12.6 Langevin and Fokker-Planck Equations 407
and
w(x,t) =
∫∞
−∞
W(x.t|x 0 .t 0 )w(x 0 ,t 0 )dx 0 ,and
w(x′,t′) =
∫∞
−∞
W(x′.t′|x 0 ,t 0 )w(x 0 ,t 0 )dx 0.We can combine these equations and arrive at the famous Einstein-Smoluchenski-Kolmogorov-
Chapman (ESKC) relation
W(xt|x 0 t 0 ) =∫∞
−∞
W(x,t|x′,t′)W(x′,t′|x 0 ,t 0 )dx′.We can replace the spatial dependence with a dependence uponsay the velocity (or momen-
tum), that is we have
W(v,t|v 0 ,t 0 ) =∫∞
−∞W(v,t|v′,t′)W(v′,t′|v 0 ,t 0 )dx′.We will now derive the Fokker-Planck equation. We start fromthe ESKC equationW(x,t|x 0 ,t 0 ) =∫∞
−∞
W(x,t|x′,t′)W(x′,t′|x 0 ,t 0 )dx′.We defines=t′−t 0 ,τ=t−t′andt−t 0 =s+τ. We have then
W(x,s+τ|x 0 ) =∫∞
−∞W(x,τ|x′)W(x′,s|x 0 )dx′.Assume now thatτis very small so that we can make an expansion in terms of a small step
xi, withx′=x−ξ, that is
W(x,s|x 0 )+∂W
∂s
τ+O(τ^2 ) =∫∞
−∞W(x,τ|x−ξ)W(x−ξ,s|x 0 )dx′.We assume thatW(x,τ|x−ξ)takes non-negligible values only whenξis small. This is just
another way of stating the Master equation!
We say thus thatxchanges only by a small amount in the time intervalτ. This means that
we can make a Taylor expansion in terms ofξ, that is we expand
W(x,τ|x−ξ)W(x−ξ,s|x 0 ) =∞
∑
n= 0(−ξ)n
n!∂n
∂xn
[W(x+ξ,τ|x)W(x,s|x 0 )].We can then rewrite the ESKC equation as
∂W
∂s
τ=−W(x,s|x 0 )+∞
∑
n= 0(−ξ)n
n!∂n
∂xn[
W(x,s|x 0 )∫∞
−∞ξnW(x+ξ,τ|x)dξ]
.
We have neglected higher powers ofτand have used that forn= 0 we get simplyW(x,s|x 0 )
due to normalization.
We say thus thatxchanges only by a small amount in the time intervalτ. This means that
we can make a Taylor expansion in terms ofξ, that is we expand
W(x,τ|x−ξ)W(x−ξ,s|x 0 ) =∞
∑
n= 0(−ξ)n
n!∂n
∂xn
[W(x+ξ,τ|x)W(x,s|x 0 )].We simplify the above by introducing the moments