FokkerPlanckEquation - crowlogic/arb4j GitHub Wiki

The Fokker-Planck equation, also known as the Kolmogorov forward equation, is a partial differential equation that describes the time evolution of the probability density function of a stochastic process. It is widely used in various fields, such as statistical physics, financial mathematics, and population biology.

The general form of the Fokker-Planck equation is given by:

$$\frac{\partial P(x,t)}{\partial t} = \frac{\partial^2}{\partial x^2} \left[ D(x)P(x,t) \right] - \frac{\partial}{\partial x} \left[ F(x)P(x,t) \right]$$

where $P(x,t)$ is the probability density function at time $t$ for a particle with position $x$, $F(x)$ is the drift velocity (the average velocity of the particle), and $D(x)$ is the diffusion coefficient (a measure of the randomness of the particle's motion). The addend on the right-hand side represents the drift of the particle, while the subtrahend represents its diffusion.

The Fokker-Planck equation can be used to model a wide range of stochastic processes, including Brownian motion, which describes the random motion of particles suspended in a fluid.

There are many techniques available for solving the Fokker-Planck equation, including numerical methods and analytical techniques such as perturbation theory and the method of characteristics. These methods can be used to study the properties of stochastic systems, such as the probability of a particle reaching a certain position or the time it takes for a particle to diffuse a certain distance.

Overall, the Fokker-Planck equation provides a powerful tool for understanding the behavior of stochastic systems, and has numerous applications in physics, chemistry, biology, and finance.