# Fokker Planck equation

### Description

 The Fokker Planck equation has the form

where  is a smooth function,  some parameter and  a probability density on . The partial differential equation is in divergence form and conserves mass. Hence, also  is a probability density on . In the case, where  has some growth at , the equilibrium solutions  are characterized by

The density  is called Gibbs distribution, which accounts to the fact, that it minimizes the (Gibbs) free energy

where  is the energy and  the (Gibbs-Boltzmann) entropy

By this formulation   plays the role of the inverse temperature. Note, that the Euler-Lagrange equation is given by

which is solved by .

### Pathwise formulation via stochastic differential equations

The stochastic process  defined by

where  is an -dimensional Brownian motion, is the pathwise formulation of the Fokker Planck equation. If distribution of the initial random vector  is given by , then at later times the solution  has distribution , where  is solution of the Fokker Planck equation. The dynamic of the pathwise formulation undergoing diffusion in the potential . The particle is solely described by its position , i.e. inertia is neglected. Then, the particle chooses the steepest descent with respect ot , which is perturbed by random fluctuation, which are assumed to be normal distributed. The coupling parameter  describing the strength of the fluctuations plays again the role of inverse temperature. In particular, the limit  describes the behaviour of the system without fluctuations.

### Ornstein-Uhlenbeck process

A prominent example, which can be explicitly solved, is the Ornstein-Uhlenbeck process, corresponding to the choice

The stationary distribution is given by the Gaussian

The underlying picture becomes particular simple in one dimension. There, the leading order behaviour of the process can characterized by its mean  and variance , which are given by

Both,  and , satisfy an ordinary differential equation not depending on 

Hence, the mean converges exponentially fast to  with rate $\gamma$ and the variance converges exponentially fast from the initial variance  to  with rate 

### Numerical example

For general potentials  explicit solutions of the Fokker Planck equation are not available. However, the pathwise formulation allows to do Monte-Carlo simulation of single realisations of the process . The case, where  is not conve is especially interessting, because at low temperature, i.e. , the process shows metastable behaviour. Therefore, we consider the potential function  looking like

The realization of the process  for increasing values of :

Monte Carlo simulation for Fokker Planck equation 

Monte Carlo simulation for Fokker Planck equation 

Monte Carlo simulation for Fokker Planck equation 

Monte Carlo simulation for Fokker Planck equation  (red )

For increasing values of  the particle goes quickly to the local minima and is trapped there longer and longer times. The transition to the global energy minimizing state happens along a path crossing the saddle. This behaviour is called metastable. For the limiting dyanmic  without any fluctuation a transition is not possible and the particle is stuck in the local minima.

### References

1. R. Jordan, D. Kinderlehrer, and F. Otto, “The Variational Formulation of the Fokker-Planck Equation,” SIAM Journal on Mathematical Analysis, vol. 29, no. 1, 1998.