Fokker–Planck equation

Overview

In statistical mechanics and information theory, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability density function of the position or velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. The equation can be generalized to other observables as well. The Fokker–Planck equation has multiple applications in information theory, graph theory, data science, finance, economics, etc. It is named after Adriaan Fokker and Max Planck, who described it in 1914 and 1917. It is also known as the Kolmogorov forward equation, after Andrey Kolmogorov, who independently discovered it in 1931. When applied to particle position distributions, it is better known as the Smoluchowski equation (after Marian Smoluchowski), and in this context it is equivalent to the convection–diffusion equation. When applied to particle position and momentum distributions, it is known as the Klein–Kramers equation. The case with zero diffusion is the continuity equation. The Fokker–Planck equation is obtained from the master equation through Kramers–Moyal expansion. The first consistent microscopic derivation of the Fokker–Planck equation in the single scheme of classical and quantum mechanics was performed by Nikolay Bogoliubov and Nikolay Krylov in 1939.

One dimension

In one spatial dimension x, for an Itô process driven by the standard Wiener process

W

t

{\displaystyle W_{t}}

and described by the stochastic differential equation (SDE)

d

X

t

= μ (

X

t

, t )

d t + σ (

X

t

, t )

d

W

t

{\displaystyle dX_{t}=\mu (X_{t},t)\,dt+\sigma (X_{t},t)\,dW_{t}}

Higher dimensions

More generally, if

d

X

t

=

μ

(

X

t

, t )

d t +

σ

(

X

t

, t )

d

W

t

,

Generalization

In general, the Fokker–Planck equations are a special case to the general Kolmogorov forward equation

∂

∂ t

ρ =

A

∗

ρ

{\displaystyle {\frac {\partial }{\partial t}}\rho ={\mathcal {A}}^{*}\rho }

where the linear operator

A

∗

{\displaystyle {\mathcal {A}}^{*}}

is the Hermitian adjoint to the infinitesimal generator for the Markov process.

Wiener process

A standard scalar Wiener process is generated by the stochastic differential equation

d

X

t

= d

W

t

.

{\displaystyle dX_{t}=dW_{t}.}

Here the drift term is zero and the diffusion coefficient is 1/2. Thus the corresponding Fokker–Planck equation is

∂ p ( x , t )

∂ t

=

1 2

∂

2

p ( x , t )

∂

x

2

Boltzmann distribution at the thermodynamic equilibrium

The overdamped Langevin equation

d

x

t

= −

γ

k

B

T

d U

d x

d t +

2 γ

d

W

t

{\displaystyle dx_{t}=-{\frac {\gamma }{k_{\text{B}}T}}{\frac {dU}{dx}}\,dt+{\sqrt {2\gamma }}dW_{t}}

gives

∂

t

Ornstein–Uhlenbeck process

The Ornstein–Uhlenbeck process is a process defined as

d

X

t

= − a

X

t

d t + σ

d

W

t

.

{\displaystyle dX_{t}=-aX_{t}\,dt+\sigma \,dW_{t}.}

with

a > 0

{\displaystyle a>0}

. Physically, this equation can be motivated as follows: a particle of mass

m

{\displaystyle m}

with velocity

Plasma physics

In plasma physics, the distribution function for a particle species

s

{\displaystyle s}

,

p

s

(

x

,

v

, t )

{\displaystyle p_{s}(\mathbf {x} ,\mathbf {v} ,t)}

, takes the place of the probability density function. The corresponding Boltzmann equation is given by

∂

p

s

∂ t

+

v

⋅

Smoluchowski diffusion equation

Consider an overdamped Brownian particle under external force

F ( r )

{\displaystyle F(r)}

:

m

r ¨

= − γ

r ˙

+ F ( r ) + σ ξ ( t )

{\displaystyle m{\ddot {r}}=-\gamma {\dot {r}}+F(r)+\sigma \xi (t)}

where the

m

r ¨

{\displaystyle m{\ddot {r}}}

term is negligible (the meaning of "overdamped"). Thus, it is just

γ

d r = F ( r )

d t + σ

d

W

Computational considerations

Brownian motion follows the Langevin equation, which can be solved for many different stochastic forcings with results being averaged (canonical ensemble in molecular dynamics). However, instead of this computationally intensive approach, one can use the Fokker–Planck equation and consider the probability

p (

v

, t )

d

v

{\displaystyle p(\mathbf {v} ,t)\,d\mathbf {v} }

of the particle having a velocity in the interval

(

v

,

v

+ d

v

)

{\displaystyle (\mathbf {v} ,\mathbf {v} +d\mathbf {v} )}

when it starts its motion with

v

0

{\displaystyle \mathbf {v} _{0}}

1-D linear potential example

Brownian dynamics in one dimension is simple.

Theory

Starting with a linear potential of the form

U ( x ) = c x

{\displaystyle U(x)=cx}

the corresponding Smoluchowski equation becomes,

∂

∂ t

P ( x , t ∣

x

0

,

t

0

) =

∂

∂ x

D

(

∂

∂ x

+ β c