Derivations

Fundamental derivations connecting Maxwell's equations to the fluid and kinetic equations used throughout pypic. Each section states the key result up front; click Derivation to expand the full mathematical development.

These derivations use normalized units (\(\mu_0 = \epsilon_0 = 1\)) unless stated otherwise. See Conventions for the normalization mapping.

Maxwell's Equations

The foundation for all plasma physics in pypic. In SI:

\[\nabla \cdot \mathbf{E} = \frac{\rho_c}{\epsilon_0}, \qquad \nabla \cdot \mathbf{B} = 0\]

\[\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}, \qquad \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\]

In normalized units (\(\mu_0 = \epsilon_0 = 1\)):

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

\[\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}, \qquad \nabla \times \mathbf{B} = \mathbf{J} + \frac{\partial \mathbf{E}}{\partial t}\]

The constraint \(\nabla \cdot \mathbf{B} = 0\) is monitored by div_b as a numerical health check. The displacement current \(\partial \mathbf{E}/\partial t\) is retained in PIC codes but dropped in MHD (\(v \ll c\)) ¹.

Vlasov-Maxwell System

PIC codes solve the Vlasov equation for the distribution function \(f_s(\mathbf{x}, \mathbf{v}, t)\) of each species \(s\):

\[\frac{\partial f_s}{\partial t} + \mathbf{v} \cdot \nabla f_s + \frac{q_s}{m_s}\left(\mathbf{E} + \mathbf{v} \times \mathbf{B}\right) \cdot \frac{\partial f_s}{\partial \mathbf{v}} = 0\]

This is a collisionless Boltzmann equation in 6D phase space. The electromagnetic fields \((\mathbf{E}, \mathbf{B})\) evolve via Maxwell's equations with sources computed from \(f_s\):

\[n_s = \int f_s \, d^3v, \qquad n_s \mathbf{V}_s = \int \mathbf{v} \, f_s \, d^3v\]

\[\rho_c = \sum_s q_s n_s, \qquad \mathbf{J} = \sum_s q_s n_s \mathbf{V}_s\]

This self-consistent loop (particles \(\to\) moments \(\to\) fields \(\to\) forces \(\to\) particles) is the Vlasov-Maxwell system.

Fluid Moment Equations

Taking velocity moments of the Vlasov equation yields the fluid equations that connect kinetic physics to the macroscopic quantities in pypic. The zeroth moment gives mass conservation; the first moment gives momentum balance.

Continuity (zeroth moment):

\[\frac{\partial n_s}{\partial t} + \nabla \cdot (n_s \mathbf{V}_s) = 0\]

Momentum (first moment):

\[m_s n_s \left(\frac{\partial \mathbf{V}_s}{\partial t} + \mathbf{V}_s \cdot \nabla \mathbf{V}_s\right) = n_s q_s \left(\mathbf{E} + \mathbf{V}_s \times \mathbf{B}\right) - \nabla \cdot \mathbf{P}_s\]

where the pressure tensor \(\mathbf{P}_s\) is the second velocity moment about the mean flow:

\[\mathbf{P}_s = m_s \int (\mathbf{v} - \mathbf{V}_s)(\mathbf{v} - \mathbf{V}_s) \, f_s \, d^3v\]

Derivation: velocity moments of the Vlasov equation

Multiply the Vlasov equation by a velocity-space weight \(\phi(\mathbf{v})\) and integrate over all velocities:

\[\int \phi \frac{\partial f_s}{\partial t} d^3v + \int \phi \, \mathbf{v} \cdot \nabla f_s \, d^3v + \frac{q_s}{m_s} \int \phi \left(\mathbf{E} + \mathbf{v} \times \mathbf{B}\right) \cdot \frac{\partial f_s}{\partial \mathbf{v}} d^3v = 0\]

Zeroth moment (\(\phi = 1\)):

