Fokker–Planck equation explained

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 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.^[1] 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.^{[2] [3]} It is also known as the Kolmogorov forward equation, after Andrey Kolmogorov, who independently discovered it in 1931.^[4] When applied to particle position distributions, it is better known as the Smoluchowski equation (after Marian Smoluchowski),^[5] 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.^[6]

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.^{[7] [8]}

One dimension

W _{t} and described by the stochastic differential equation (SDE)$$dX\_t\; =\; \backslash mu(X\_t,\; t)\; \backslash ,dt\; +\; \backslash sigma(X\_t,\; t)\; \backslash ,dW\_t$$with drift \mu(X _{t,} t) and diffusion coefficient D(X _{t,} t) = \sigma ^{2 (X} _{t,} t)/ 2 , the Fokker–Planck equation for the probability density p(x, t) of the random variable X _{t} is In the following, use \sigma = \sqrt{ 2 D} .l{L} (the following can be found in Ref.^[9]):$$\backslash mathcalp(X\_t)\; =\; \backslash lim\_\; \backslash frac1\backslash left(\backslash mathbb\backslash big[p(X\_\{t\; +\; \backslash Delta\; t\})\; \backslash mid\; X\_t\; =\; x\; \backslash big]\; -\; p(x)\backslash right).$$

The transition probability

P _{t,} (x \mid x') , the probability of going from (t', x') to (t, x) , is introduced here; the expectation can be written as$$\backslash mathbb(p(X\_)\; \backslash mid\; X\_t\; =\; x)\; =\; \backslash int\; p(y)\; \backslash ,\; \backslash mathbb\_(y\; \backslash mid\; x)\; \backslash ,dy.$$Now we replace in the definition of l{L} , multiply by P _{t,} (x \mid x') and integrate over dx . The limit is taken on$$\backslash int\; p(y)\; \backslash int\; \backslash mathbb\_(y\; \backslash mid\; x)\backslash ,\backslash mathbb\_(x\; \backslash mid\; x\text{'})\; \backslash ,dx\; \backslash ,dy\; -\; \backslash int\; p(x)\; \backslash ,\; \backslash mathbb\_(x\; \backslash mid\; x\text{'})\; \backslash ,dx.$$Note now that$$\backslash int\; \backslash mathbb\_(y\; \backslash mid\; x)\; \backslash ,\; \backslash mathbb\_(x\; \backslash mid\; x\text{'})\; \backslash ,dx\; =\; \backslash mathbb\_(y\; \backslash mid\; x\text{'}),$$which is the Chapman–Kolmogorov theorem. Changing the dummy variable y to x , one gets$$\backslash begin\; \backslash int\; p(x)\; \backslash lim\_\; \backslash frac1\; \backslash left(\backslash mathbb\_(x\; \backslash mid\; x\text{'})\; -\; \backslash mathbb\_(x\; \backslash mid\; x\text{'})\; \backslash right)\; \backslash ,dx,\backslash end$$which is a time derivative. Finally we arrive to$$\backslash int\; [\backslash mathcal\{L\}p(x)]\; \backslash mathbb\_(x\; \backslash mid\; x\text{'})\; \backslash ,dx\; =\; \backslash int\; p(x)\; \backslash ,\; \backslash partial\_t\; \backslash mathbb\_(x\; \backslash mid\; x\text{'})\; \backslash ,dx.$$From here, the Kolmogorov backward equation can be deduced. If we instead use the adjoint operator of l{L} , l{L} ^{\dagger} , defined such that$$\backslash int\; [\backslash mathcal\{L\}p(x)]\; \backslash mathbb\_(x\; \backslash mid\; x\text{'})\; \backslash ,dx\; =\; \backslash int\; p(x)\; [\backslash mathcal\{L\}^\backslash dagger\; \backslash mathbb\{P\}\_\{t,\; t\text{'}\}(x\; \backslash mid\; x\text{'})]\; \backslash ,dx,$$then we arrive to the Kolmogorov forward equation, or Fokker–Planck equation, which, simplifying the notation p(x, t) = P _{t,} (x \mid x') , in its differential form reads$$\backslash mathcal^\backslash dagger\; p(x,\; t)\; =\; \backslash partial\_t\; p(x,\; t).$$

