Explanation of Boltzmann equation

The Boltzmann equation is a fundamental equation in statistical mechanics that describes the behaviour of a system of particles in equilibrium. In the context of plasma physics, the Boltzmann equation is used to understand the statistical properties and dynamics of charged particles (ions and electrons) in a plasma.

It is important to note that the Boltzmann equation is a highly complex partial differential equation, and its solution is often challenging. Various approximations and simplifications are used to make progress in solving the Boltzmann equation and obtaining meaningful results for practical plasma systems.

The first term on LHS is the rate of change of number of particles due to fields and depend explicitly on time.

The second term on LHS is variation of number of particles due to their motion

The third term on LHS is variation of number of particles due to acceleration which causes velocity to change.

The term on RHS collision effect causing the variation of particles

Now, multiplying equation 1 by mv and integrating over velocity space, we obtain first order equation

Let us consider separate the v into average fluid velocity u and a thermal velocity w i.e

then second term becomes

This is fluid equation of motion. This equation describes the flow of momentum i.e the force balance in this component of plasma which yield conservation of momentum.

The Boltzmann equation is a kinetic equation that provides a microscopic description of the behaviour of particles in a plasma. By solving the Boltzmann equation, one can obtain information about various plasma properties, such as the distribution of particle velocities, temperature, density, and transport coefficients.

Second order moment: Energy conservation equation

Boltzmann equation is

Multiplying equation 1 by mv²/2 and integrating over hte velocity space we obtain second order moment.

Putting all these terms in equation 1, we get

• ∂(𝜌u^2/2 + U)/∂t represents the time derivative of the total energy density, which includes the kinetic energy (𝜌u^2/2) and internal energy (U) of the plasma.

• ∇·(𝜌u^2u/2 + Uu + p + q) represents the divergence of the energy flux vector, which consists of contributions from the kinetic energy flux (𝜌u^2u/2), internal energy flux (Uu), pressure flux (p), and heat flux (q).

• nqE·u represents the work done by the electric field (E) on the plasma due to the motion of charged particles with number density (n) and electric charge (q) in the direction of the velocity vector (u).

• 𝜌g·u represents the work done by the gravitational field (g) on the plasma, where 𝜌 is the density and u is the velocity vector.

• E_ij represents the components of the electric field tensor.

This equation describes the conservation of energy in the plasma, accounting for various terms that contribute to the change in energy density over time. The terms on the right-hand side represent energy input or output due to external fields (electric and gravitational) and the work done by them. The left-hand side represents the change in the total energy density with time and the divergence of energy fluxes within the plasma.

It's important to note that the specific form and interpretation of the equation can vary depending on the context and assumptions made in the plasma physics problem you are considering.

This note is a part of the Physics Repository.