The first term gives \(\partial n_s / \partial t\). The second gives \(\nabla \cdot (n_s \mathbf{V}_s)\). The third vanishes by integration by parts (the Lorentz force is divergence-free in velocity space, and \(f_s \to 0\) as \(|\mathbf{v}| \to \infty\)):

\[\frac{\partial n_s}{\partial t} + \nabla \cdot (n_s \mathbf{V}_s) = 0\]

First moment (\(\phi = m_s \mathbf{v}\)):

Decompose \(\mathbf{v} = \mathbf{V}_s + \mathbf{w}\) where \(\mathbf{w}\) is the random (thermal) velocity with \(\langle \mathbf{w} \rangle = 0\). The convective term produces the pressure tensor divergence:

\[\int m_s \mathbf{v} (\mathbf{v} \cdot \nabla f_s) \, d^3v = \nabla \cdot (m_s n_s \mathbf{V}_s \mathbf{V}_s + \mathbf{P}_s)\]

The Lorentz term integrates by parts to give \(n_s q_s (\mathbf{E} + \mathbf{V}_s \times \mathbf{B})\) (the magnetic force contribution from random velocities vanishes because \(\mathbf{P}_s\) is symmetric while \(\mathbf{v} \times \mathbf{B}\) is antisymmetric). Combining and using the continuity equation to simplify the convective derivative:

\[m_s n_s \frac{d \mathbf{V}_s}{dt} = n_s q_s (\mathbf{E} + \mathbf{V}_s \times \mathbf{B}) - \nabla \cdot \mathbf{P}_s\]

where \(d/dt = \partial/\partial t + \mathbf{V}_s \cdot \nabla\) is the convective derivative.

Each moment introduces the next-higher moment (the closure problem): the continuity equation involves \(\mathbf{V}_s\), the momentum equation involves \(\mathbf{P}_s\), and so on. MHD closes the hierarchy by assuming an equation of state (\(P = P(\rho, s)\)); PIC sidesteps it by evolving \(f_s\) directly.

Generalized Ohm's Law

The electric field in a plasma can be decomposed into physical contributions by manipulating the electron momentum equation. The result is the generalized Ohm's law:

\[\mathbf{E} = \underbrace{-\mathbf{V} \times \mathbf{B}}_{\text{ideal}} + \underbrace{\frac{\mathbf{J} \times \mathbf{B}}{n_e |q_e|}}_{\text{Hall}} - \underbrace{\frac{\nabla \cdot \mathbf{P}_e}{n_e |q_e|}}_{\text{electron pressure}} + \underbrace{\frac{m_e}{|q_e|} \frac{d\mathbf{V}_e}{dt}}_{\text{electron inertia}}\]

Each term has a characteristic length scale where it becomes important: the ideal term dominates at MHD scales (\(L \gg d_i\)), the Hall term at ion scales (\(d_e \ll L \lesssim d_i\)), and the pressure and inertia terms at electron scales (\(L \sim d_e\)). The non-ideal electric field \(\mathbf{E}' = \mathbf{E} + \mathbf{V} \times \mathbf{B}\) vanishes in ideal MHD and is non-zero where the frozen-in condition breaks down.

See E_prime, E_Hall, E_ideal in the equations reference ²³.

Derivation: from the two-fluid momentum equations

Start with the electron momentum equation:

\[m_e n_e \frac{d\mathbf{V}_e}{dt} = -n_e |q_e| (\mathbf{E} + \mathbf{V}_e \times \mathbf{B}) - \nabla \cdot \mathbf{P}_e\]

(using \(q_e = -|q_e|\) for electrons). Solve for \(\mathbf{E}\):

\[\mathbf{E} = -\mathbf{V}_e \times \mathbf{B} - \frac{\nabla \cdot \mathbf{P}_e}{n_e |q_e|} + \frac{m_e}{|q_e|} \frac{d\mathbf{V}_e}{dt}\]

