Анищенко В.С. Нелинейные эффекты в хаотических и стохастических системах
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k
+
k
−
x
y
x y
0
0.5
1
1.5
2
x
0
0.5
1
1.5
y
0.5
1
V(x,y)
V (x, y) A = 0.3 L = 1 F = 0 k = 0.8
V (x, y)
x ∈ [0, 1] y ∈ [0, L]
V (x, y) = V
0
(x, y) − F y = {1 − A sin(2πy/L)}U(x) −F y.
U(x)
k ∈ (0, 1)
k = 1/2 U(x) U(x)
y A
˙x = f
x
(x, y) +
√
2Dξ
x
(t), ˙y = f
y
(x, y) +
√
2Dξ
y
(t).
ξ
i
f
i
(x, y)
p(x, y, t)
p
s
(x, y) = p
0
(x + 1, y) , p
s
(x, y) = p
s
(x, y + L)
p
s
(x, y)
J
x
(x, y) = f
x
(x, y)p
s
(x, y) − D
x
∂
∂x
p
s
(x, y),
J
y
(x, y) = f
y
(x, y)P
0
(x, y) − D
y
∂
∂y
P
0
(x, y).
hv
x
i
x
hv
x
i = J
tot
x
=
Z
1
0
Z
L
0
J
x
(x, y)dx dy.
F 6= 0
y
x
[0, 1] × [0, L]
x
0 1
0.5
1
1.5
x
y
0
0.15
0.3
0.45
0.6
0.75
0.9
1.05
1.2
1.35
1.5
1.65
1.8
0 1
0.5
1
1.5
0 1
0.5
1
1.5
0.7
x
y
0
0.15
0.3
0.45
0.6
0.75
0.9
1.05
1.2
1.35
1.5
1.65
1.8
0 1
0.5
1
1.5
[J
x
(x, y), J
y
(x, y)]
D = 0.2 A = 0.8 L = 1
F = 5 k = 0.8 D
x
= 0.3
D
y
= 0.01 F = 1 L = 1 k = 0.8 A = 0.8
x = 0.8 y = 0.75
V
0
(x, y) y
F → −F F J
tot
x
(F )
O(∝ F
2
) F
J
tot
x
F ' 5.5
y F
y
˙y = F
x y
D
x
D
y
D
y
= 0 D
x
= D
y
D
x
D
y
= 0.01
F ' 5 F
10 D
x
= 0.3
D
y
= 0.01
F
y
V
0
0 5 10 15
F
−0.08
−0.06
−0.04
−0.02
0.00
<v
x
>
0.0 5.0 10.0
F
−0.11
−0.06
−0.01
0.04
J
x
Flashing 1D
2D, Dy=0.01
2D, Dy=Dx
x
F
D
x
= 0.3
k = 0.8 A = 0.8
y
y
D
x
D
y
F
F
F
W
σ
(i → i ±1)
σ = ±1
i = 1, 2, 3
i → i + 1 1
1
W
σ
(1 → 2) = W
σ
(2 → 3) = k
σ
W
σ
(3 → 4) = k
−2
σ
= 1/[W
σ
(1 → 2) × W
σ
(2 → 3)],
W
σ
(i → i − 1) =
1
W
σ
(i − 1 → i)
.
σ = 1
σ = −1
γ
σ = 1
k < 1 σ = −1 k = 1
k γ
L = 1
σ(t)
γ
P
σ
(j, t)
j σ
∂P
σ
(j, t)
∂t
= W
σ
(j + 1 → j)P
σ
(j + 1, t) + W
σ
(j − 1 → j)P
σ
(j − 1, t)
−[W
σ
(j → j + 1) + W
σ
(j → j − 1)] P
σ
(j, t) − γP
σ
(j, t) + γP
−σ
(j, t) .
W
σ
(i → j)
k
σ
= exp
−
σ(t) + 1
2
∆U/D
.
γ
i
γ
i
-2
∆U
∆U
23
∆U
U U U
1 2 3
12
31
U∆
0
1 1 1
k
k
k
-1
-1
k
2
k
k
σ = +1 σ = −1
1 ↔ 3
˙
P = W P ,
P W
W
˙
P = 0
J(i, t) i i + 1
J(i, t) = [ W
+1
(i → i + 1) P
+1
(i, t) − W
+1
(i + 1 → i) P
+1
(i + 1, t) ]
+[ W
−1
(i → i + 1) P
−1
(i, t) − W
−1
(i + 1 → i) P
−1
(i + 1, t) ] .
t → ∞ J
0
J
0
= J(1) = J(2) = J(3)
γ k
0 5 10 15 20
γ
−0.010
−0.008
−0.006
−0.004
−0.002
0.000
v
γ
k = 0.1 k = 0.2
k = 0.4
ω ≈ 2 k ≈ 0.2
σ = −1
k = 1
σ = +1
k → 0
D
vt
t
v D
P e =
|v|L
D
.
L