Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
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˙x = −
∂V (x, t)
∂x
+
√
2Dξ(t) .
V (x, y) = U (x) + W (x, t)
U(x + L) = U (x)
F
0
W = −F
0
x y(t) y(t)
ξ(t)
F
0
= 0
F
0
0.0 1.0 2.0
x
-0.20
-0.10
0.00
0.10
0.20
V(x)
U(x) L = 1
U(x)
τ
ξ(t)
U(x)
y(t)
U(x) 6= U(−x)
x
U(x) y(t)
W (x, t) = −F
0
y(t)
|∆U|
x
V(x)
∆U
right
∆U
left
x
V(x)
∆U
right
∆U
left
V (x, t) = U (x) − F
0
y(t) y
U(x)
y(t)
y(t)
˙x = −
∂
∂x
U(x) y(t) +
√
2D ξ(t),
y(t) = 1, 0
y(t)
D
D 6= 0
x
V(x)
potential = on
P(x,t)
x
V(x)
potential = off
P(x,t)
x
V(x)
potential = on
P(x,t)
y(t)
t → ∞
∂
∂t
P (x, t) = −
∂
∂x
−
∂V (x, t)
∂x
P (x, t)
+ D
∂
2
∂x
2
P (x, t).
U(x) = U(x+L)
P
asy
(x, t)
∂P
asy
/∂t = 0
[0, L]
y(t)
J(x, t) = −
∂V (x, t)
∂x
P (x, t) − D
∂
∂x
P (x, t),
x J(x, t) → J
asy
(t)
x
h˙x(t)i
L
=
L
Z
0
J(x, t) dx .
h˙xi
L
= LJ
asy
(t)
y(t)
y(t)
P
asy
(x, y) = P
0
(x, y)
C(y) −
J
asy
(y)
D
x
Z
0
dx
0
1
P
0
(x
0
, y)
P
0
(x, y) = exp[−Φ(x, y)] Φ(x, t) = V (x, y)/D
J
asy
(y) C(y)
y
J
asy
(y) = C(y) D
1 −
P
0
(0, y)
P
0
(L, y)
Z
L
0
dx
1
P
0
(x, y)
−1
.
C(y)
C(y) =
Z
L
0
dx P
0
(x, y)
Z
x+L
x
dx
0
1
P
0
(x
0
, y)
−1
Z
L
0
dx
1
P
0
(x, y)
.
h˙x(y)i
L
L D
=
1 −
P (0, y)
P (L, y)
Z
L
0
dx P
0
(x, y)
Z
x+L
x
dx
0
1
P
0
(x
0
, y)
−1
.
hh˙xi
L
i
y
=
Z
dy h˙x(y)i
L
ρ(y),
ρ(y)
y(t) = A cos(ωt + ϕ
0
)
hh˙xi
L
i
y
=
1
2 π
Z
2π
0
dϕ
0
h˙x(A cos(ωt + ϕ
0
))i
L
,
ω
0
0.2
0.4
0.6
0.8
1
-1 0 1 2 3 4
U
x
f(x) = −
∂
∂x
U(x) =
(
a
1
,
nL < x < nL + x
1
,
−a
2
, nL − x
2
< x < nL,
A U
0
=
a
1
x
1
= a
2
x
2
x
1
+ x
2
= L
max y(t)
a
1
a
2
a
2
− a
1
min
1,
D
U
0
,
h˙xi ≈
U
2
0
(a
2
− a
1
)A
2
8D
2
a
1
a
2
sinh
2
(U
0
/2D)
U
2
0
4D
2
sinh
2
(U
0
/2D)
+
U
0
2D tanh(U
0
/2D)
− 2
,
h˙xi/A
2
D/U
0
A
A < min(a
1
, a
2
) A > min(a
1
, a
2
)
D/U
0
A