Now express \(\mathbf{V}_e\) in terms of the bulk (ion) velocity and current. For a quasi-neutral plasma (\(n_e \approx n_i \equiv n\)):

\[\mathbf{J} = n|q_e|(\mathbf{V}_i - \mathbf{V}_e) \quad \Longrightarrow \quad \mathbf{V}_e = \mathbf{V}_i - \frac{\mathbf{J}}{n|q_e|}\]

Since \(m_i \gg m_e\), the center-of-mass velocity \(\mathbf{V} \approx \mathbf{V}_i\). Substituting into the \(\mathbf{V}_e \times \mathbf{B}\) term:

\[\mathbf{V}_e \times \mathbf{B} = \left(\mathbf{V} - \frac{\mathbf{J}}{n|q_e|}\right) \times \mathbf{B} = \mathbf{V} \times \mathbf{B} - \frac{\mathbf{J} \times \mathbf{B}}{n|q_e|}\]

Substituting back:

\[\mathbf{E} = -\mathbf{V} \times \mathbf{B} + \frac{\mathbf{J} \times \mathbf{B}}{n|q_e|} - \frac{\nabla \cdot \mathbf{P}_e}{n|q_e|} + \frac{m_e}{|q_e|}\frac{d\mathbf{V}_e}{dt}\]

This identifies the four terms:

Term	Expression	Dominant scale
Ideal (convective)	\(-\mathbf{V} \times \mathbf{B}\)	\(L \gg d_i\)
Hall	\(\mathbf{J} \times \mathbf{B} / (n\|q_e\|)\)	\(d_e \ll L \lesssim d_i\)
Electron pressure	\(-\nabla \cdot \mathbf{P}_e / (n\|q_e\|)\)	\(L \sim d_e\)
Electron inertia	\((m_e/\|q_e\|) \, d\mathbf{V}_e/dt\)	\(L \sim d_e\)

Dropping all but the first term recovers ideal MHD: \(\mathbf{E} = -\mathbf{V} \times \mathbf{B}\).

Ideal MHD Equations

Ideal MHD treats the plasma as a single conducting fluid. It follows from the two-fluid equations by (1) taking the single-fluid limit (\(m_e \to 0\), charge neutrality), (2) dropping the displacement current (\(v \ll c\)), and (3) using the ideal Ohm's law (\(\mathbf{E} = -\mathbf{V} \times \mathbf{B}\)).

The closed system in normalized units (\(\mu_0 = 1\)):

\[\frac{\partial \rho_m}{\partial t} + \nabla \cdot (\rho_m \mathbf{V}) = 0 \qquad \text{(mass)}\]

\[\rho_m \frac{d\mathbf{V}}{dt} = (\nabla \times \mathbf{B}) \times \mathbf{B} - \nabla P \qquad \text{(momentum)}\]

\[\frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{V} \times \mathbf{B}) \qquad \text{(induction)}\]

\[\frac{d}{dt}\left(\frac{P}{\rho_m^\gamma}\right) = 0 \qquad \text{(entropy/energy)}\]

Derivation: from two-fluid to single-fluid MHD

Single-fluid variables. Define the mass density, bulk velocity, and total pressure:

\[\rho_m = \sum_s n_s m_s \approx n_i m_i, \qquad \mathbf{V} = \frac{\sum_s n_s m_s \mathbf{V}_s}{\rho_m} \approx \mathbf{V}_i, \qquad P = \sum_s P_s\]

Mass conservation. Sum the species continuity equations:

\[\frac{\partial \rho_m}{\partial t} + \nabla \cdot (\rho_m \mathbf{V}) = 0\]

Momentum. Sum the species momentum equations. The internal electric forces cancel by quasi-neutrality (\(\sum_s n_s q_s = 0\)). Using Ampere's law without displacement current (\(\mathbf{J} = \nabla \times \mathbf{B}\)):

\[\rho_m \frac{d\mathbf{V}}{dt} = \mathbf{J} \times \mathbf{B} - \nabla P = (\nabla \times \mathbf{B}) \times \mathbf{B} - \nabla P\]

The magnetic force can be decomposed via the vector identity:

\[(\nabla \times \mathbf{B}) \times \mathbf{B} = (\mathbf{B} \cdot \nabla)\mathbf{B} - \nabla\left(\frac{B^2}{2}\right)\]

revealing magnetic tension \((\mathbf{B} \cdot \nabla)\mathbf{B}\) (field lines resist bending) and magnetic pressure \(-\nabla(B^2/2)\) (field lines resist compression). The plasma beta \(\beta = 2P/B^2\) measures their relative importance.