Remains the issue of defining explicitly

l{L} . This can be done taking the expectation from the integral form of the Itô's lemma:$$\backslash mathbb\backslash big(p(X\_t)\backslash big)\; =\; p(X\_0)\; +\; \backslash mathbb\backslash left(\backslash int\_0^t\; \backslash left(\backslash partial\_t\; +\; \backslash mu\backslash partial\_x\; +\; \backslash frac\backslash partial\_x^2\; \backslash right)\; p(X\_)\; \backslash ,dt\text{'}\backslash right).$$

The part that depends on

dW _{t} vanished because of the martingale property.

Then, for a particle subject to an Itô equation, using$$\backslash mathcal\; =\; \backslash mu\backslash partial\_x\; +\; \backslash frac\backslash partial\_x^2,$$it can be easily calculated, using integration by parts, that$$\backslash mathcal^\backslash dagger\; =\; -\backslash partial\_x(\backslash mu\; \backslash cdot)\; +\; \backslash frac12\; \backslash partial\_x^2(\backslash sigma^2\; \backslash cdot),$$which bring us to the Fokker–Planck equation:$$\backslash partial\_t\; p(x,\; t)\; =\; -\backslash partial\_x\; \backslash big(\backslash mu(x,\; t)\; \backslash cdot\; p(x,\; t)\backslash big)\; +\; \backslash partial\_x^2\backslash left(\backslash frac\; \backslash ,\; p(x,t)\backslash right).$$

While the Fokker–Planck equation is used with problems where the initial distribution is known, if the problem is to know the distribution at previous times, the Feynman–Kac formula can be used, which is a consequence of the Kolmogorov backward equation.

The stochastic process defined above in the Itô sense can be rewritten within the Stratonovich convention as a Stratonovich SDE:$$dX\_t\; =\; \backslash left[\backslash mu(X\_t,\; t)\; -\; \backslash frac\{1\}\{2\}\; \backslash frac\{\backslash partial\}\{\backslash partial\; X\_t\}D(X\_t,\; t)\backslash right]\; \backslash ,dt\; +\; \backslash sqrt\; \backslash circ\; dW\_t.$$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 Itô SDE.

The zero-drift equation with constant diffusion can be considered as a model of classical Brownian motion:$$\backslash frac\; p(x,\; t)\; =\; D\_0\backslash frac\backslash left[p(x,\; t)\backslash right].$$

This model has discrete spectrum of solutions if the condition of fixed boundaries is added for

\{ 0 \leq x \leq L\} :

It has been shown^[10] that in this case an analytical spectrum of solutions allows deriving a local uncertainty relation for the coordinate-velocity phase volume:$$\backslash Delta\; x\; \backslash ,\; \backslash Delta\; v\; \backslash geq\; D\_0.$$Here

D _{0} is a minimal value of a corresponding diffusion spectrum D _{j} , while \Delta x and \Delta v represent the uncertainty of coordinate–velocity definition.

Higher dimensions

More generally, if

$$d\backslash mathbf\_t\; =\; \backslash boldsymbol(\backslash mathbf\_t,t)\backslash ,dt\; +\; \backslash boldsymbol(\backslash mathbf\_t,t)\backslash ,d\backslash mathbf\_t,$$

where

X _{t} and \boldsymbol{\mu}( X _{t,t)} are -dimensional vectors, \boldsymbol{\sigma}( X _{t,t)} is an N x M matrix and W _{t} is an M-dimensional standard Wiener process, the probability density p( x ,t) for X _{t} satisfies the Fokker–Planck equationwith drift vector \boldsymbol{\mu} = (\mu _{1,\ldots,\mu} _{N)} and diffusion tensor $\backslash mathbf\; =\; \backslash frac\; \backslash boldsymbol^\backslash mathsf$, i.e.$$D\_(\backslash mathbf,t)\; =\; \backslash frac\backslash sum\_^M\; \backslash sigma\_(\backslash mathbf,t)\; \backslash sigma\_(\backslash mathbf,t).$$

