Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
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γ = const.
v(x, E) = ±
p
2[E − U(x)],
˙x = v(x, E) ,
˙
E = −γv
2
(x, E) +
√
2D v(x, E) ξ(t),
v
y
E,x
=
√
2D v(x, E) ξ(t)
K
x
1
= v(x, E) , K
E
1
= −γv
2
(x, E) + D , K
E,E
2
= D v
2
(x, E).
p
2
=
p
2
(x, E, t|x
0
, E
0
, t
0
)
∂p
2
∂t
= −
∂
∂x
v(x, E) p
2
+
∂
∂E
γv
2
(x, E) − D
p
2
+ D
∂
2
∂E
2
v
2
(x, E)p
2
.
p
s
(x, E) =
N
|v(x, E)|
exp
−
γE
D
.
x E ≥ U(x)
p
s
(E) ∝ T (E) exp
−
γ E
D
,
T (E)
E
d
p
s
(E) ∝ E
d
2
−1
exp
−
γ E
D
.
dx
dt
= v , m
dv
dt
= −γv −
∂U
∂x
,
U(x)
U(x) = −
a
2
x
2
+
b
4
x
4
,
f(x) = −
∂U (x)
∂x
= ax − bx
3
.
v
0
= 0 , x
1,3
= ±
r
a
b
, a , b > 0 .
v
0
= 0 , x
2
= 0.
γ < 8a
γ → 0
γ → ∞
m → 0 v = 0
v(t)
x(t)
dx
dt
=
a
γ
x −
b
γ
x
3
.
γ > 0
d
dt
E =
d
dt
m
2
v
2
+ U(x)
= −γv
2
≤ 0,
v
dx
dt
= v , m
dv
dt
= −γv −
∂U (x)
∂x
+
√
2D ξ(t).
∂
∂t
p
2
(x, v, t|x
0
, v
0
, t
0
) + v
∂p
2
∂x
−
1
m
∂U (x)
∂x
∂p
2
∂v
=
γ
m
∂
∂v
v p
2
+
D
m
2
∂
2
p
2
∂v
2
.
p
s
(x, v)
p
s
= p
s
(E(x, v))
p
s
(x, v) = N
−1
exp
−
γm
2D
v
2
−
γU (x)
D
.
D = γk
B
T
x
2
= v
0
= 0
v
x
x
2
∆U = U(x = x
2
) −
U(x = x
1
) > 0
p
s
(x = x
2
, v = v
0
)
p
s
(x = x
1
, v = v
0
)
= exp
−
γ∆U
D
.
D
D
1
˙x = v , ˙v = ε v (α − βx
2
− δv
2
) − ω
2
0
x +
√
2D ξ(t) ,
βδ > 0 ε = 1 α
∝
√
α
ε ·α 1
ω
0
˙
A = ε
A
2
α − b A
2
+ y
A
, ˙ϕ = y
ϕ
,
x
v
p
s
(x,v)
p
s
(x, v)
b = (β + 3δω
2
0
)/4 y
A
y
φ
p
s
(A, ϕ) = N exp
2 ε ω
2
0
D
α
2
A
2
−
b
4
A
4
.
α, ε >
0
γ(x, v) =
γ(E) β = δω
2
0
p
2
(E, t|E
0
, t
0
) =
Z
2π
0
dϕ
Z
∞
0
d δ
E −
A
2
ω
2
0
2
p
2
(A, ϕ|A
0
, ϕ
0
),
∂
∂t
p
2
(E, t|E
0
, t
0
) = −ε
∂
∂E
2 E
α −
b
2ω
2
0
E
p
2
+ D
∂
∂E
E
∂
∂E
p
2
.
p
s
(E) = N exp
2 ε
D
E
α −
b E
ω
2
0
.
ε < 0 E
s
= A
2
s
ω
2
0
/2 → ∞
D/(2|ε|α)
hE
n
i = n!hEi
n
h∆E
2
i
hEi
2
= 1.
h∆E
2
i
hEi
2
=
π
2
− 1 < 1
E
s
=
αω
2
0
/2b
p
s
(E) = N exp
(
−
2 b ε
D ω
2
0
E −
ω
2
0
α
2 b
2
)
h∆E
2
i
hEi
2
=
D b
ε ω
2
0
α
2
.
εω
2
0
α
2
/Db
E
s
= ω
2
0
α/2 b
D → 0
δ
j
s
x, v
r
r =
j
s
n
0
,
n
0
∆U
γ
W
Ω
(t|x
0
, t
0
) =
Z
Ω
dx p
2
(x, t|x
0
, t
0
)
t t
0
x
0
Ω
W
Ω
w
Ω
= dW
Ω
/dt t −t
0
p
2
(x, t|x
0
, t
0
)
hT
n
Ω
i
w
Ω
(t − t
0
|x
0
)
hT i
Ω
∝
1
r
.
x
1
D = γ k
B
T
p
eq
= N exp
−
v
2
2 k
B
T
−
U(x)
k
B
T
.
p
s
(x, v)
v > 0 x = 0
j
eq
=
Z
∞
0
dv v p
eq
(x = 0 , v) .
k
eq
n
0
=
Z
0
−∞
dx
Z
∞
−∞
dv p
eq
(x , v)
x
1
k
B
T
∆U
x
1
−∞
∞
U(x) = U(x
1
) +
1
2
∂
2
U(x)
∂x
2
|
x=x
1
(x − x
1
)
2
+ o
(x − x
1
)
3
.
ω
2
0
= 2a
r
eq
=
ω
0
2π
exp
−
∆U
k
B
T
.
c
v,v
(τ) = −d
2
c
x,x
(τ)/dτ
2
∆U
r
eq
Rice
=
1
2π
s
c
v,v
(0)
c
x,x
(0)
exp
−
∆U
c
v,v
(0)
.
τ = 0
γ
γ ω
b
ω
2
b
ω
2
b
= (1/m)|∂
2
U(x)/∂x
2
|
x
b
= x
2
= 0 ω
2
b
= a/m