1 Introduction to Nonequilibrium Statistical Physics and Its Foundations
21
formally as contributions from a probability drift and from the diffusion of probability, with
J (z, t) = D
(1)
(z, t)W (z, t) −
∂
∂z
D
(2
(z, t)W (z, t)
(1.62)
representing a probability current. The Fokker-Planck equation is thus a balance
equation, analogous to those of conserved quantities in thermodynamcs, since probability is conserved.
For more general cases, in which the number of stochastic variables is N , we can
merge the equations of motion in the following system:
˙
z = h + G
(1.63)
where
˙
z =
⎛
⎜
⎜
⎜
⎝
˙
z 1
˙
z 2
. . .
˙
z N
⎞
⎟
⎟
⎟
⎠
, h =
⎛
⎜
⎜
⎜
⎝
h 1
h 2
. . .
h N
⎞
⎟
⎟
⎟
⎠
, G =
⎛
⎜
⎜
⎜
⎝
g 1,1 g 1,2 · · · g 1,N
g 2,1 g 2,2 · · · g 2,N
. . .
. . .
. . .
. . .
g N ,1 g N ,2 · · · g N ,N
⎞
⎟
⎟
⎟
⎠
, =
⎛
⎜
⎜
⎜
⎝
1
2
. . .
N
⎞
⎟
⎟
⎟
⎠
Therefore, we will have N different D
(1) ’s and N
2 different D
(2) ’s.
1.3.2 Exercise
Consider now the case of a Brownian particle in a potential
⎧
⎪ ⎨
⎪ ⎩
¨
x + γ ˙
x + f
(x) = σ(t)
E[(t)] = 0
E[(t))(t
)] = δ(t − t
)
(1.64)
To use the Fokker-Planck equation, we need first order equations. To remove the
second order term ¨
x let us use a system of two equations, defining ˙
x as v
˙
x(t) = v(t)
˙
v(t) = −γv(t) − f
(x(t)) + σ(t)
(1.65)
Therefore,
h x = v(t) g xx = 0 g xv = 0
h v = −γv(t) − f
(x(t)) g vx = 0 g vv = σ