Vlasov-Maxwell equations

Note

This discussion is based on the Nättilä (2019) paper, where an extended discussion can be found.

Let us discuss the special-relativistic formulation of the Vlasov/Boltzmann equation. The spatial coordinate location vector is \(\mathbf{x} \equiv (x,y,z)\) and coordinate time is denoted by \(t\). The coordinate velocity (three-velocity) is \(\mathbf{v} \equiv d_t \mathbf{x}\), and its individual Cartesian components are written as \(\mathbf{v} = (v_x, v_y, v_z)\). The proper time (measured with a co-moving clock), \(\tau\), is related to the coordinate time through the Lorentz factor \(\gamma \equiv d_{\tau} t\). The proper velocity (the spatial components of the four-velocity) is \(\mathbf{u} \equiv d_{\tau} \mathbf{x} = \gamma \mathbf{v}\). The Lorentz factor and the velocities are related by the expression

\[\gamma^2 = 1 + (u/c)^2 = (1-(v/c)^2)^{-1},\]

where \(c\) is the speed of light, \(u = |\mathbf{u}|\) and \(v = |\mathbf{v}|\). Acceleration is denoted with \(\mathbf{a} \equiv d_t \mathbf{u}\).

The six-dimensional phase-space density distribution for particle species s is \(f_s \equiv f_s(\mathbf{x}, \mathbf{u}; t)\). Thus, \(f_s\, d^3 x \, d^3 u\) is the number of particles in the six-dimensional differential phase-space volume between \(\mathbf{x}\), \(\mathbf{u}\) and \(\mathbf{x} + d\mathbf{x}\), \(\mathbf{u} + d\mathbf{u}\).

The evolution of \(f_s\) is governed by the relativistic Boltzmann/Vlasov equation

\[\frac{\partial f_s}{\partial t} + \mathbf{v} \cdot \nabla_{\mathbf{x}} f_s + \mathbf{a}_{\mathrm{s}} \cdot \nabla_{\mathbf{u}} f_s = C,\]

where \(\nabla_{\mathbf{x}} = \frac{d}{d \mathbf{x}}\) and \(\nabla_{\mathbf{u}} = \frac{d}{d \mathbf{u}}\) are the spatial and momentum parts of the differential operator \(\nabla\), respectively. The term on the right-hand side, defined as \(C \equiv \partial_t f_s ~|_{\mathrm{coll}}\), is the collision operator. For the Vlasov equation \(C = 0\), i.e., the plasma is collisionless.

The acceleration of a charged particle, \(\mathbf{a}_{\mathrm{s}}\), is governed by the Lorentz force

\[\mathbf{a}_{\mathrm{s}} \equiv \mathrm{d}_t \mathbf{u} = \frac{q_{\mathrm{s}} }{ m_{\mathrm{s}} } \left(\mathbf{E} + \frac{\mathbf{v}}{c} \times \mathbf{B} \right) = \frac{q_{\mathrm{s}} }{ m_{\mathrm{s}} } \left(\mathbf{E} + \frac{\mathbf{u}}{\gamma c} \times \mathbf{B} \right),\]

where \(\mathbf{E}\) and \(\mathbf{B}\) are the electric and magnetic fields, \(q_{\mathrm{s}}\) is the charge, and \(m_{\mathrm{s}}\) is the mass of the particle of species s.

Moments of the distribution function define macroscopic (bulk) quantities of the plasma. The zeroth moment of the distribution function \(f_s\) defines the charge density of species s as

\[\rho_{\mathrm{s}} \equiv q_{\mathrm{s}} \int f_s \, \mathrm{d} \mathbf{u}.\]

The total charge density is \(\rho = \sum_{\mathrm{s}} \rho_{\mathrm{s}}\). The first moment defines the current (charge flux) vector as

\[\mathbf{J}_{\mathrm{s}} \equiv q_{\mathrm{s}} \int f_s \mathbf{v} \, \mathrm{d} \mathbf{u} = q_{\mathrm{s}} \int f_s \, \frac{ \mathbf{u}}{\gamma} ~\mathrm{d} \mathbf{u}.\]

The total current is obtained by summing over all plasma species, \(\mathbf{J} = \sum_{\mathrm{s}} \mathbf{J}_{\mathrm{s}}\).

Maxwell’s equations

The evolution of the electric field \(\mathbf{E}\) and magnetic field \(\mathbf{B}\) is governed by Maxwell’s equations. These are Gauss’s law,

\[\nabla \cdot \mathbf{E} = 4\pi \rho,\]

Gauss’s law for magnetism,

\[\nabla \cdot \mathbf{B} = 0,\]

Faraday’s law,

\[\nabla \times \mathbf{E} = -\frac{1}{c}\frac{\partial \mathbf{B}}{\partial t},\]

and Ampère’s law,

\[\nabla \times \mathbf{B} = \frac{4\pi}{c}\mathbf{J} +\frac{1}{c}\frac{\partial \mathbf{E}}{\partial t}.\]

Charge conservation follows from these by taking the divergence of Ampère’s law and substituting Gauss’s law to get

\[\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{J} = 0.\]