Physics Equations Reference

Equation reference for the derived quantities in pypic. For canonical field names and data types, see SCHEMA. For detailed convention discussions, see conventions.md.

Normalization: All computation uses code (normalized) units. In PIC normalization, \(\mu_0 = \epsilon_0 = 1\). In MHD normalization, \(\mu_0 = 1\). Physical constants vanish from the normalized forms, reappearing only at SI conversion boundaries. Temperatures are in energy units throughout (\(T = P/n\), not \(T = P/(nk_B)\)); divide by \(k_B\) to convert to Kelvin.

SI conversion factors

To convert a quantity from code units to SI: \(x_{SI} = x_{code} \times f\), where \(f\) is the SI factor for that quantity type. The eight primitive references (\(n_{ref}\), \(m_{ref}\), \(q_{ref}\), \(v_{ref}\), \(l_{ref}\), \(t_{ref}\), \(B_{ref}\), \(E_{ref}\)) are set by the normalization system (PIC, MHD, or custom). They normalize the primitive canonical fields (\(n_s\), plus per-species \(m_s\)/\(q_s\), \(\mathbf{V}\), \(\mathbf{B}\), \(\mathbf{E}\)). SI factors for compound canonical fields (\(\rho_m\), \(\rho_c\), \(\mathbf{J}\), \(P\), energy and flux densities) are products of the primitives — there is no \(\rho_{ref}\) because \(\rho_m\) and \(\rho_c\) are derived (\(\sum_s n_s m_s\) and \(\sum_s n_s q_s\)), not primitive.

Quantity type	SI factor \(f\)	SI unit	Used by
density	\(n_{ref}\)	m\(^{-3}\)	\(n_e\), \(n_i\)
mass_density	\(n_{ref} m_{ref}\)	kg/m\(^3\)	\(\rho_m\)
charge_density	\(q_{ref} n_{ref}\)	C/m\(^3\)	\(\rho_c\)
velocity	\(v_{ref}\)	m/s	\(V\), \(v_A\), \(v_{th}\)
b_field	\(B_{ref}\)	T	\(B\)
e_field	\(E_{ref}\)	V/m	\(E\)
pressure	\(n_{ref} m_{ref} v_{ref}^2\)	Pa	\(P\), \(P_e\), \(P_i\) 1
temperature	\(m_{ref} v_{ref}^2\)	J	\(T\) (energy units)
energy_density	\(n_{ref} m_{ref} v_{ref}^2\)	J/m\(^3\)	\(e_k\), \(e_{th}\), \(e_B\)
specific_energy	\(v_{ref}^2\)	J/kg	\(h\), \(e_{int}\)
current_density	\(q_{ref} n_{ref} v_{ref}\)	A/m\(^2\)	\(J\)
frequency	\(1/t_{ref}\)	rad/s	\(\omega_p\), \(\omega_c\)
length	\(l_{ref}\)	m	\(d_e\), \(d_i\), \(r_i\)
poynting_flux	\(E_{ref} B_{ref} / \mu_0\)	W/m\(^2\)	\(S\) (EM flux)
energy_flux	\(n_{ref} m_{ref} v_{ref}^3\)	W/m\(^2\)	EF, KEF, HF, EHF, \(q\)
power_density	\(n_{ref} m_{ref} v_{ref}^2 / t_{ref}\)	W/m\(^3\)	\(J \cdot E\)

poynting_flux and energy_flux share SI units (W/m\(^2\)) but differ in normalization: EM flux scales with field references, particle energy flux with matter references (\(n_{ref} m_{ref} v_{ref}^3\)).

The \(1/\mu_0\) on poynting_flux is what carries \(\mathbf{S}\) from pypic's SI-rationalized code units, where derived.poynting_flux returns a bare \(\mathbf{E} \times \mathbf{B}\), back to the SI \(\mathbf{E} \times \mathbf{B} / \mu_0\). It follows from that definition alone. The two factors coincide whenever the reference set satisfies \(B_{ref}^2 = \mu_0 n_{ref} m_{ref} v_{ref}^2\) — which the built-in constructors do by construction — but an independently chosen \(B_{ref}\) need not, so the field-reference form is the one implemented.

1. Densities and Moments

Per-species e/i suffix names throughout this document are pypic library-side aliases ¹. Two stylistic patterns appear, forced by identifier readability:

• Uppercase prefix → no underscore: Pe, Pi, Ve, Vi, Te, Ti, and the multi-letter forms EFe, KEFi, HFi, EHFe — alias the canonical _s0 / _s1 per-species form.
• Lowercase prefix → underscore mandatory: n_e, n_i, s_e, s_i, beta_e, beta_i, q_e, q_i, P_par_e, s_gyro_e, ... — same alias relationship, written with _e / _i because betae / se would be unreadable.

The cross-tool canonical form uses 0-based species indices: P_s{N}, V_s{N}, T_s{N}, n_s{N}, s_s{N}, beta_s{N}, q_s{N} (see schema.md § Per-species naming). See Aliases for the full registration list.

