
√
2Dξ(t) , y
ϕ
= −
cos(ω
0
t + ϕ)
Aω
0
√
2Dξ(t) .
T = 2π/ω
0
A ϕ
T
f
A
(a, ϕ) f
ϕ
(a, ϕ)
y
A
y
φ
K
A
1
∝ D
cos
2
(ω
0
t + ϕ)
ω
2
0
, K
ϕ
1
∝ −D
cos(ω
0
t + ϕ) sin(ω
0
t + ϕ)
A
2
ω
2
0
.
T K
A
1
D/(2ω
2
0
)
K
A,ϕ
2
= 0
K
A,A
2
=
D
2ω
2
0
, K
ϕ,ϕ
2
=
D
2ω
2
0
A
2
.
p
2
(A, ϕ, t|A
0
, ϕ
0
, t
0
∂
∂t
p
2
= −
∂
∂A
f
A
+
D
2 ω
2
0
p
2
−
∂
∂ϕ
f
ϕ
p
2
+
D
2ω
2
0
∂
2
p
2
∂A
2
+
D
2ω
2
0
A
2
∂
2
p
2
∂ϕ
2
.
γ = const
ω
0
p
s
(A, ϕ) =
1
2π
A
σ
2
exp
−
A
2
2σ
2
,
σ
2
= D/ω
2
0
γ