Induction. Substitute the ideal Ohm's law \(\mathbf{E} = -\mathbf{V} \times \mathbf{B}\) into Faraday's law:

\[\frac{\partial \mathbf{B}}{\partial t} = -\nabla \times \mathbf{E} = \nabla \times (\mathbf{V} \times \mathbf{B})\]

This is the frozen-in flux equation: magnetic field lines move with the fluid. It breaks down wherever the non-ideal terms in Ohm's law become significant (reconnection sites, current sheets).

Energy closure. Assuming adiabatic evolution (no heat conduction or radiation), the entropy is conserved along fluid elements:

\[\frac{d}{dt}\left(\frac{P}{\rho_m^\gamma}\right) = 0\]

This is equivalent to the entropy \(s = \ln(P/\rho_m^\gamma)\) being a Lagrangian invariant.

Energy Conservation: Poynting's Theorem

The electromagnetic energy balance connects the field energy densities (\(e_B\), \(e_E\)), the Poynting flux (\(\mathbf{S}\)), and the energy conversion rate (\(\mathbf{J} \cdot \mathbf{E}\)):

\[\frac{\partial}{\partial t}\left(\frac{B^2 + E^2}{2}\right) + \nabla \cdot (\mathbf{E} \times \mathbf{B}) = -\mathbf{J} \cdot \mathbf{E}\]

Identifying: \(e_B = B^2/2\), \(e_E = E^2/2\), \(\mathbf{S} = \mathbf{E} \times \mathbf{B}\) (normalized Poynting flux), and \(-\mathbf{J} \cdot \mathbf{E}\) as the sink term transferring electromagnetic energy to particle kinetic energy.

See e_B, e_E, S, J_dot_E in the equations reference.

Derivation: from Maxwell's equations

Take the dot product of Faraday's law with \(\mathbf{B}\) and Ampere's law with \(\mathbf{E}\):

\[\mathbf{B} \cdot (\nabla \times \mathbf{E}) = -\mathbf{B} \cdot \frac{\partial \mathbf{B}}{\partial t} = -\frac{1}{2}\frac{\partial B^2}{\partial t}\]

\[\mathbf{E} \cdot (\nabla \times \mathbf{B}) = \mathbf{E} \cdot \mathbf{J} + \mathbf{E} \cdot \frac{\partial \mathbf{E}}{\partial t} = \mathbf{J} \cdot \mathbf{E} + \frac{1}{2}\frac{\partial E^2}{\partial t}\]

Subtract the first from the second:

\[\mathbf{E} \cdot (\nabla \times \mathbf{B}) - \mathbf{B} \cdot (\nabla \times \mathbf{E}) = \mathbf{J} \cdot \mathbf{E} + \frac{1}{2}\frac{\partial}{\partial t}(E^2 + B^2)\]

The left side equals \(-\nabla \cdot (\mathbf{E} \times \mathbf{B})\) by the vector identity \(\nabla \cdot (\mathbf{A} \times \mathbf{B}) = \mathbf{B} \cdot (\nabla \times \mathbf{A}) - \mathbf{A} \cdot (\nabla \times \mathbf{B})\). Rearranging:

\[\frac{\partial}{\partial t}\underbrace{\left(\frac{E^2 + B^2}{2}\right)}_{e_E + e_B} + \nabla \cdot \underbrace{(\mathbf{E} \times \mathbf{B})}_{\mathbf{S}} = -\mathbf{J} \cdot \mathbf{E}\]

Physical interpretation: \(\mathbf{J} \cdot \mathbf{E} > 0\) means the electromagnetic field loses energy to particle motion (acceleration) ¹. In reconnection, this identifies the energy conversion region ⁴. Per-species decomposition \(\mathbf{J}_s \cdot \mathbf{E}\) reveals which species is energized.