If instead of an Itô SDE, a Stratonovich SDE is considered,

$$d\backslash mathbf\_t\; =\; \backslash boldsymbol(\backslash mathbf\_t,t)\backslash ,dt\; +\; \backslash boldsymbol(\backslash mathbf\_t,t)\backslash circ\; d\backslash mathbf\_t,$$

the Fokker–Planck equation will read:^[9]

$$\backslash frac\; =\; -\backslash sum\_^N\; \backslash frac\; \backslash left[\backslash mu\_i(\backslash mathbf\{x\},t)\; \backslash ,\; p(\backslash mathbf\{x\},t)\; \backslash right]\; +\; \backslash frac\; \backslash sum\_^M\; \backslash sum\_^\; \backslash frac\; \backslash left\backslash$$

Generalization

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

$$\backslash partial\_t\; \backslash rho\; =\; \backslash mathcal^*\backslash rho$$

where the linear operator

l{A} ^{*} is the Hermitian adjoint to the infinitesimal generator for the Markov process.^[11]

Examples

Wiener process

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

$$\backslash frac\; =\; \backslash frac\; \backslash frac,$$

which is the simplest form of a diffusion equation. If the initial condition is

p(x, 0 ) = \delta(x) , the solution is

$$p(x,t)\; =\; \backslash frace^.$$

Boltzmann distribution at the thermodynamic equilibrium

The overdamped Langevin equation$$dx\_t\; =\; -\backslash frac\; (\backslash nabla\_x\; U)dt\; +\; dW\_t$$gives $\backslash partial\_t\; p\; =\; \backslash frac\; 1\; 2\; \backslash nabla\backslash cdot\; \backslash left(\backslash frac\; \backslash nabla\; U\; +\; \backslash nabla\; p\backslash right)$. The Boltzmann distribution

p(x) \propto e ^{- U(x)/k _{B} T} is an equilibrium distribution, and assuming U grows sufficiently rapidly (that is, the potential well is deep enough to confine the particle), the Boltzmann distribution is the unique equilibrium.

Ornstein–Uhlenbeck process

The Ornstein–Uhlenbeck process is a process defined as

$$dX\_t\; =\; -a\; X\_t\; \backslash ,\; dt\; +\; \backslash sigma\; \backslash ,\; dW\_t.$$

with

a> 0 . Physically, this equation can be motivated as follows: a particle of mass m with velocity V _{t} moving in a medium, e.g., a fluid, will experience a friction force which resists motion whose magnitude can be approximated as being proportional to particle's velocity - a V _{t} with a = constant . Other particles in the medium will randomly kick the particle as they collide with it and this effect can be approximated by a white noise term; \sigma (d W _{t/dt)} . Newton's second law is written as

$$m\; \backslash frac=-a\; V\_t\; +\backslash sigma\; \backslash frac.$$

Taking

m = 1 for simplicity and changing the notation as V _{t →} X _{t} leads to the familiar form dX _{t} = - a X _{t} dt + \sigma dW _{t} .

The corresponding Fokker–Planck equation is$$\backslash frac\; =\; a\; \backslash frac\backslash left(x\; \backslash ,p(x,t)\backslash right)\; +\; \backslash frac\; \backslash frac,$$

The stationary solution (

\partial _{t} p = 0 ) is$$p\_(x)\; =\; \backslash sqrt\; e^.$$

Plasma physics

In plasma physics, the distribution function for a particle species

s , p _{s} ( x , v ,t) , takes the place of the probability density function. The corresponding Boltzmann equation is given by

$$\backslash frac\; +\; \backslash mathbf\; \backslash cdot\; \backslash boldsymbol\; p\_s\; +\; \backslash frac\; \backslash left(\backslash mathbf\; +\; \backslash mathbf\; \backslash times\; \backslash mathbf\; \backslash right)\; \backslash cdot\; \backslash boldsymbol\_v\; p\_s\; =\; -\backslash frac\; \backslash left(p\_s\; \backslash langle\backslash Delta\; v\_i\backslash rangle\backslash right)\; +\; \backslash frac\; \backslash frac\; \backslash left(p\_s\; \backslash langle\backslash Delta\; v\_i\; \backslash ,\; \backslash Delta\; v\_j\backslash rangle\backslash right),$$

where the third term includes the particle acceleration due to the Lorentz force and the Fokker–Planck term at the right-hand side represents the effects of particle collisions. The quantities

\langle\Delta v _{i\rangle} and \langle\Delta v _{i} \Delta v _{j\rangle} are the average change in velocity a particle of type s experiences due to collisions with all other particle species in unit time. Expressions for these quantities are given elsewhere.^[12] If collisions are ignored, the Boltzmann equation reduces to the Vlasov equation.

