Vlasov equation

The Vlasov–Maxwell system of equations

Rather than relying on a collision-based kinetic description for the interaction of charged particles in plasma, Vlasov utilized a self-consistent collective field created by the charged plasma particles. Such a description uses the distribution functions ${\displaystyle f_e(\mathbf{r},\mathbf{p},t)}$ and ${\displaystyle f_i(\mathbf{r},\mathbf{p},t)}$ for electrons and plasma cations. The distribution function ${\displaystyle f_{\alpha }(\mathbf{r},\mathbf{p},t)}$ for a species α describes the number of particles of the species α having approximately the momentum ${\displaystyle \mathbf{p}}$ near the position ${\displaystyle \mathbf{r}}$ at time t. Instead of the Boltzmann equation, the following system of equations was proposed as a description of the charged components of a plasma:

$${\displaystyle \frac{\mathrm{\partial }{f}_{\alpha }}{\mathrm{\partial }t}+{\mathbf{v}}_{\alpha }\cdot \mathrm{\nabla }{f}_{\alpha }+q_{\alpha }\left(\mathbf{E}+{\mathbf{v}}_{\alpha }\times \mathbf{B}\right)\cdot \frac{\mathrm{\partial }{f}_{\alpha }}{\mathrm{\partial }\mathbf{p}}=0}$$

$${\displaystyle \begin{array}{cccc}\mathrm{\nabla }\times \mathbf{B}& ={\mu }_0\mathbf{j}+{\mu }_0{\epsilon }_0\frac{\mathrm{\partial }\mathbf{E}}{\mathrm{\partial }t},& \mathrm{\nabla }\cdot \mathbf{B}& =0,\\ \mathrm{\nabla }\times \mathbf{E}& =-\frac{\mathrm{\partial }\mathbf{B}}{\mathrm{\partial }t},& \mathrm{\nabla }\cdot \mathbf{E}& =\frac{\rho }{{\epsilon }_0},\end{array}}$$

$${\displaystyle \begin{array}{cc}\rho & =\sum _{\alpha }{q}_{\alpha }\int f_{\alpha }({\mathbf{r}}_{\alpha },{\mathbf{p}}_{\alpha },t){\mathrm{d}}^3\mathbf{p},\\ \mathbf{j}& =\sum _{\alpha }{q}_{\alpha }\int f_{\alpha }({\mathbf{r}}_{\alpha },{\mathbf{p}}_{\alpha },t){\mathbf{v}}_{\alpha }{\mathrm{d}}^3\mathbf{p},\\ {\mathbf{v}}_{\alpha }& =\frac{{\mathbf{p}}_{\alpha }/m_{\alpha }}{\sqrt{1+p_{\alpha }^2/{\left(m_{\alpha }c\right)}^2}}\end{array}}$$

Here c is the speed of light, q_α is the charge of each species, m_α is the mass of each species, ${\displaystyle \mathbf{E}(\mathbf{r},t)}$ and ${\displaystyle \mathbf{B}(\mathbf{r},t)}$ represent collective self-consistent electromagnetic field created in the point ${\displaystyle \mathbf{r}}$ at time moment t by all plasma particles. The essential difference of this system of equations from equations for particles in an external electromagnetic field is that the self-consistent electromagnetic field depends in a complex way on the distribution functions of electrons and ions ${\displaystyle f_e(\mathbf{r},\mathbf{p},t)}$ and ${\displaystyle f_i(\mathbf{r},\mathbf{p},t)}$.

The Vlasov–Poisson equation

The Vlasov–Poisson equations are an approximation of the Vlasov–Maxwell equations in the non-relativistic zero-magnetic field limit: $${\displaystyle \frac{\mathrm{\partial }{f}_{\alpha }}{\mathrm{\partial }t}+{\mathbf{v}}_{\alpha }\cdot \frac{\mathrm{\partial }{f}_{\alpha }}{\mathrm{\partial }\mathbf{x}}+\frac{q_{\alpha }\mathbf{E}}{m_{\alpha }}\cdot \frac{\mathrm{\partial }{f}_{\alpha }}{\mathrm{\partial }\mathbf{v}}=0,}$$

and Poisson's equation for self-consistent electric field: $${\displaystyle {\mathrm{\nabla }}^2\varphi +\frac{\rho }{\epsilon }=0.}$$

Here q_α is the particle's electric charge, m_α is the particle's mass, ${\displaystyle \mathbf{E}(\mathbf{x},t)}$ is the self-consistent electric field, ${\displaystyle \varphi (\mathbf{x},t)}$ the self-consistent electric potential, ρ is the electric charge density, and ${\displaystyle \epsilon }$ is the electric permitivity.

