The derivation of the backward fokker planck equation

science_boy

1. The problem statement, all variables and given/known data
Starting with the F.P. equation, where the probability has been defined in terms of the stationary probability and another probability function, Q(x,t), i am trying to derive the backward F.P. equation in terms of Q(x,t).

2. Relevant equations
The F.P. equation provided: [d(P(x,t)]/dt=-{d[f(x)P(x,t)]/dx}+D{[d^2P(x,t)]/dx^2}
where f(x) is the applied force

P(x,t) = Ps(x)Q(x,t), where Ps(x) is the stationary distribution function of the F.P. eqn.

I know the end result, ie the backward F.P. eqn is:

[d(Q(x,t)]/dt=f(x){d[Q(x,t)]/dx}+D{[d^2Q(x,t)]/dx^2}

In my attempt i have made use of equations:

Fick’s law: J(x,t)=-D*del[n(x,t)]

conservation of mass

and the diffusion equation:

d[n(x,t)]/dt=D*del^2[(n(x,t)]

3. The attempt at a solution
So far i have managed to gain the backward F.P. eqn. However i made use of this equation:

d[Ps(x)f(x)]/dx-D[d^2(Ps(x))/dx^2]=0 (eq.1)

Whilst familiar, i cannot seem to obtain this equation form. Instead of just stating it, i really need to understand where it comes from. Through research i seem to believe it comes from Fick's law with an external force. Where making use of Fick's law along with the conservation of mass law i can obtain a similar equation. This is the diffusion equation equalling to zero. Still, i am unconvinced of that method to gain (eq.1).

Does anyone recognise Eq.1. or can you seem to get a more conclusive derivation of Eq.1 than me! Any advice would be much appreciated.

Thanks

Science_boy