Fokker–Planck equation

(Redirected from Fokker-Planck equation)
A solution to the one-dimensional Fokker–Planck equation, with both the drift and the diffusion term. The initial condition is a Dirac delta function at x = 1, and the distribution drifts towards the left.

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 case with zero diffusion is known in statistical mechanics as Liouville 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]

One dimension

In one spatial dimension x, for an Itō process given by the stochastic differential equation

$dX_t = \mu(X_t,t)dt + \sqrt{2 D(X_t,t)}dW_t$

with drift $\mu(X_t,t)$ and diffusion coefficient $D(X_t,t)$, the Fokker–Planck equation for the probability density $f(x,t)$ of the random variable $X_t$ is

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

The link between stochastic differential equations and partial differential equations is given by the Feynman-Kac formula.

The stochastic process defined above in the Ito sense can be rewritten within the Stratonovich convention as a Stratonovich SDE:

$dX_t = \left[\mu(X_t,t) - \frac{1}{2} \frac{\partial}{\partial X_t}D(X_t,t)\right]dt + \sqrt{2 D(X_t,t)} \circ dW_t.$

It includes an added noise-induced drift term due to diffusion gradient effects if the noise is state-dependent. This convention is more often used in physical applications. Indeed, it is well known that any solution to the Stratonovich SDE is a solution to the Ito SDE.

Many dimensions

More generally, if $\mathbf{X}_t$ is an N-dimensional random vector and $\mathbf{W}_t$ is an M-dimensional standard Wiener process,

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

the probability density $f(\mathbf{x},t)$ for the random vector $\mathbf{X}_t$ satisfies the Fokker–Planck equation

$\frac{\partial f(\mathbf{x},t)}{\partial t} = -\sum_{i=1}^N \frac{\partial}{\partial x_i} \left[ \mu_i(\mathbf{x}) f(\mathbf{x},t) \right] + \sum_{i=1}^{N} \sum_{j=1}^{N} \frac{\partial^2}{\partial x_i \, \partial x_j} \left[ D_{ij}(\mathbf{x}) f(\mathbf{x},t) \right],$

with drift vector $\boldsymbol{\mu} = (\mu_1,\ldots,\mu_N)$ and diffusion tensor

$D_{ij}(\mathbf{x},t) = \frac{1}{2} \sum_{k=1}^M \sigma_{ik}(\mathbf{x},t) \sigma_{jk}(\mathbf{x},t).$

Examples

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

$dX_t = dW_t.$

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

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

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

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

Alternatively, in plasma physics, the distribution function for a particle species $s$, $f_{s} \left(\vec{x},\vec{v},t\right)$, takes the place of the probability density function. The corresponding Fokker-Planck equation is given by

$\frac{\partial f_{s}}{\partial t} + \vec{v} \cdot \vec{\nabla} f_{s} + \frac{Z_{s} e}{m_{s}} \left( \vec{E} + \vec{v} \times \vec{B} \right) \cdot \vec{\nabla}_{v} f_{s} = \sum_{s'} C\left[f_{s},f_{s'}\right]$,

where the third term includes the particle acceleration due to the Lorentz force and the right-hand side represents the effects of particle collisions. If collisions are ignored the Fokker-Planck equation reduces to the Vlasov equation.

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

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

The x-derivatives here only act on the $\delta$-function, not on $f(x,t)$. 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.

Notes and references

1. ^ Leo P. Kadanoff (2000). Statistical Physics: statics, dynamics and renormalization. World Scientific. ISBN 981-02-3764-2.
2. ^ A. D. Fokker, Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld, Ann. Phys. 348 (4. Folge 43), 810–820 (1914).
3. ^ M. Planck, Sitz.ber. Preuß. Akad. (1917).
4. ^ Andrei Kolmogorov, "On Analytical Methods in the Theory of Probability", 448-451, (1931), (in German).
5. ^ N. N. Bogolyubov (jr) and D. P. Sankovich (1994). "N. N. Bogolyubov and statistical mechanics". Russian Math. Surveys 49(5): 19—49.
6. ^ 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).
7. ^ Zinn-Justin, Jean (1996). Quantum field theory and critical phenomena. Oxford: Clarendon Press. ISBN 0-19-851882-X.
8. ^ 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. Bibcode:1976ZPhyB..23..377J. doi:10.1007/BF01316547.