Vlasov–Poisson equations are used to describe various phenomena in plasma, in particular Landau damping and the distributions in a double layer plasma, where they are necessarily strongly non-Maxwellian, and therefore inaccessible to fluid models.

Moment equations

In fluid descriptions of plasmas (see plasma modeling and magnetohydrodynamics (MHD)) one does not consider the velocity distribution. This is achieved by replacing ${\displaystyle f(\mathbf{r},\mathbf{v},t)}$ with plasma moments such as number density n, flow velocity u and pressure p. They are named plasma moments because the n-th moment of ${\displaystyle f}$ can be found by integrating ${\displaystyle v^{n}f}$ over velocity. These variables are only functions of position and time, which means that some information is lost. In multifluid theory, the different particle species are treated as different fluids with different pressures, densities and flow velocities. The equations governing the plasma moments are called the moment or fluid equations.

Below the two most used moment equations are presented (in SI units). Deriving the moment equations from the Vlasov equation requires no assumptions about the distribution function.

Continuity equation

The continuity equation describes how the density changes with time. It can be found by integration of the Vlasov equation over the entire velocity space. $${\displaystyle \int \frac{\mathrm{d}f}{\mathrm{d}t}{\mathrm{d}}^3\mathbf{v}=\int \left(\frac{\mathrm{\partial }f}{\mathrm{\partial }t}+\mathbf{v}\cdot \frac{\mathrm{\partial }f}{\mathrm{\partial }\mathbf{r}}+\mathbf{a}\cdot \frac{\mathrm{\partial }f}{\mathrm{\partial }\mathbf{v}}\right){\mathrm{d}}^3\mathbf{v}=0}$$

After some calculations, one ends up with $${\displaystyle \frac{\mathrm{\partial }n}{\mathrm{\partial }t}+\mathrm{\nabla }\cdot (n\mathbf{u})=0.}$$

The number density n, and the momentum density nu, are zeroth and first order moments: $${\displaystyle \begin{array}{cc}n& =\int f(\mathbf{r},\mathbf{v},t){\mathrm{d}}^3\mathbf{v}\\ n\mathbf{u}& =\int \mathbf{v}f(\mathbf{r},\mathbf{v},t){\mathrm{d}}^3\mathbf{v}\end{array}}$$

Momentum equation

The rate of change of momentum of a particle is given by the Lorentz equation: $${\displaystyle m\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}=q(\mathbf{E}+\mathbf{v}\times \mathbf{B})}$$

By using this equation and the Vlasov Equation, the momentum equation for each fluid becomes $${\displaystyle mn\frac{\mathrm{D}}{\mathrm{D}t}\mathbf{u}=-\mathrm{\nabla }\cdot \mathcal{P}+qn\mathbf{E}+qn\mathbf{u}\times \mathbf{B},}$$ where ${\displaystyle \mathcal{P}}$ is the pressure tensor. The material derivative is $${\displaystyle \frac{\mathrm{D}}{\mathrm{D}t}=\frac{\mathrm{\partial }}{\mathrm{\partial }t}+\mathbf{u}\cdot \mathrm{\nabla }.}$$

The pressure tensor is defined as the particle mass times the covariance matrix of the velocity: $${\displaystyle \begin{array}{cc}\mathcal{P}& =m\int (\mathbf{v}-\mathbf{u})(\mathbf{v}-\mathbf{u})f(\mathbf{r},\mathbf{v},t){\mathrm{d}}^3\mathbf{v}\\ P_{ij}& =m\int (v_i-u_i)(v_j-u_j)f(\mathbf{r},\mathbf{v},t){\mathrm{d}}^3\mathbf{v}.\end{array}}$$

The frozen-in approximation

As for ideal MHD, the plasma can be considered as tied to the magnetic field lines when certain conditions are fulfilled. One often says that the magnetic field lines are frozen into the plasma. The frozen-in conditions can be derived from Vlasov equation.

We introduce the scales T, L, and V for time, distance and speed respectively. They represent magnitudes of the different parameters which give large changes in ${\displaystyle f}$. By large we mean that $${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }t}T\sim f\left|\frac{\mathrm{\partial }f}{\mathrm{\partial }\mathbf{r}}\right|L\sim f\left|\frac{\mathrm{\partial }f}{\mathrm{\partial }\mathbf{v}}\right|V\sim f.}$$

We then write $${\displaystyle t^{\prime }=\frac{t}{T},{\mathbf{r}}^{\prime }=\frac{\mathbf{r}}{L},{\mathbf{v}}^{\prime }=\frac{\mathbf{v}}{V}.}$$