Smoluchowski diffusion equation

Consider an overdamped Brownian particle under external force

F(r) :^[13] $$m\backslash ddot\; =\; -\; \backslash gamma\; \backslash dot\; +\; F(r)\; +\; \backslash sigma\; \backslash xi(t)$$where the m\ddot r term is negligible (the meaning of "overdamped"). Thus, it is just \gamma dr = F(r) dt + \sigma dW _{t} . The Fokker–Planck equation for this particle is the Smoluchowski diffusion equation:$$\backslash partial\_t\; P(r,t|\; r\_0,\; t\_0)\; =\; \backslash nabla\; \backslash cdot\; [D\; (\backslash nabla\; -\; \backslash beta\; F(r))\; P(r,t|\; r\_0,\; t\_0)]$$Where D is the diffusion constant and \beta = (1)/(k _{B} T) . The importance of this equation is it allows for both the inclusion of the effect of temperature on the system of particles and a spatially dependent diffusion constant.

Starting with the Langevin Equation of a Brownian particle in external field

F(r) , where \gamma is the friction term, \xi is a fluctuating force on the particle, and \sigma is the amplitude of the fluctuation.

$$m\backslash ddot\; =\; -\; \backslash gamma\; \backslash dot\; +\; F(r)\; +\; \backslash sigma\; \backslash xi(t)$$

At equilibrium the frictional force is much greater than the inertial force,

\left\vert \gamma • r \right\vert \gg \left\vert m \ddot{r} \right\vert . Therefore, the Langevin equation becomes,

$$\backslash gamma\; \backslash dot\; =\; F(r)\; +\; \backslash sigma\; \backslash xi(t)$$

Which generates the following Fokker–Planck equation,

$$\backslash partial\_t\; P(r,t|r\_0,t\_0)\; =\; \backslash left(\backslash nabla^2\backslash frac\; -\; \backslash nabla\; \backslash cdot\; \backslash frac\backslash right)\; P(r,t|r\_0,t\_0)$$

Rearranging the Fokker–Planck equation,

$$\backslash partial\_t\; P(r,t|r\_0,t\_0)=\; \backslash nabla\; \backslash cdot\; \backslash left(\backslash nabla\; D-\; \backslash frac\backslash right)\; P(r,t|r\_0,t\_0)$$

Where

D = (\sigma ^{2})/(2 \gamma ^{2}) . Note, the diffusion coefficient may not necessarily be spatially independent if \sigma or \gamma are spatially dependent.

Next, the total number of particles in any particular volume is given by,

$$N\_V\; (t|\; r\_0,\; t\_0)\; =\; \backslash int\_V\; dr\; P(r,t|r\_0,t\_0)$$

Therefore, the flux of particles can be determined by taking the time derivative of the number of particles in a given volume, plugging in the Fokker–Planck equation, and then applying Gauss's Theorem.

$$\backslash partial\_t\; N\_V\; (t|r\_0,\; t\_0)\; =\; \backslash int\_V\; dV\; \backslash nabla\; \backslash cdot\backslash left(\backslash nabla\; D-\; \backslash frac\backslash right)\; P(r,t|r\_0,\; t\_0)=\; \backslash int\_\; d\backslash mathbf\; \backslash cdot\; j(r,t|r\_0,\; t\_0)$$

$$j(r,t|r\_0,\; t\_0)\; =\; \backslash left(\backslash nabla\; D-\; \backslash frac\backslash right)P(r,t|r\_0,\; t\_0)$$

