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Fokker-Planck equation
 
 
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The Fokker–Planck equation describes the time evolution
Time evolution

Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state . In this formulation, time is not required to be a continuous parameter, but may be discrete time or even wiktionary:finite....
 of the probability density function
Probability density function

In mathematics, a probability density function is a function that represents a probability distribution in terms of integrals.Formally, a probability distribution has density ƒ, if ƒ is a non-negative Lebesgue integration function such that the probability of the interval [a, b] is given by...
 of the position of a particle, and can be generalized to other observables as well. It is named after Adriaan Fokker
Adriaan Fokker

Adriaan Dani?l Fokker , was a Netherlands physicist and musician.Fokker was born in Buitenzorg, Dutch East Indies ; he was a cousin of the Aeronautics engineer Anthony Fokker....
 and Max Planck
Max Planck

Karl Ernst Ludwig Marx Planck, better known as Max Planck was a Germany physicist. He is considered to be the founder of the Quantum mechanics, and one of the most important physicists of the twentieth century....
 and is also known as the Kolmogorov
Andrey Kolmogorov

Andrey Nikolaevich Kolmogorov was a Soviet Union Russian mathematician, preeminent in the 20th century who advanced various scientific fields ....
 forward equation. The first use of the Fokker–Planck equation was the statistical description of Brownian motion
Brownian motion

Brownian motion is the seemingly random movement of particles suspended in a liquid or gas or the mathematical model used to describe such random movements, often called a particle theory....
 of a particle in a fluid
Fluid

A fluid is defined as a substance that continually deforms under an applied shear stress. All liquids and all gases are fluids. Fluids are a subset of the Phase and include liquids, gas, Plasma physics and, to some extent, plasticity ....
.

In one spatial dimension x, the Fokker–Planck equation for a process with drift D1(x,t) and diffusion D2(x,t) is

More generally, the time-dependent probability distribution may depend on a set of macrovariables .






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The Fokker–Planck equation describes the time evolution
Time evolution

Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state . In this formulation, time is not required to be a continuous parameter, but may be discrete time or even wiktionary:finite....
 of the probability density function
Probability density function

In mathematics, a probability density function is a function that represents a probability distribution in terms of integrals.Formally, a probability distribution has density ƒ, if ƒ is a non-negative Lebesgue integration function such that the probability of the interval [a, b] is given by...
 of the position of a particle, and can be generalized to other observables as well. It is named after Adriaan Fokker
Adriaan Fokker

Adriaan Dani?l Fokker , was a Netherlands physicist and musician.Fokker was born in Buitenzorg, Dutch East Indies ; he was a cousin of the Aeronautics engineer Anthony Fokker....
 and Max Planck
Max Planck

Karl Ernst Ludwig Marx Planck, better known as Max Planck was a Germany physicist. He is considered to be the founder of the Quantum mechanics, and one of the most important physicists of the twentieth century....
 and is also known as the Kolmogorov
Andrey Kolmogorov

Andrey Nikolaevich Kolmogorov was a Soviet Union Russian mathematician, preeminent in the 20th century who advanced various scientific fields ....
 forward equation. The first use of the Fokker–Planck equation was the statistical description of Brownian motion
Brownian motion

Brownian motion is the seemingly random movement of particles suspended in a liquid or gas or the mathematical model used to describe such random movements, often called a particle theory....
 of a particle in a fluid
Fluid

A fluid is defined as a substance that continually deforms under an applied shear stress. All liquids and all gases are fluids. Fluids are a subset of the Phase and include liquids, gas, Plasma physics and, to some extent, plasticity ....
.

In one spatial dimension x, the Fokker–Planck equation for a process with drift D1(x,t) and diffusion D2(x,t) is

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

where is the drift vector and the diffusion
Diffusion

Molecular diffusion, often called simply diffusion, is a net transport of molecules from a region of higher concentration to one of lower concentration by random molecular motion....
 tensor
Tensor

A tensor is an object which extends the notion of Scalar , Vector , and Matrix . The term has slightly different meanings in mathematics and physics....
; 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 densities of stochastic differential equation
Stochastic differential equation

A stochastic differential equation is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process....
s. Consider the Ito
Ito calculus

Ito calculus, named after Kiyoshi Ito, extends the methods of calculus to stochastic processes such as Brownian motion . It has important applications in mathematical finance and stochastic differential equations....
 stochastic differential equation

where is the state and is a standard M-dimensional Wiener process
Wiener process

In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called Brownian motion, after Robert Brown ....
. If the initial distribution is , then the probability density
Probability density

