# Small time solution to Fokker-Planck equation

In reference to this note, a specific Focker-Planck equation with initial condition $$W(\rho, t=0)=\delta(\rho-1)$$ have the solution

$$W\left(\rho,t\right)=\dfrac{e^{-\frac{t}{4}}}{\sqrt{\pi}t^{\frac{3}{2}}}\intop_{\mathrm{arccosh}\left(\sqrt{\rho}\right)}^{\infty}\frac{d\left(y^{2}\right)e^{-\left(\frac{y^{2}}{t}\right)}}{\sqrt{\left(\cosh\right)-\rho}}.$$

The equation describes the probability distribution of resistance, $$\rho$$, of a one-dimensional disordered system as the system size $$t$$ is varied. They have provided the long $$t$$ behavior for the same in the notes. If one interested in looking at the short $$t$$ behavior, how should one proceed? What will be the approximate $$W$$ in the small-time, $$t<<1$$?

## 1 Answer

One way to approach the problem is to use the equivalence of Fokker-Planck and Langevin equation (for Gaussian noise). Deduce the appropriate Langevin equation that corresponds to the desired Fokker-Plank (make sure to take into account treatment of multiplicative noise, if there is any - Ito vs Stratonovich). Use standard techniques to get the the time dependance of distribution moments (mean, variance, etc). Numerical solution and statistics can give great intuition about the process (but no analytical expression, obviously).

• I am actually looking for an analytical expression for the mean and variance of the inverse resistance. Is it possible to get that? – Boa_Constrictor Jun 6 at 4:27
• It's possible for linear Fokker-Planck equation (the operator is independent of density)., You can find the standard example here - physics.gu.se/~frtbm/joomla/media/mydocs/LennartSjogren/… Don't know about the non linear case. – Alexander Jun 6 at 12:05