The Fokker-Planck equation describes the time evolution of the probability density function of position and velocity of a particle. 8:17 am, August 6, 1945, Japanese time. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... Velocity (symbol: v) is a vector measurement of the rate and direction of motion. ...

The first use of the Fokker-Planck equation was the statistical description of Brownian motion of a particle in a fluid. An example of 1000 simulated steps of Brownian motion in two dimensions. ... A subset of the phases of matter, fluids include liquids, gases, plasmas and, to some extent, plastic solids. ...

Brownian motion follows the Langevin equation, which can be solved for many different stochastic forcings with results being averaged (the Monte Carlo Method). In statistical physics, a Langevin equation is a stochastic differential equation describing Brownian motion in a potential. ... Monte Carlo methods are algorithms for solving various kinds of computational problems by using random numbers (or more often pseudo-random numbers), as opposed to deterministic algorithms. ...

However, instead of this computationally intensive approach, one can use the Fokker-Planck equation and consider , that is, the probability density function of the particle having a velocity in the interval , when it starts its motion with at time t₀.

The general form of the Fokker-Planck equation for N variables is

where D¹ is the drift vector and D² the diffusion tensor, the latter of which results from the presence of the stochastic force. In physics and engineering, the word vector typically refers to a quantity that has close relationship to the spatial coordinates, informally described as an object with a magnitude and a direction. The word vector is also now used for more general concepts (see also vector and generalizations below), but this... Diffusion is the spontaneous spreading of something such as particles, heat, or momentum. ... In mathematics, a tensor is a certain kind of geometrical entity, or alternatively generalized quantity. The tensor concept includes the ideas of scalar, vector and linear operator. ...