Kinetic Vlasov Equation | Month of Equations

Time and again, researchers try to study the different kinds of time-evolving physical systems. For this purpose, they use certain mathematical equations that describe the dynamics of the particles/components constituting these systems. Today, I am going to discuss a famous equation of statistical mechanics-The Kinetic Vlasov Equation. This equation helps to model the dynamics of a variety of physical systems, specifically plasmas.

"Plasma is a gas of charged and neutral particles that interact via long-range Coulomb interactions and exhibit quasi-neutrality as well as collective behavior. "

To model the dynamics of a particular plasma system, we mainly use two approaches-the fluid approach and the kinetic approach. Where the fluid approach deals with the averaged quantities such as number densities of charged species, average velocities of a particular charged fluid, electrostatic potential, etc., the kinetic approach is based on tracking the evolution of a probability distribution function for the particles. But in certain scenarios, we sometimes prefer to use the fluid theory while studying plasmas whereas, in other times, a plasma kinetic theory is best. Let me explain a bit more about the importance of both these approaches.

When we wish to study the collective dynamics of a plasma species, i.e., electrons or ions, it is better and often easier to take the fluid approach. On the other hand, single-particle effects such as resonant interactions of plasma particles with waves can only be studied via the kinetic approach to study the plasma system. It is where the Vlasov equation comes handy.

Kinetic Vlasov Equation
The Kinetic Vlasov Equation

The origin of the Vlasov Equation

The Vlasov equation originates from a simple principle of statistical mechanics:

The probability may flow throughout the phase space {x,v}, but cannot disappear. In other words, there is neither a "source" generating, not a "sink" absorbing the probability.

This means that the derivative of f (x,v,t) with time equates to zero. By applying the chain rule to this principle, the kinetic Vlasov equation is obtained. To model plasma effects kinetically, one requires a kinetic Vlasov equation for each plasma species and a probability distribution function.

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The probability distribution function

The probability distribution function f(x, v, t) describes the probability of finding a particle at a certain point (x,v) in the phase space at an instant 't'. Now, is there a form of expression for this distribution function described for plasmas? or can we exactly define such a distribution function for the different plasma species? The answer is No! However, we have the kinetic Vlasov equation that describes the dynamics of this distribution function and help us obtain an exact analytical form of this distribution function.

Graphical form of Maxwellian distribution function

There are various analytical forms of this distribution function. One of these is a simple Maxwellian distribution whereas there are also other non-Maxwellian forms of this distribution function.

Now, we all know well about the Maxwellian distribution function that has an exponential form, but the non-Maxwellian distributions may have a power-law type decay. The various non-Maxwellian distributions widely used to study space plasma environments are:

Various spacecraft observations have detected that space plasmas possess a non-Maxwellian type velocity distribution rather than a Maxwellian distribution. It is because of the high temperatures and low densities of these plasma environments. Various research studies show that the non-Maxwellian distributions are better to fit the observed spacecraft data of plasma particles. Altogether, the kinetic Vlasov equation along with a suitable probability distribution for particles helps to model plasma dynamics and the important phenomena such as Landau damping.

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Author's message:

As a plasma physicist, I study different plasma phenomena such as waves, instabilities as well as Landau damping. The purpose of this article was to make you familiar with the theoretical approaches used to study these plasma phenomena. On one hand, fluid theory helps study all the important linear and nonlinear plasma phenomena. On the other hand, to study phenomena such as particle acceleration kinetic theory is required. I hope this article gives you an insight into how do the researchers model a physical system. Happy reading!!

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