In equilibrium, it is assumed that the flux goes to zero. Therefore, Boltzmann statistics can be applied for the probability of a particles location at equilibrium, where

F(r) = - \nabla U(r) is a conservative force and the probability of a particle being in a state r is given as P(r,t|r _{0,} t _{0)} = (e ^{-\beta})/(Z) .

$$j(r,t|r\_0,\; t\_0)\; =\; \backslash left(\backslash nabla\; D-\; \backslash frac\backslash right)\backslash frac\; =\; 0$$

$$\backslash Rightarrow\; \backslash nabla\; D\; =\; F(r)\; \backslash left(\backslash frac\; -\; D\; \backslash beta\backslash right)$$

This relation is a realization of the fluctuation–dissipation theorem. Now applying

\nabla ⋅ \nabla to D P(r,t|r _{0,} t _{0)} and using the Fluctuation-dissipation theorem,

Rearranging,

$$\backslash Rightarrow\; \backslash nabla\; \backslash cdot\; \backslash left(\backslash nabla\; D-\; \backslash frac\backslash right)P(r,t|r\_0,t\_0)=\backslash nabla\; \backslash cdot\; D(\backslash nabla-\backslash beta\; F(r))\; P(r,t|r\_0,t\_0)$$

Therefore, the Fokker–Planck equation becomes the Smoluchowski equation,$$\backslash partial\_t\; P(r,t|\; r\_0,\; t\_0)\; =\; \backslash nabla\; \backslash cdot\; D\; (\backslash nabla\; -\; \backslash beta\; F(r))\; P(r,t|\; r\_0,\; t\_0)$$for an arbitrary force

F(r) .

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 of the particle having a velocity in the interval ( v , v + d v ) when it starts its motion with v _{0} at time 0.

1-D linear potential example

Brownian dynamics in one dimension is simple.^[14]

Theory

Starting with a linear potential of the form

U(x) = cx the corresponding Smoluchowski equation becomes,

$$\backslash partial\_t\; P(x,t|\; x\_0,\; t\_0)\; =\; \backslash partial\_x\; D\; (\backslash partial\_x\; +\; \backslash beta\; c)\; P(x,t|\; x\_0,\; t\_0)$$

Where the diffusion constant,

D , is constant over space and time. The boundary conditions are such that the probability vanishes at x → \pm inf in with an initial condition of the ensemble of particles starting in the same place, P(x,t|x _{0,t} _{0)=} \delta (x - x _{0)} .

Defining

\tau = D t and b = \beta c and applying the coordinate transformation,

$$y\; =\; x\; +\backslash tau\; b,\backslash \; \backslash \; \backslash \; y\_0=\; x\_0\; +\; \backslash tau\_0\; b$$

With

P(x, t, |x _{0,} t _{0)} = q(y, \tau|y _{0,} \tau _{0)} the Smoluchowki equation becomes,$$\backslash partial\_\backslash tau\; q(y,\; \backslash tau|\; y\_0,\; \backslash tau\_0)\; =\backslash partial\_y^2\; q(y,\; \backslash tau|\; y\_0,\; \backslash tau\_0)$$

Which is the free diffusion equation with solution,$$q(y,\; \backslash tau|\; y\_0,\; \backslash tau\_0)=\; \backslash frac\; e^$$

And after transforming back to the original coordinates,$$P(x,\; t\; |\; x\_0,\; t\_0)=\; \backslash frac\; \backslash exp$$

Simulation

The simulation on the right was completed using a Brownian dynamics simulation.^{[15] [16]} Starting with a Langevin equation for the system,$$m\backslash ddot\; =\; -\; \backslash gamma\; \backslash dot\; -c\; +\; \backslash sigma\; \backslash xi(t)$$where

\gamma is the friction term, \xi is a fluctuating force on the particle, and \sigma is the amplitude of the fluctuation. At equilibrium the frictional force is much greater than the inertial force, \left| \gamma • x \right| \gg \left| m \ddot{x} \right| . Therefore, the Langevin equation becomes,$$\backslash gamma\; \backslash dot\; =\; -c\; +\; \backslash sigma\; \backslash xi(t)$$

For the Brownian dynamic simulation the fluctuation force

\xi(t) is assumed to be Gaussian with the amplitude being dependent of the temperature of the system $\backslash sigma\; =\; \backslash sqrt$. Rewriting the Langevin equation,

$$\backslash frac=-D\; \backslash beta\; c\; +\; \backslash sqrt\backslash xi(t)$$where $D\; =\; \backslash frac$ is the Einstein relation. The integration of this equation was done using the Euler–Maruyama method to numerically approximate the path of this Brownian particle.

