Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
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dv/dt
˙x = v = −
1
γ
∂U (x)
∂x
+
s
2k
B
T
γ
ξ(t) ,
p
2
(x, t|x
0
, t
0
)
∂p
2
(x, t|x
0
, t
0
)
∂t
= −
∂
∂x
−
1
γ
∂U
∂x
p
2
+
k
B
T
γ
∂
2
p
2
∂x
2
.
p
s
j
s
x = x
A
> x
2
= 0
p
s
(x
A
) = 0
j
s
x
1
0 = −
d
dx
−
1
γ
∂U
∂x
p
s
+
k
B
T
γ
d
2
p
s
dx
2
+ j
s
δ(x − x
1
) .
x
1
p
s
(x < x
1
) = p
eq
(x)
x
1
lim
ε→+0
p
s
(x
1
+ ε) = p
eq
(x
1
)
x
1
x
A
x
2
[x
1
, x
A
]
−
1
γ
∂U
∂x
p
s
k
B
T
γ
d
dx
p
s
= j
s
Θ(x − x
1
) .
1/p
eq
(x)
d
dx
p
s
(x)
p
eq
(x)
= −j
s
1
p
eq
(x)
γ
k
B
T
Θ(x − x
1
).
p
s
(x) = −j
s
γ
k
B
T
p
eq
(x)
Z
x
dx
0
1
p
eq
(x
0
)
Θ(x
0
− x
1
) .
p
s
1 = j
s
γ
k
B
T
Z
x
A
x
1
dx
0
1
p
eq
(x
0
)
.
r
K
=
k
B
T
γ
R
0
−∞
dx
0
p
s
(x
0
)
1
R
x
A
x
1
dx
0
(p
eq
(x
0
))
−1
.
k
TP
x = x
1
p
s
p
eq
x
1
x
x
2
= 0
x
2
U(x) = U(x
b
) −(m/2)ω
2
b
(x −x
b
)
2
±∞
r
K
=
ω
b
γ
ω
0
2π
exp
−
∆U
k
B
T
.
∝ 1/γ γ >> ω
b
x(t)
x
0
t
0
= 0
x
0
∈ (a, b)
P
a,b
(t, x
0
) =
b
Z
a
p
2
(x, t|x
0
) dx
x(t)
(a, b) t > 0
x = a x = b
p
2
(x, t|a) = p
2
(x, t|b) = 0 , x ∈ (a, b).
P
a,b
(t, a) = P
a,b
(t, b) = 0.
j(x, t)
j(x = a, t) = 0
x t > t
0
Z
dx p
2
(y, t
y
|x, t
x
) L
F
x
p
2
(x, t
x
|x
0
, t
0
)
=
Z
dx p
2
(x, t
x
|x
0
, t
0
) L
B
x
p
2
(y, t
y
|x, t
x
),
−∞
b
p
2
(y, t
y
|x, t
x
) j(x, t) = p
2
(x, t
x
|x
0
, t
0
)
∂
∂x
p
2
(y, t
y
, |x, t
x
),
x a j(a, t) = 0
∂p
2
(x, t|x
0
, t
0
)
∂x
0
x
0
=a
= 0 .
∂P
a,b
(t, x
0
)
∂x
0
x
0
=a
= 0 .
P
a,b
(t, x
0
)
x
0
P
a,b
τ
lim
τ→0
1
τ
(1 − P
a,b
(τ, x
0
) ) = 0, x
0
6= a, b .
