A solution to the one-dimensional Fokker–Planck equation, with both the drift and the diffusion term. In this case the initial condition is a Dirac delta function centered away from zero velocity. Over time the distribution widens due to random impulses.
The transition probability, the probability of going from to , is introduced here; the expectation can be written as
Now we replace in the definition of , multiply by and integrate over . The limit is taken on
Note now that
which is the Chapman–Kolmogorov theorem. Changing the dummy variable to , one gets
which is a time derivative. Finally we arrive to
From here, the Kolmogorov backward equation can be deduced. If we instead use the adjoint operator of , , defined such that
then we arrive to the Kolmogorov forward equation, or Fokker–Planck equation, which, simplifying the notation , in its differential form reads
Remains the issue of defining explicitly . This can be done taking the expectation from the integral form of the Itô's lemma:
The part that depends on vanished because of the martingale property.
Then, for a particle subject to an Itô equation, using
it can be easily calculated, using integration by parts, that
which bring us to the Fokker–Planck equation:
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:
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:
This model has discrete spectrum of solutions if the condition of fixed boundaries is added for :
It has been shown that in this case an analytical spectrum of solutions allows deriving a local uncertainty relation for the coordinate-velocity phase volume:
Here is a minimal value of a corresponding diffusion spectrum , while and represent the uncertainty of coordinate–velocity definition.
with . Physically, this equation can be motivated as follows: a particle of mass with velocity 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 with . 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; . Newton's second law is written as
Taking for simplicity and changing the notation as leads to the familiar form .
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 and are the average change in velocity a particle of type experiences due to collisions with all other particle species in unit time. Expressions for these quantities are given elsewhere. If collisions are ignored, the Boltzmann equation reduces to the Vlasov equation.
The Smoluchowski Diffusion equation is the Fokker-Planck equation restricted to Brownian particles affected by an external force .
Where is the diffusion constant and . 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.
Derivation of the Smoluchowski Equation from the Fokker-Planck Equation
Starting with the Langevin Equation of a Brownian particle in external field , where is the friction term, is a fluctuating force on the particle, and is the amplitude of the fluctuation.
At equilibrium the frictional force is much greater than the inertial force, . Therefore the Langevin equation becomes,
Which generates the following Fokker-Planck equation,
Rearranging the Fokker-Planck equation,
Where . Note, the diffusion coefficient may not necessarily be spatially independent if or are spatially dependent.
Next, the total number of particles in any particular volume is given by,
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.
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 is a conservative force and the probability of a particle being in a state is given as .
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 of the particle having a velocity in the interval when it starts its motion with at time 0.
Brownian Dynamics simulation for particles in 1-D linear potential compared with the solution of the Fokker-Planck equation.
Starting with a linear potential of the form the corresponding Smoluchowski equation becomes,
Where the diffusion constant, , is constant over space and time. The boundary conditions are such that the probability vanishes at with an initial condition of the ensemble of particles starting in the same place, .
Defining and and applying the coordinate transformation,
With the Smoluchowki equation becomes,
Which is the free diffusion equation with solution,
And after transforming back to the original coordinates,
The simulation on the right was completed using a Brownian dynamics simulation. Starting with a Langevin equation for the system,
Where is the friction term, is a fluctuating force on the particle, and is the amplitude of the fluctuation. At equilibrium the frictional force is much greater than the inertial force, . Therefore the Langevin equation becomes,
For the Brownian dynamic simulation the fluctuation force is assumed to be Gaussian with the amplitude being dependent of the temperature of the system . Rewriting the Langevin equation,
Where 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.
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.
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 for volatility smile modeling of options via local volatility, 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–Planck equation given by a mixture model. More information is available also in Fengler (2008), Gatheral (2008), and Musiela and Rutkowski (2008).
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. 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 is as follows. Start by inserting a delta function and then integrating by parts:
The -derivatives here only act on the -function, not on . Integrate over a time interval ,
This equation expresses as functional of . Iterating times and performing the limit gives a path integral with action
The variables conjugate to are called "response variables".
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.
^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).
^Bruno Dupire (1994) Pricing with a Smile. Risk Magazine, January, 18–20.
^Bruno Dupire (1997) Pricing and Hedging with Smiles. Mathematics of Derivative Securities. Edited by M.A.H. Dempster and S.R. Pliska, Cambridge University Press, Cambridge, 103–111. ISBN0-521-58424-8.
Pavliotis, Grigorios A. (2014). Stochastic Processes and Applications: Diffusion Processes, the Fokker–Planck and Langevin Equations. Springer Texts in Applied Mathematics. Springer. ISBN978-1-4939-1322-0.
Risken, Hannes (1996). The Fokker–Planck Equation: Methods of Solutions and Applications. Springer Series in Synergetics (2nd ed.). Springer. ISBN3-540-61530-X.