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. Furthermore, in the case of overdamped dynamics when the Fokker–Planck equation contains second partial derivatives with respect to all spatial variables, the equation can be written in the form of a master equation that can easily be solved numerically.^[17] In many applications, one is only interested in the steady-state probability distribution

p _{0(x)} , which can be found from $\backslash frac\; =\; 0$.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}( 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}( 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.^{[18] [19]} Brigo and Mercurio (2002, 2003) propose a solution in parametric form via a particular local volatility {\sigma}( X _{t,t)} consistent with a solution of the Fokker–Planck equation given by a mixture model.^{[20] [21]} More information is available also in Fengler (2008),^[22] Gatheral (2008),^[23] and Musiela and Rutkowski (2008).^[24]

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.^[25] This is used, for instance, in critical dynamics.

A derivation of the path integral is possible in a similar way as in quantum mechanics. The derivation for a Fokker–Planck equation with one variable

x is as follows. Start by inserting a delta function and then integrating by parts:

The

x -derivatives here only act on the \delta -function, not on p(x,t) . Integrate over a time interval \varepsilon ,

$$p(x\text{'},\; t\; +\; \backslash varepsilon)\; =\backslash int\_^\backslash infty\; \backslash ,\; \backslash mathrmx\backslash left(\backslash left(1+\backslash varepsilon\; \backslash left[D\_1(x,t)\; \backslash frac\; \backslash partial\; \{\backslash partial\; x\}\; +\; D\_2(x,t)\; \backslash frac\{\backslash partial^2\}\{\backslash partial\; x^2\}\backslash right]\backslash right)\; \backslash delta(x\text{'}\; -\; x)\; \backslash right)\; p(x,t)+O(\backslash varepsilon^2).$$

Insert the Fourier integral

$$\backslash delta\; =\; \backslash int\_^\; \backslash frac\; e^$$

for the

\delta -function,

This equation expresses

p(x', t + \varepsilon) as functional of p(x,t) . Iterating (t' - t)/\varepsilon times and performing the limit \varepsilon → 0 gives a path integral with action

$$S=\backslash int\; \backslash mathrmt\backslash left[\backslash tilde\{x\}\; D\_1\; (x,t)\; +\backslash tilde\{x\}^2\; D\_2\; (x,t)\; -\backslash tilde\{x\}\backslash frac\{\backslash partial\; x\}\{\backslash partial\; t\}\; \backslash right].$$

The variables

\tilde{x} conjugate to x are called "response variables".^[26]

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
• Langevin equation
• Vlasov equation
• Master equation
• Mean-field game theory
• Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy of equations
• Ornstein–Uhlenbeck process
• Convection–diffusion equation
• Klein–Kramers equation

Further reading

• Book: Frank, Till Daniel . 2005 . Nonlinear Fokker–Planck Equations: Fundamentals and Applications . Springer Series in Synergetics . Springer . 3-540-21264-7 .
• Book: Gardiner, Crispin . Crispin Gardiner . 2009 . Stochastic Methods . 4th . Springer . 978-3-540-70712-7 .
• Book: Pavliotis, Grigorios A. . Stochastic Processes and Applications: Diffusion Processes, the Fokker–Planck and Langevin Equations . 2014 . Springer Texts in Applied Mathematics . Springer . 978-1-4939-1322-0 .
• Book: Risken, Hannes . The Fokker–Planck Equation: Methods of Solutions and Applications . 2nd . 1996 . Springer Series in Synergetics . Springer . 3-540-61530-X .