−
∂
∂t
0
P
a,b
=
∂
∂t
P
a,b
(t, x
0
) = K
1
(x
0
)
∂
∂x
0
P
a,b
+ K
2
(x
0
)
∂
2
∂x
2
0
P
a,b
.
t
P
a,b
(t, x
0
)
P
a,b
(t → ∞, x
0
) = 0
W
a,b
(t, x
0
) = 1 − P
a,b
(t, x
0
)
W
a,b
(t → ∞, x
0
) = W
as
a,b
((x
0
) = 1
x
0
b
a b
W
as
a,b
(b) = 1 a
b
W
as
a,b
(a) = 0
∂W
as
a,b
(x
0
)/∂t = 0
W
as
a,b
(x
0
) =
x
0
R
a
dx exp
−
R
x
K
1
K
2
dx
0
b
R
a
dx exp
−
R
x
K
1
K
2
dx
0
.
t
W
a,b
(t, x
0
)
[a, b]
t
0
t
w
a,b
(x
0
, t) ∆t =
∂W
a,b
(t, x
0
)
∂t
∆t = −
∂P
a,b
∂t
∆t .
∂
∂t
w
a,b
= K
1
(x
0
)
∂
∂x
0
w
a,b
+ K
2
(x
0
)
∂
2
∂x
2
0
w
a,b
,
w
a,b
(x
0
, t = 0) = 0 , x
0
∈ (a, b) , w
a,b
(a, t) = w
a,b
(b, t) = δ(t),
a
∂w
a,b
(x
0
, t)
∂x
0
(x
0
=a)
= 0.
T
a,b
(x
0
) =
∞
Z
0
tw
a,b
(x
0
, t) dt.
T
a,b
(x
0
) =
∞
Z
0
P
a,b
(t, x
0
) dt.
[a, b]
hT
a,b
i
x
0
=
∞
Z
0
P
a,b
(t, x
0
) P (x
0
) dt dx
0
.
T
a,b
(x
0
)
t
∞
Z
0
∂
∂t
P
a,b
dt = P
a,b
(∞, x
0
) − P
a,b
(0, x
0
) = −1 .
−1 = K
1
(x
0
)
∂
∂x
0
T
a,b
(x
0
) + K
2
(x
0
)
∂
2
∂x
2
0
T
a,b
(x
0
) .
[a, b]
T
a,b
(a) = T
a,b
(b) = 0 .
x = a
∂T
a,b
(x
0
)
∂x
0
x
0
=a
= 0.
T
(n)
a,b
= −
∞
Z
0
t
n
∂
∂T
W
a,b
(t, x
0
, t
0
) dt.
T
(0)
a,b
= 1 T
a,b
=
T
(1)
a,b
−nT
(n−1)
a,b
= K
1
(x
0
)
∂
∂x
0
T
(n)
a,b
(x
0
) + K
2
(x
0
)
∂
2
∂x
2
0
T
(n)
a,b
(x
0
)
T
(1)
a,b
u
(n)
a,b
=
dT
(n)
a,b
dx
0
,
a < b
u
(n)
a,b
(x
0
) = exp ( − Φ(x
0
))
C
n
− n
x
Z
a
T
(n−1)
a,b
(x
0
)
K
2
(x
0
)
exp (Φ(x
0
))dx
0
,
C
n
Φ(x) =
Z
x
K
1
(y)
K
2
(y)
dy .
x = a
b u
(n)
a,b
(a) = 0 T
(n)
a,b
(b) = 0
T
(n)
a,b
(x
0
) = n
b
Z
x
0
exp ( − Φ(x))
Z
x
a
T
(n−1)
a,b
K
2
(x
0
)
exp (Φ(x
0
))dx
0
dx .
T
(1)
a,b
(x
0
) =
b
Z
x
0
dx exp ( − Φ(x))
x
R
a
dx
0
exp (Φ(x
0
))
K
2
(x
0
)
.
a b
T
(n)
a,b
(x
0
) = C
n
x
0
Z
a
exp ( − Φ(x))dx −
n
x
0
Z
a
exp ( − Φ(x))
Z
x
a
T
(n−1)
a,b
K
2
(x
0
)
exp (Φ(x
0
))dx
0
dx,
C
n
T
(n)
a,b
(b) = 0
(−∞, b) x = b
a
−∞
x = ∞
∆U k
B
T
x = x
1
x
0
x
2
T
−∞,b
(x
0
) = Q(b)
1
r
K
,
Q(b)
b x
2
b = x
2
Q = 1/2
