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Derive the Fokker-Planck equation by requiring conservation of probability:
∫∂VJ⋅dS=-d/dt∫Vp(r,t)dV
The flux can be written as a sum of convective and diffusive terms
J=p(r,t)v(r,t)-D(r,t)∇p(r,t)
and substitution of this with use of the divergence theorem yields
∂tp(x,t)=-∂x[p(x,t)v(x,t)]+∂x[D(x,t)∂xp(x,t)]
where I have moved to one dimension for simplicity.
However the form found here
https://en.wikipedia.org/wiki/Fokker–Planck_equation
is given as
∂tp(x,t)=-∂x[p(x,t)v(x,t)]+∂x2[D(x,t)p(x,t)]
I was wondering if anybody would be able to help me account for this difference. Thanks!
∫∂VJ⋅dS=-d/dt∫Vp(r,t)dV
The flux can be written as a sum of convective and diffusive terms
J=p(r,t)v(r,t)-D(r,t)∇p(r,t)
and substitution of this with use of the divergence theorem yields
∂tp(x,t)=-∂x[p(x,t)v(x,t)]+∂x[D(x,t)∂xp(x,t)]
where I have moved to one dimension for simplicity.
However the form found here
https://en.wikipedia.org/wiki/Fokker–Planck_equation
is given as
∂tp(x,t)=-∂x[p(x,t)v(x,t)]+∂x2[D(x,t)p(x,t)]
I was wondering if anybody would be able to help me account for this difference. Thanks!