Notes and References

1. Book: Statistical Physics: statics, dynamics and renormalization. Leo P. Kadanoff. World Scientific. 978-981-02-3764-6. 2000.
2. Fokker. A. D.. 1914. Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld. Ann. Phys.. 348. 4. Folge 43. 810–820. 1914AnP...348..810F. 10.1002/andp.19143480507.
3. Planck. M.. 1917. Über einen Satz der statistischen Dynamik und seine Erweiterung in der Quantentheorie. Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin. 24. 324–341.
4. Andrei . Kolmogorov . Über die analytischen Methoden in der Wahrscheinlichkeitstheorie . . 104 . 1 . On Analytical Methods in the Theory of Probability . 415–458 [pp. 448–451] . 1931 . de . 10.1007/BF01457949 . 119439925 .
5. Book: Dhont, J. K. G.. An Introduction to Dynamics of Colloids. Elsevier. 1996. 978-0-08-053507-4. 183.
6. Book: Wolfgang . Paul . Jörg . Baschnagel . A Brief Survey of the Mathematics of Probability Theory . Stochastic Processes . 17–61 [esp. 33–35] . Springer . 2013 . 978-3-319-00326-9. 10.1007/978-3-319-00327-6_2 .
7. [Nikolay Boglyubov Jr.|N. N. Bogolyubov Jr.]
8. [Nikolay Bogoliubov|N. N. Bogoliubov]
9. Book: Öttinger, Hans Christian. Stochastic Processes in Polymeric Fluids. 1996. Springer-Verlag. Berlin-Heidelberg. 978-3-540-58353-0. 75.
10. Kamenshchikov . S. . Clustering and Uncertainty in Perfect Chaos Systems. Journal of Chaos . 2014 . 1–6 . 2014 . 10.1155/2014/292096. 1301.4481 . 17719673 . free .
11. Book: Pavliotis, Grigorios A. . 2014 . Stochastic Processes and Applications : Diffusion Processes, the Fokker-Planck and Langevin Equations . Springer . 978-1-4939-1322-0 . 38–40 . 10.1007/978-1-4939-1323-7_2 .
12. Rosenbluth . M. N. . Fokker–Planck Equation for an Inverse-Square Force . Physical Review . 107 . 1. 1–6 . 1957 . 10.1103/physrev.107.1. 1957PhRv..107....1R .
13. Web site: Smoluchowski Diffusion Equation. Ioan. Kosztin. Spring 2000. Non-Equilibrium Statistical Mechanics: Course Notes.
14. Web site: The Brownian Dynamics Method Applied. Kosztin. Ioan. Spring 2000. Non-Equilibrium Statistical Mechanics: Course Notes.
15. Web site: Brownian Dynamics. Koztin. Ioan. Non-Equilibrium Statistical Mechanics: Course Notes. 2020-05-18. 2020-01-15 . https://web.archive.org/web/20200115202424/http://www.ks.uiuc.edu/~kosztin/PHYCS498NSM/. dead.
16. Web site: The Brownian Dynamics Method Applied. Kosztin. Ioan . Non-Equilibrium Statistical Mechanics: Course Notes. 2020-05-18. 2020-01-15 . https://web.archive.org/web/20200115202424/http://www.ks.uiuc.edu/~kosztin/PHYCS498NSM/. dead.
17. Holubec Viktor, Kroy Klaus, and Steffenoni Stefano . Physically consistent numerical solver for time-dependent Fokker–Planck equations . Phys. Rev. E . 99 . 4. 032117 . 2019 . 10.1103/PhysRevE.99.032117. 30999402 . 1804.01285 . 2019PhRvE..99c2117H . 119203025 .
18. [Bruno Dupire]
19. [Bruno Dupire]
20. 10.1142/S0219024902001511. 2002. Brigo . D.. Mercurio. Fabio. Lognormal-Mixture Dynamics and Calibration to Market Volatility Smiles. International Journal of Theoretical and Applied Finance. 5. 4. 427–446. 10.1.1.210.4165.
21. 10.1088/1469-7688/3/3/303. Alternative asset-price dynamics and volatility smile. 2003. Brigo . D.. Mercurio . F.. Sartorelli . G.. Quantitative Finance. 3. 3. 173–183. 154069452.
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