Vlasov Equation

Vlasov Equation Explained

Vlasov equation is a differential equation that is used to model how plasma behaves and evolves over time.

The equation has applications in fields where we have to deal with hot, charged particles, which interact with each other. An example is in stars in outer space, which are hot gases of charged particles.

Prerequisites

To see the context of Vlasov’s equations, we need to know some terms beforehand. This section outlines some of them.

Plasma

Plasma is the fourth state of matter. It consists of hot gas of charged ions and electrons. Plasma occurs in situations like lightning, discharge tubes and neon lights. It is observed in fully ionised form in the stars.

Plasma globes, a tool which help demonstrate the curious fourth state of matter. (Source

Phase Space

This is a way to describe the state of a system. It is often used in Hamiltonian and Lagrangian mechanics.

It is like a coordinate system that describes a point in space by using 3 coordinates, for the x, y and z axes. However, in the context of Vlasov’s equation, phase space has 6 coordinates, 3 corresponding to the position and 3 corresponding to respective momenta.

An example of phase space in a different context. It has only 3 dimensions, two of which are spatial and one is related to momentum.
An example of phase space in a different context. It has only 3 dimensions, two of which are spatial and one is related to momentum. (Source)

Distribution Function

A distribution function does just what its name says: it shows how particles are distributed. In this context, it describes the number (or equivalently, fraction) of particles that occupy specific positions in the phase space. As we can imagine, plasma consists of many particles, each with their own position and momentum. The distribution function accounts for this and explains their relative distribution.

The Equation

What Does The Equation Do?

The equation lets us calculate the future state of a system, which has a given state in the present. Here, by state, we are referring to the distribution function of the system. As such, this idea is not new. Using distribution functions to describe multi-particle systems was quite common in physics, especially thermodynamics.

The Need For The Equation

Prior to the development of the Vlasov equation, the prevailing method to deal with distribution functions was with Boltzmann’s equation. This equation was used to get the distribution function of a system, given its physical conditions.

It had success in many thermodynamic systems such as fluids with temperature gradients and so forth. However, this had a collision term that involved assuming that the particles involved kept colliding. While this worked in most cases which it was previously applied to, it could not satisfactorily be extended to plasma.

One reason was because plasma consists of charged particles, which interact at long ranges via Coulomb interaction. It also ended up going against experimental evidence that other scientists came up with.

The Modifications Vlasov Made

Anatoly Vlasov realized that the Boltzmann equation was not appropriate for studying plasmas. Thus, what he did was simple: he eliminated the term corresponding to the collisions, which was the problem. Thus, at the beginning, it was known as collisionless Boltzmann equation but later came to be known as Vlasov equation.

Vlasov Equation

Where the derivatives with respect to r and p correspond to the spatial position and momentum. f is simply the distribution function corresponding to the plasma.

The Extensions

Vlasov equation as stated above is the simplest, most basic form. As such, the form of the distribution function can change. The basic principle remains the same, we try to arrive at the solution for f at any later point in time.

One example is using it for non-relativistic particles which are not under the influence of gravity. This gives us, using Hamilton’s equations:

Hamilton's equations

Where the respective derivatives are replaced by their equivalents in Hamiltonian mechanics.

We can also expand it into a system of equations that will give us more context based on the parameters of the system. 

So, that was a general overview of Vlasov equation. Hope you found it useful.

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