In MHD, \(e_E\) is negligible (\(v \ll c\)), and Poynting's theorem reduces to the magnetic energy budget:

\[\frac{\partial}{\partial t}\left(\frac{B^2}{2}\right) + \nabla \cdot (\mathbf{E} \times \mathbf{B}) = -\mathbf{J} \cdot \mathbf{E}\]

MHD Wave Speeds

Linearizing the ideal MHD equations around a uniform equilibrium (\(\rho_0\), \(P_0\), \(\mathbf{B}_0 = B_0 \hat{z}\), \(\mathbf{V}_0 = 0\)) yields three wave families. Their phase speeds are the characteristic speeds used throughout pypic.

Alfvén wave (incompressible, transverse) ⁵⁶:

\[v_A = \frac{B_0}{\sqrt{\rho_0}}\]

Sound wave (compressible, longitudinal, no magnetic coupling) ⁶:

\[c_s = \sqrt{\frac{\gamma P_0}{\rho_0}}\]

Fast and slow magnetosonic waves (compressible, coupled) ⁷:

\[v_{f,s}^2 = \frac{1}{2}\left[(v_A^2 + c_s^2) \pm \sqrt{(v_A^2 + c_s^2)^2 - 4 v_A^2 c_s^2 \cos^2\theta}\right]\]

At perpendicular propagation (\(\theta = 90°\)), the fast speed reduces to \(v_{ms} = \sqrt{v_A^2 + c_s^2}\), which is the v_ms stored by pypic.

See v_A, c_s, v_ms in the equations reference.

Derivation: linearization of ideal MHD

Perturb all quantities around a uniform equilibrium: \(\rho = \rho_0 + \rho_1\), \(\mathbf{V} = \mathbf{V}_1\), \(\mathbf{B} = B_0 \hat{z} + \mathbf{B}_1\), \(P = P_0 + P_1\), with \(P_1 = c_s^2 \rho_1\) from the adiabatic equation of state.

Substitute into the ideal MHD equations and keep only first-order terms. Seek plane-wave solutions \(\propto \exp[i(\mathbf{k} \cdot \mathbf{x} - \omega t)]\) with wavevector \(\mathbf{k}\) at angle \(\theta\) to \(\mathbf{B}_0\).

The linearized momentum equation becomes:

\[-\omega^2 \rho_0 \mathbf{V}_1 = -k^2 c_s^2 (\hat{k} \cdot \mathbf{V}_1)\hat{k} + (\mathbf{k} \times (\mathbf{k} \times (\mathbf{V}_1 \times \mathbf{B}_0))) \times \mathbf{B}_0\]

using the linearized induction equation \(\mathbf{B}_1 = -(\mathbf{k} \times (\mathbf{V}_1 \times \mathbf{B}_0))/\omega\) and Ampere's law.

Alfvén mode. For perturbations \(\mathbf{V}_1\) perpendicular to both \(\mathbf{k}\) and \(\mathbf{B}_0\) (shear polarization), the pressure force vanishes (\(\hat{k} \cdot \mathbf{V}_1 = 0\), so \(\rho_1 = 0\)) and the dispersion relation reduces to:

\[\omega^2 = k_\parallel^2 v_A^2 = k^2 v_A^2 \cos^2\theta\]

This is the Alfvén wave: an incompressible transverse oscillation of field lines, propagating only along \(\mathbf{B}_0\), with phase speed \(v_A = B_0/\sqrt{\rho_0}\).