Vlasov equation can now be written $${\displaystyle \frac{1}{T}\frac{\mathrm{\partial }f}{\mathrm{\partial }{t}^{\prime }}+\frac{V}{L}{\mathbf{v}}^{\prime }\cdot \frac{\mathrm{\partial }f}{\mathrm{\partial }{\mathbf{r}}^{\prime }}+\frac{q}{mV}(\mathbf{E}+V{\mathbf{v}}^{\prime }\times \mathbf{B})\cdot \frac{\mathrm{\partial }f}{\mathrm{\partial }{\mathbf{v}}^{\prime }}=0.}$$

So far no approximations have been done. To be able to proceed we set ${\displaystyle V=R{\omega }_g}$, where ${\displaystyle {\omega }_g=qB/m}$ is the gyro frequency and R is the gyroradius. By dividing by ω_g, we get $${\displaystyle \frac{1}{{\omega }_{g}T}\frac{\mathrm{\partial }f}{\mathrm{\partial }{t}^{\prime }}+\frac{R}{L}{\mathbf{v}}^{\prime }\cdot \frac{\mathrm{\partial }f}{\mathrm{\partial }{\mathbf{r}}^{\prime }}+\left(\frac{\mathbf{E}}{VB}+{\mathbf{v}}^{\prime }\times \frac{\mathbf{B}}{B}\right)\cdot \frac{\mathrm{\partial }f}{\mathrm{\partial }{\mathbf{v}}^{\prime }}=0}$$

If ${\displaystyle 1/{\omega }_g\ll T}$ and ${\displaystyle R\ll L}$, the two first terms will be much less than ${\displaystyle f}$ since ${\displaystyle \mathrm{\partial }f/\mathrm{\partial }{t}^{\prime }\sim f,v^{\prime }\lesssim 1}$ and ${\displaystyle \mathrm{\partial }f/\mathrm{\partial }{\mathbf{r}}^{\prime }\sim f}$ due to the definitions of T, L, and V above. Since the last term is of the order of ${\displaystyle f}$, we can neglect the two first terms and write $${\displaystyle \left(\frac{\mathbf{E}}{VB}+{\mathbf{v}}^{\prime }\times \frac{\mathbf{B}}{B}\right)\cdot \frac{\mathrm{\partial }f}{\mathrm{\partial }{\mathbf{v}}^{\prime }}\approx 0\Rightarrow (\mathbf{E}+\mathbf{v}\times \mathbf{B})\cdot \frac{\mathrm{\partial }f}{\mathrm{\partial }\mathbf{v}}\approx 0}$$

This equation can be decomposed into a field aligned and a perpendicular part: $${\displaystyle {\mathbf{E}}_{\parallel }\frac{\mathrm{\partial }f}{\mathrm{\partial }{\mathbf{v}}_{\parallel }}+({\mathbf{E}}_{\perp }+\mathbf{v}\times \mathbf{B})\cdot \frac{\mathrm{\partial }f}{\mathrm{\partial }{\mathbf{v}}_{\perp }}\approx 0}$$

The next step is to write ${\displaystyle \mathbf{v}={\mathbf{v}}_0+\mathrm{\Delta }\mathbf{v}}$, where $${\displaystyle {\mathbf{v}}_0\times \mathbf{B}=-{\mathbf{E}}_{\perp }}$$

It will soon be clear why this is done. With this substitution, we get $${\displaystyle {\mathbf{E}}_{\parallel }\frac{\mathrm{\partial }f}{\mathrm{\partial }{\mathbf{v}}_{\parallel }}+(\mathrm{\Delta }{\mathbf{v}}_{\perp }\times \mathbf{B})\cdot \frac{\mathrm{\partial }f}{\mathrm{\partial }{\mathbf{v}}_{\perp }}\approx 0}$$

If the parallel electric field is small, $${\displaystyle (\mathrm{\Delta }{\mathbf{v}}_{\perp }\times \mathbf{B})\cdot \frac{\mathrm{\partial }f}{\mathrm{\partial }{\mathbf{v}}_{\perp }}\approx 0}$$

This equation means that the distribution is gyrotropic. The mean velocity of a gyrotropic distribution is zero. Hence, ${\displaystyle {\mathbf{v}}_0}$ is identical with the mean velocity, u, and we have $${\displaystyle \mathbf{E}+\mathbf{u}\times \mathbf{B}\approx 0}$$

To summarize, the gyro period and the gyro radius must be much smaller than the typical times and lengths which give large changes in the distribution function. The gyro radius is often estimated by replacing V with the thermal velocity or the Alfvén velocity. In the latter case R is often called the inertial length. The frozen-in conditions must be evaluated for each particle species separately. Because electrons have much smaller gyro period and gyro radius than ions, the frozen-in conditions will more often be satisfied.