The term "BGK mode" refers to a specific mode of behaviour in plasma physics known as the Bhatnagar-Gross-Krook (BGK) approximation. The BGK approximation is a simplification of the Boltzmann equation, which describes the behaviour of a dilute gas or plasma in terms of the distribution function of particles in phase space.

In the BGK approximation, the collision term in the Boltzmann equation is replaced with a relaxation term that models the effects of collisions on the distribution function. This relaxation term is expressed in a simpler form and assumes a local equilibrium state. The BGK approximation is often used to study the behaviour of plasmas with relatively weak collisions or in situations where the full complexity of the Boltzmann equation is not necessary.

The BGK mode refers to a specific solution or mode of behaviour that emerges from the BGK approximation. It is characterized by the relaxation of the distribution function toward a local Maxwellian equilibrium state. In this mode, the particles in the plasma undergo collisions and exchange energy, momentum, and other properties, leading to a gradual relaxation of the system towards a new equilibrium.

The BGK mode is particularly relevant in the context of kinetic simulations and models of plasmas, where the Boltzmann equation is approximated using the BGK approach. It provides a simplified description of the particle interactions and allows for efficient numerical simulations of plasma behaviour.

It is important to note that the BGK approximation is an approximation and does not capture all the details of the true kinetic behaviour of a plasma. However, it is a widely used and effective method for studying many plasma phenomena, including the evolution of plasma waves, particle transport, and plasma instabilities.

We have seen that Landau damping is directly connected to the requirement that fo(γ) be initially uniform in space. On other hand, one could adjust f(v, t=0) such that the particles neither gain nor loose energy by adjusting the relative number trapped and untrapped particles. In such cases the density a constant along each trajectory such a wave is BGK mode

Van Kampen mode

The BGK mode, as mentioned earlier, arises from the Bhatnagar-Gross-Krook (BGK) approximation, which simplifies the Boltzmann equation for the distribution function in plasma. The BGK mode represents the relaxation of the distribution function towards a local Maxwellian equilibrium.

On the other hand, the Van Kampen mode emerges as a small amplitude limit of the BGK mode. It is a linear wave solution to the BGK equation in the limit of small perturbations. The Van Kampen mode describes the behaviour of small-amplitude waves in plasmas and is particularly relevant for studying the linear stability of plasmas and wave-particle interactions.

In Van - Kampen mode is only the particles with velocity V = V𝜙. The number of trapped particles can be changed by adding to f(v, t-0) a term proportional to 𝛿(v- v𝜙) i.e

𝛿(v- v𝜙) = 0 ; V ≠ V𝜙

= 1 ; V = V𝜙

Adding particles with V = V𝜙 will not cause damping, there are just many particles gaining energy or loosing energy.

The Van Kampen mode helps to understand how small-amplitude waves propagate and interact with particles in a plasma. It provides insights into the dispersion properties, resonances, and wave-particle coupling in a linear regime. It is a useful tool for studying various plasma phenomena, including wave heating, wave absorption, and particle acceleration.

This note is a part of the Physics Repository.

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