Fokker–Planck equation

The Fokker–Planck equation describes the time evolution of the probability density function of the velocity of a particle, and can be generalized to other observables as well.[1] It is named after Adriaan Fokker[2] and Max Planck[3] and is also known as the Kolmogorov forward equation (diffusion), named after Andrey Kolmogorov, who first introduced it in a 1931 paper.[4] When applied to particle position distributions, it is better known as the Smoluchowski equation.

The first consistent microscopic derivation of the Fokker–Planck equation in the single scheme of classical and quantum mechanics was performed[5] by Nikolay Bogoliubov and Nikolay Krylov.[6]

In one spatial dimension x, the Fokker–Planck equation for a process \( dX_t = D_1(X_t,t)dt + \sqrt{2 D_2(X_t,t)}dW_t \) with Itō drift D1(x,t) and diffusion D2(x,t) is:

\( \frac{\partial}{\partial t}f(x,t)=-\frac{\partial}{\partial x}\left[ D_{1}(x,t)f(x,t)\right] +\frac{\partial^2}{\partial x^2}\left[ D_{2}(x,t)f(x,t)\right]. \)

Often, however, in physical applications one considers the more relevant Stratonovich process (written in Itō form):

\( dX_t = \left[D_1(X_t,t) + \frac{1}{2} \frac{\partial}{\partial x}D_2(X_t,t)\right]dt + \sqrt{2 D_2(X_t,t)}dW_t \)

Which includes an added drift term due to diffusion gradient effects.

More generally, the time-dependent probability distribution may depend on a set of N macrovariables x_i. The general form of the Fokker–Planck equation is then

\( \frac{\partial f}{\partial t} = -\sum_{i=1}^N \frac{\partial}{\partial x_i} \left[ D_i^1(x_1, \ldots, x_N) f \right] + \sum_{i=1}^{N} \sum_{j=1}^{N} \frac{\partial^2}{\partial x_i \, \partial x_j} \left[ D_{ij}^2(x_1, \ldots, x_N) f \right], \)

where \( D^1 \) is the drift vector and \( D^2 \) the diffusion tensor; the latter results from the presence of the stochastic force.

Relationship with stochastic differential equations

The Fokker–Planck equation can be used for computing the probability density for a stochastic process described by a stochastic differential equation. Consider the Itō stochastic differential equation

\( \mathrm{d}\mathbf{X}_t = \boldsymbol{\mu}(\mathbf{X}_t,t) \,\mathrm{d}t + \boldsymbol{\sigma}(\mathbf{X}_t,t)\, \mathrm{d}\mathbf{W}_t, \)

where \( \mathbf{X}_t \) is an N-dimensional random variable and \( \mathbf{W}_t \) is a standard M-dimensional Wiener process. The probability density f(\mathbf{x},t) for the random variable \( \mathbf{X}_t \) satisfies the Fokker–Planck equation with the drift and diffusion terms

\( D^1_i(\mathbf{x},t) = \mu_i(\mathbf{x},t) \)

\( D^2_{ij}(\mathbf{x},t) = \frac{1}{2} \sum_k \sigma_{ik}(\mathbf{x},t) \sigma_{jk}(\mathbf{x},t). \)

Similarly, a Fokker–Planck equation can be derived for the Stratonovich stochastic differential equations that physicists and engineers generally prefer. In this case, noise-induced drift terms appear if the noise strength is state-dependent.
Examples

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

\( \ \mathrm{d}X_t = \mathrm{d}W_t. \)

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

\( \frac{\partial f(x,t)}{\partial t} = \frac{1}{2} \frac{\partial^2 f(x,t)}{\partial x^2}, \)

that is the simplest form of diffusion equation. If the initial condition is \( f(x,0) = \delta(x) \) , the solution is

\( f(x,t) = \frac{1}{\sqrt{2 \pi t}}e^{-{x^2}/({2t})}. \)

Computational considerations

Brownian motion follows the Langevin equation, which can be solved for many different stochastic forcings with results being averaged (the Monte Carlo method, canonical ensemble in molecular dynamics). However, instead of this computationally intensive approach, one can use the Fokker–Planck equation and consider the probability \( f(\mathbf{v}, t)d\mathbf{v} \) of the particle having a velocity in the interval \( (\mathbf{v}, \mathbf{v} + d\mathbf{v}) \) when it starts its motion with \( \mathbf{v}_0 \) at time 0.
Solution

Being a partial differential equation, the Fokker–Planck equation can be solved analytically only in special cases. A formal analogy of the Fokker–Planck equation with the Schrödinger equation allows the use of advanced operator techniques known from quantum mechanics for its solution in a number of cases. In many applications, one is only interested in the steady-state probability distribution \( f_0(x) \) , which can be found from \( \dot{f}_0(x)=0 \) . The computation of mean first passage times and splitting probabilities can be reduced to the solution of an ordinary differential equation which is intimately related to the Fokker–Planck equation.
Particular cases with known solution and inversion

In mathematical finance for volatility smile modeling of options via local volatility, one has the problem of deriving a diffusion coefficient \( {\sigma}(\mathbf{X}_t,t) \) consistent with a probability density obtained from market option quotes. The problem is therefore an inversion of the Fokker Planck–equation: Given the density \( f(x,t) \) of the option underlying X deduced from the option market, one aims at finding the local volatility \( {\sigma}(\mathbf{X}_t,t) \) consistent with f. This is an inverse problem that has been solved in general by Dupire (1994, 1997) with a non-parametric solution. Brigo and Mercurio (2002, 2003) propose a solution in parametric form via a particular local volatility\( {\sigma}(\mathbf{X}_t,t) \) consistent with a solution of the Fokker–Planck equation given by a mixture model. More information is available also in Fengler (2008), Gatheral (2008) and Musiela and Rutkowski (2008).
Fokker–Planck equation and path integral

Every Fokker–Planck equation is equivalent to a path integral. The path integral formulation is an excellent starting point for the application of field theory methods.[7] This is used, for instance, in critical dynamics.

