Fokker–Planck equation
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.
It is named after Adriaan Fokker and Max Planck, who described it in 1914 and 1917.
It includes an added noise-induced drift term due to diffusion gradient effects if the noise is state-dependent.
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[11] that in this case an analytical spectrum of solutions allows deriving a local uncertainty relation for the coordinate-velocity phase volume:
is an M-dimensional standard Wiener process, the probability density
is the Hermitian adjoint to the infinitesimal generator for the Markov process.
[12] A standard scalar Wiener process is generated by the stochastic differential equation
grows sufficiently rapidly (that is, the potential well is deep enough to confine the particle), the Boltzmann distribution is the unique equilibrium.
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
In plasma physics, the distribution function for a particle species
experiences due to collisions with all other particle species in unit time.
Consider an overdamped Brownian particle under external force
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
Therefore, Boltzmann statistics can be applied for the probability of a particles location at equilibrium, where
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
with an initial condition of the ensemble of particles starting in the same place,
is assumed to be Gaussian with the amplitude being dependent of the temperature of the system
The integration of this equation was done using the Euler–Maruyama method to numerically approximate the path of this Brownian particle.
consistent with a probability density obtained from market option quotes.
consistent with f. This is an inverse problem that has been solved in general by Dupire (1994, 1997) with a non-parametric solution.
[19][20] Brigo and Mercurio (2002, 2003) propose a solution in parametric form via a particular local volatility
The path integral formulation is an excellent starting point for the application of field theory methods.
A derivation of the path integral is possible in a similar way as in quantum mechanics.
Start by inserting a delta function and then integrating by parts:
[27] 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.
