In a fully ionized plasma, the collisional is the effect of many small coulomb collisions and usual procedure is to use Fokker planck equation for the collision term.

The Fokker-Planck equation is a partial differential equation used to describe the time evolution of the probability distribution function in systems that experience random forces or collisions. It is commonly employed in the study of various physical systems, including plasma physics, condensed matter physics, and statistical mechanics.

Let 𝜓 (v, ∆v) be the probability that a particle initially with velocity v undergoing many micro collisions acquires an increment of velocity ∆v in time ∆t, f(r,v,t) can be written as

Using Taylor's theorem, we can expand the product of f𝜓 to second order as

So we can write

Interpretation

- ∆v is change of velocity in a collision

- The first term on RHS contain a term -<∆v>/∂t called negative acceleration. It defines the frictional force that causes rapid particles to slow down and slow particles to accelerate. This term acts as narrowing of the distribution.

- The second term is a coefficient of diffusion of velocity-space. It describes the fact that a narrow velocity distribution will broaden as a result of collision.

The two term thus operate in opposite senses are in balance for an equilibrium condition. There is no net change in f because dynamic friction balances diffusion in velocity space.

The Fokker-Planck equation incorporates various physical processes, including advection (movement of particles), diffusion, and drift (particle motion under the influence of forces). It describes how the probability distribution function evolves over time due to these processes.

The Fokker-Planck equation is particularly useful in analyzing systems that involve interactions, random forces, or collisions, such as plasmas. It allows for the study of the statistical behaviour of particles and the evolution of their probability distribution in such systems.

Solving the Fokker-Planck equation requires appropriate initial and boundary conditions, as well as assumptions about the specific form of the diffusion coefficient and potential energy. Depending on the complexity of the system, analytical solutions may not always be possible, and numerical methods are often employed to approximate the solution.

This note is a part of the Physics Repository.

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