# Derivation of the Fokker-Planck Equation by Continuity

• I
• fayled
It seems like the Wikipedia formula is for a specific case, while the formula derived in the conversation is more general.
fayled
Derive the Fokker-Planck equation by requiring conservation of probability:
∂VJ⋅dS=-d/dt∫Vp(r,t)dV
The flux can be written as a sum of convective and diffusive terms
J=p(r,t)v(r,t)-D(r,t)p(r,t)
and substitution of this with use of the divergence theorem yields
tp(x,t)=-∂x[p(x,t)v(x,t)]+∂x[D(x,t)∂xp(x,t)]
where I have moved to one dimension for simplicity.

However the form found here
https://en.wikipedia.org/wiki/Fokker–Planck_equation
is given as
tp(x,t)=-∂x[p(x,t)v(x,t)]+∂x2[D(x,t)p(x,t)]

I was wondering if anybody would be able to help me account for this difference. Thanks!

You might have better luck in a math forum.

## 1. What is the Fokker-Planck equation?

The Fokker-Planck equation is a partial differential equation that describes the temporal evolution of the probability density function of a stochastic process. It is widely used in statistical physics and stochastic processes to model the behavior of a large number of particles or systems.

## 2. How is the Fokker-Planck equation derived?

The Fokker-Planck equation can be derived using the principle of continuity and the Chapman-Kolmogorov equation. This involves considering the changes in the probability density function over small time intervals and then taking the limit as the time interval approaches zero.

## 3. What is the role of the continuity equation in deriving the Fokker-Planck equation?

The continuity equation is a fundamental equation in physics that describes the conservation of a quantity. In the context of the Fokker-Planck equation, it is used to ensure that the integral of the probability density function over all possible states remains constant.

## 4. What are some applications of the Fokker-Planck equation?

The Fokker-Planck equation has many applications in various fields such as physics, chemistry, biology, and economics. It is commonly used to model diffusion processes, Brownian motion, and other stochastic processes. It is also used in the study of financial markets and in the analysis of biological systems.

## 5. Are there any limitations to the Fokker-Planck equation?

Like any mathematical model, the Fokker-Planck equation has its limitations. It assumes that the underlying stochastic process is Markovian and that the probability density function is smooth and continuous. It also does not take into account any external influences or fluctuations that may affect the system. Therefore, it may not accurately describe some real-world systems and may need to be modified or combined with other equations to achieve a more accurate representation.

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