Magnetosonic modes. For perturbations in the \((\mathbf{k}, \mathbf{B}_0)\) plane, the pressure and magnetic forces couple. The dispersion relation becomes a quadratic in \(\omega^2/k^2\):

\[\left(\frac{\omega^2}{k^2}\right)^2 - (v_A^2 + c_s^2)\frac{\omega^2}{k^2} + v_A^2 c_s^2 \cos^2\theta = 0\]

with solutions:

\[v_{f,s}^2 = \frac{\omega^2}{k^2} = \frac{1}{2}\left[(v_A^2 + c_s^2) \pm \sqrt{(v_A^2 + c_s^2)^2 - 4 v_A^2 c_s^2 \cos^2\theta}\right]\]

The \(+\) root is the fast magnetosonic wave (magnetic pressure and gas pressure reinforce). The \(-\) root is the slow magnetosonic wave (they oppose). At perpendicular propagation (\(\theta = 90°\), \(\cos\theta = 0\)):

\[v_f = \sqrt{v_A^2 + c_s^2} \equiv v_{ms}, \qquad v_s = 0\]

At parallel propagation (\(\theta = 0°\)): \(v_f = \max(v_A, c_s)\) and \(v_s = \min(v_A, c_s)\).

CGL Double-Adiabatic Invariants

In a collisionless magnetized plasma, the pressure tensor is not isotropic. The CGL theory ⁸ provides closure by conserving two adiabatic invariants along fluid elements:

\[\frac{d}{dt}\left(\frac{P_\perp}{n B}\right) = 0, \qquad \frac{d}{dt}\left(\frac{P_\parallel B^2}{n^3}\right) = 0\]

These govern the gyrotropic entropy \(s_{gyro} = \ln(P_\parallel P_\perp^2 / n^5)\) and the firehose/mirror instability thresholds.

Derivation: from magnetic moment conservation and flux tube geometry

First invariant (\(P_\perp / nB\)): the magnetic moment \(\mu = m v_\perp^2 / (2B)\) is an adiabatic invariant for individual particles gyrating in a slowly varying field. For a distribution of particles, the perpendicular thermal energy per particle is \(P_\perp / n\), giving:

\[\frac{P_\perp}{nB} \propto \frac{\langle v_\perp^2 \rangle}{B} \propto \mu = \text{const}\]

Second invariant (\(P_\parallel B^2 / n^3\)): consider a flux tube of cross-section \(A\) and length \(L\). Magnetic flux conservation gives \(BA = \text{const}\), and particle conservation in the tube gives \(nAL = \text{const}\), so \(nL/B = \text{const}\), i.e. \(L \propto B/n\).

For the parallel degree of freedom, the particles bounce between mirror points. The parallel adiabatic invariant \(J = \oint m v_\parallel \, dl\) gives \(v_\parallel L = \text{const}\), so \(P_\parallel / n \propto v_\parallel^2 \propto 1/L^2 \propto n^2/B^2\). Therefore:

\[\frac{P_\parallel B^2}{n^3} = \text{const}\]

Gyrotropic entropy. Eliminating \(B\) between the two invariants:

• From the first: \(B \propto P_\perp / n\)
• Substituting into the second: \(P_\parallel (P_\perp / n)^2 / n^3 = P_\parallel P_\perp^2 / n^5 = \text{const}\)

Hence \(s_{gyro} = \ln(P_\parallel P_\perp^2 / n^5)\) is conserved. The exponent 5 arises from the geometric constraint (\(3\) from parallel + \(2\) from perpendicular), not from the adiabatic index \(\gamma = 5/3\). See Conventions for this distinction.

Instability thresholds. When the CGL invariants are violated (collisions, wave-particle interactions), the pressure anisotropy drives kinetic instabilities:

• Firehose (\(P_\parallel > P_\perp + B^2/2\)): excess parallel pressure bends field lines. Threshold: \((P_\parallel - P_\perp)/(B^2/2) > 1\) ⁹.
• Mirror (\(P_\perp / P_\parallel > 1 + 1/\beta_\perp\)): excess perpendicular pressure creates density compressions along field lines ¹⁰¹¹.

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