Probability density may refer to:* Probability density function in probability theory* Probability amplitude in quantum mechanics...
  of the state is given by the Fokker–Planck equation with the drift and diffusion terms

Similarly, a Fokker–Planck equation can be derived for Stratonovich
Stratonovich integral

In stochastic processes, the Stratonovich integral is a stochastic integral, the most common alternative to the Ito calculus. While the Ito integral is...
 stochastic differential equation
Stochastic differential equation

A stochastic differential equation is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process....
s. In this case, noise-induced drift terms appear if the noise strength is state-dependent.

Examples

A standard scalar Wiener process
Wiener process

In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called Brownian motion, after Robert Brown ....
 is generated by the stochastic differential equation
Stochastic differential equation

A stochastic differential equation is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process....


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

that is the simplest form of diffusion equation
Diffusion equation

The diffusion equation is a partial differential equation which describes density fluctuations in a material undergoing diffusion. It is also used to describe processes exhibiting diffusive-like behaviour, for instance the 'diffusion' of alleles in a population in population genetics....
.

A simple algebraic substitution shows that
is a solution to this equation.

Computational considerations

Brownian motion follows the Langevin equation
Langevin equation

In statistical physics, a Paul Langevin equation is a stochastic differential equation describing Brownian motion in a potential.The first Langevin equations to be studied were those in which the potential is constant, so that the acceleration of a Brownian particle of mass is expressed as the sum of a viscous force , a noise term...
, which can be solved for many different stochastic forcings with results being averaged (the Monte Carlo method
Monte Carlo method

Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used when computer simulation physics and mathematics systems....
, canonical ensemble in molecular dynamics
Molecular dynamics

Molecular dynamics is a form of computer simulation in which atoms and molecules are allowed to interact for a period of time by approximations of known physics,...
). However, instead of this computationally intensive approach, one can use the Fokker–Planck equation and consider , that is, the probability of the particle having a velocity in the interval when it starts its motion with at time 0.

Solution

Being a partial differential equation
Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a Relation involving an unknown Function of several independent variables and its partial derivatives with respect to those variables....
, the Fokker–Planck equation can be solved analytically only in special cases. A formal analogy of the Fokker-Planck equation with the Schrodinger 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 , which can be found from . 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
Mathematical finance

Mathematical finance is the branch of applied mathematics concerned with the financial markets.The subject has a close relationship with the discipline of financial economics, which is concerned with much of the underlying theory....
 for volatility smile
Volatility Smile

In finance, the volatility smile is a long-observed pattern in which at-the-money option tend to have lower Implied volatility than in- or out-of-the-money options....
 modeling of options via local volatility
Local volatility

In mathematical finance and financial engineering, the assets St underlying financial derivatives typically follow particular stochastic differential equations of the type...
, one has the problem of deriving a diffusion coefficient 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 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 consistent with a solution of the Fokker-Plank equation given by a mixture model
Mixture model

In mathematics, the term mixture model is a model in which independent variables are fractions of a total....
. More information is available also in Fengler (2008), Gatheral (2008) and Musiela and Rutkowski (2008).

See also

  • Kolmogorov backward equation
    Kolmogorov backward equation

    The Kolmogorov backward equation and its Adjoint of an operator the Kolmogorov forward equation are partial differential equations that arise in the theory of continuous-time continuous-state Markov processes....
  • Boltzmann equation
    Boltzmann equation

    The Boltzmann equation, also often known as the Boltzmann transport equation, devised by Ludwig Boltzmann, describes the Probability distribution of one particle in a fluid....
  • Navier–Stokes equations
  • Vlasov equation
    Vlasov equation

    The Vlasov equation is a system of non-linear integro-differential equations describing dynamics of plasma consisting of charged particles with long-range interaction....
  • Master equation
    Master equation

    In physics, a master equation is a phenomenological set of first-order differential equations describing the time evolution of the probability of a system to occupy each one of a discrete set of state :...


External links

  • on the


Books

  • Hannes Risken, "The Fokker–Planck Equation: Methods of Solutions and Applications", 2nd edition, Springer Series in Synergetics, Springer, ISBN 3-540-61530-X.
  • Crispin W. Gardiner, "Handbook of Stochastic Methods", 3rd edition (paperback), Springer, ISBN 3-540-20882-8.