Deriving FP eq from Langevin equation. Fokker-Planck equation works for Markov processes in space, so it is derived from the Langevin equation that ignores inertia.

$\partial_t P(\vec{r},t)+\vec{\nabla}\cdot[\vec{v}(\vec{r})P(\vec{r},t)-D\vec{\nabla}P(\vec{r},t)] = 0$

where:

Detailed balance and equilibrium

Setting $\partial_t P(\vec{r},t) = 0$ and $\vec{J}=0$, and using Einstein's relation, we get Boltzmann Distribution.

N-non-interacting particles

We get Smoluchowski equation.

N interacting particles

We get BBGKY hierarchy, as in Kinetic theory

Backwards Fokker-Planck equation Tells you how likely different initial conditions is to arrive at a certain fixed point in the future.

Applications of Fokker-Planck equation

First-passage time Calculation of the mean time required to leave a region.

Kramers rate theory The rate at which fluctuations push particles over a barrier.

Survival probability Crucial argument: reflecting parts of the trajectory leaves same probability See also here for nice derivation from boundary conditions

Stationary solution of 1D FP equation

Brownian ratchets

Assume a periodic potential $U(x)$ with a bias:

$V(x)=U(x)-Fx$

and assume the solution is periodic:

$P(x+L)=P(x)$

This is not the equilibrium solution (which would be an exponential growing P to compensate bias, just as the exponential growth of density in gravity or constant electric field). Therefore $J \neq 0$ even though it is stationary $\partial_t P=0$. If we integrate this from $0$ to $L$ taking this periodicity into account:

The easiest way to calculate escape time from one well to the next is to assume there is one particle per well:

The average drift velocity is $v_{drift} \equiv \frac{L}{T_{esc}}=JL$.

Fluctuation-driver transport

Analogous to AC rectification in diodes!

Mathematical properties of FP eq

Quantum mechanical analogy

See video, and the lecture notes!

Also applicable in Path integrals for stochastic processes

Stochastic quantization and path integral formulation of Fokker-Planck equation