A derivation of the path integral is possible in the same way as in quantum mechanics, simply because the Fokker–Planck equation is formally equivalent to the Schrödinger equation. Here are the steps for a Fokker–Planck equation with one variable x. Write the FP equation in the form

\( \frac{\partial }{\partial t}f\left( x^{\prime },t\right) =\int_{-\infty}^\infty dx\left( \left[ D_{1}\left( x,t\right) \frac{\partial }{\partial x}+D_2 \left( x,t\right) \frac{\partial^2}{\partial x^2}\right] \delta\left( x^{\prime }-x\right) \right) f\left( x,t\right). \)

Integrate over a time interval \varepsilon,

\( f\left( x^\prime ,t+\varepsilon \right) =\int_{-\infty }^\infty \, dx\left(\left( 1+\varepsilon \left[ D_{1}\left(x,t\right) \frac{\partial }{\partial x}+D_{2}\left( x,t\right) \frac{\partial ^{2}}{\partial x^{2}}\right]\right) \delta \left( x^\prime - x\right) \right) f\left( x,t\right)+O\left( \varepsilon ^{2}\right) \) .

Insert the Fourier integral

\( \delta \left( x^{\prime }-x\right) =\int_{-i\infty }^{i\infty} \frac{d \tilde{x}}{2\pi i }e^{\tilde{x}\left( x-x^{\prime}\right)} \)

for the \delta-function,

\( \begin{align} f\left( x^{\prime },t+\varepsilon \right) & = \int_{-\infty }^\infty dx\int_{-i\infty }^{i\infty } \frac{d\tilde{x}}{2\pi i} \left(1+\varepsilon \left[ \tilde{x}D_{1}\left( x,t\right) +\tilde{x}^{2}D_{2}\left( x,t\right) \right] \right) e^{\tilde{x}\left(x-x^{\prime }\right) }f\left( x,t\right) +O\left( \varepsilon ^{2}\right) \\ & =\int_{-\infty }^\infty dx\int_{-i\infty }^{i\infty }\frac{d\tilde{x}}{2\pi i}\exp \left( \varepsilon \left[ -\tilde{x}\frac{\left( x^{\prime}-x\right) }{\varepsilon }+\tilde{x}D_{1}\left( x,t\right) +\tilde{x}^{2}D_{2}\left( x,t\right) \right] \right) f\left( x,t\right) +O\left(\varepsilon ^{2}\right). \end{align} \)

This equation expresses \( f\left( x^\prime ,t+\varepsilon \right) \) as functional of \( f\left( x,t\right) \) . Iterating \( \left( t^\prime -t\right)/\varepsilon \) times and performing the limit \( \varepsilon \longrightarrow 0 \) gives a path integral with Lagrangian

\( L=\int dt\left[ \tilde{x}D_1 \left( x,t\right) +\tilde{x}^{2}D_2 \left( x,t\right) -\tilde{x}\frac{\partial x}{\partial t}\right]. \)

The variables \( \tilde{x} \) conjugate to x are called "response variables".[8]

Although formally equivalent, different problems may be solved more easily in the Fokker–Planck equation or the path integral formulation. The equilibrium distribution for instance may be obtained more directly from the Fokker–Planck equation.
See also

Kolmogorov backward equation
Boltzmann equation
Navier–Stokes equations
Vlasov equation
Master equation
Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy of equations
Ornstein–Uhlenbeck process

Fokker-Planck equation: methods of solution and applications, The , Hannes Risken

Fokker-Planck equation and the exit problem in applications, Asymptotic methods for the , Johan Grasman, Onno A. Herwaarden

Fokker-Planck equations: fundamentals and applications, Nonlinear, Till Daniel Frank

Notes and references

^ Leo P. Kadanoff (2000). Statistical Physics: statics, dynamics and renormalization. World Scientific. ISBN 981-02-3764-2.
^ A. D. Fokker, Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld, Ann. Phys. 348 (4. Folge 43), 810–820 (1914).
^ M. Planck, Sitz.ber. Preuß. Akad. (1917).
^ Andrei Kolmogorov, "On Analytical Methods in the Theory of Probability", 448-451, (1931), (in German).
^ N. N. Bogolyubov (jr) and D. P. Sankovich (1994). "N. N. Bogolyubov and statistical mechanics". Russian Math. Surveys 49(5): 19—49.
^ N. N. Bogoliubov and N. M. Krylov (1939). Fokker–Planck equations generated in perturbation theory by a method based on the spectral properties of a perturbed Hamiltonian. Zapiski Kafedry Fiziki Akademii Nauk Ukrainian SSR 4: 81–157 (in Ukrainian).
^ Zinn-Justin, Jean (1996). Quantum field theory and critical phenomena. Oxford: Clarendon Press. ISBN 0-19-851882-X.
^ Janssen, H. K. (1976). "On a Lagrangean for Classical Field Dynamics and Renormalization Group Calculation of Dynamical Critical Properties". Z. Physik B23 (4): 377–380. doi:10.1007/BF01316547.