Name	Description	Normalized	SI
n_s	Number density (species \(s\))	\(n_s / n_{ref}\)	--
rho_c	Charge density	\(\sum_s n_s q_s\)	--
rho_m	Mass density	\(\sum_s n_s m_s\)	--
J	Current density	\(\sum_s n_s q_s \mathbf{V}_s\)	--
V	Ion/fluid bulk velocity	1st moment of \(f_i\) (kinetic) or fluid velocity	--
Ve	Electron bulk velocity	1st moment of \(f_e\)	--
P	Total scalar pressure	\(P_e + P_i\) or \(\frac{1}{3}\mathrm{Tr}(\mathbf{P})\) or fluid \(P\)	--
Pe	Electron scalar pressure	\(P_e = n_e T_e\)	--
Pi	Ion scalar pressure	\(P_i = n_i T_i\)	--
T	Temperature (generic)	\(T = P / n\)	\(T^{SI} = T \cdot m_{ref} v_{ref}^2\)
Te	Electron temperature	\(T_e = P_e / n_e\)	\(T_e^{SI} = T_e \cdot m_{ref} v_{ref}^2\)
Ti	Ion temperature	\(T_i = P_i / n_i\)	\(T_i^{SI} = T_i \cdot m_{ref} v_{ref}^2\)

2. Thermodynamic Quantities

Name	Description	Normalized	SI
h	Specific enthalpy	\(\gamma P / ((\gamma - 1) \rho_m)\)	\(h \cdot v_{ref}^2\) [J/kg]
h_rel	Relativistic specific enthalpy3	\(c^2 + \gamma P / ((\gamma - 1) \rho_m)\)	\(h_{rel} \cdot v_{ref}^2\) [J/kg]
s	Specific entropy (single-fluid, isotropic)2	\(\ln(P / \rho_m^\gamma)\)	--
s_e	Electron entropy1	\(\ln(P_e / n_e^\gamma)\)	--
s_i	Ion entropy	\(\ln(P_i / n_i^\gamma)\)	--
s_gyro	Gyrotropic entropy	\(\ln(P_{\parallel,s} P_{\perp,s}^2 / n_s^5)\)	--
e_int	Specific internal energy	\(P / ((\gamma - 1) \rho_m)\)	\(e_{int} \cdot v_{ref}^2\) [J/kg]
gamma_eos	Adiabatic index	\(\gamma = c_p / c_v\)	--

3. Energy and Flux

Name	Description	Normalized	SI
S	Poynting flux	\(\mathbf{E} \times \mathbf{B}\)	\(\mathbf{E} \times \mathbf{B} / \mu_0\)
e_B	Magnetic energy density	\(B^2 / 2\)	\(B^2 / (2\mu_0)\)
e_E	Electric energy density	\(E^2 / 2\)	\(\epsilon_0 E^2 / 2\)
e_k	Kinetic energy density	\(\frac{1}{2}\rho_m V^2\)	--
e_th	Thermal energy density	\(P / (\gamma - 1)\)	--
e_th_trace	Thermal energy density (tensor)	\(\frac{1}{2}\mathrm{Tr}(\mathbf{P}) = \frac{1}{2}(P_{11}+P_{22}+P_{33})\)	--

Energy flux decomposition

The total energy flux per species (third-order velocity moment) decomposes into kinetic, enthalpy, and conductive heat flux contributions:

\[\mathbf{EF}_s = \underbrace{\tfrac{1}{2} n_s m_s |\mathbf{V}_s|^2 \mathbf{V}_s}_{\mathbf{KEF}_s} + \underbrace{\tfrac{\gamma}{\gamma-1} P_s \mathbf{V}_s}_{\mathbf{EHF}_s} + \underbrace{\mathbf{q}_s}_{\text{heat flux}}\]

Name	Description	Normalized	Available when
EF	Total energy flux4	\(\tfrac{1}{2} m \int f\, \mathbf{v}\, v^2\, d^3v\)	EF in output (PIC, multi-moment MHD)
KEF	Kinetic energy flux	$\tfrac{1}{2} n\, m\,	\mathbf{V}
HF	Total thermal flux	\(\mathbf{EF} - \mathbf{KEF}\)	EF in output
EHF	Enthalpy flux (adiabatic)	\(\tfrac{\gamma}{\gamma-1} P\, \mathbf{V}\)	Any code with \(P\), \(\mathbf{V}\)
q	Conductive heat flux	\(\mathbf{HF} - \mathbf{EHF}\)	EF in output

4. Pressure Tensor and Field-Aligned Decomposition

4.1 Pressure tensor

Name	Description	Normalized	SI
P	Isotropic scalar pressure9	\(P = \frac{1}{3}\mathrm{Tr}(\mathbf{P}) = \frac{1}{3}(P_{11} + P_{22} + P_{33})\)	--
P_par	Parallel pressure	\(P_\parallel = \hat{b} \cdot \mathbf{P} \cdot \hat{b}\)	--
P_perp	Perpendicular pressure	\(P_\perp = (\mathrm{Tr}(\mathbf{P}) - P_\parallel) / 2\)	--
Pij	Full pressure tensor	6 independent components: P_11, P_12, P_13, P_22, P_23, P_33	--
agyrotropy	Agyrotropy measure8	\(Q = 1 - 4 I_2 / [(I_1 - P_\parallel)(I_1 + 3 P_\parallel)]\)	--

4.2 Field-aligned vector decomposition

Projects a vector \(\mathbf{A}\) onto \(\hat{b} = \mathbf{B}/|\mathbf{B}|\). Generic form, same algebra for \(\mathbf{J}\), \(\mathbf{V}\), \(\mathbf{E}\), and the non-ideal residual \(\mathbf{E}' = \mathbf{E} + \mathbf{V}\times\mathbf{B}\). All returns are NaN where \(|B| = 0\).

Name	Description	Normalized
J_par, V_par, E_par	Signed parallel projection	\(A_\parallel = \mathbf{A}\cdot\hat{b}\)
J_perp_1/2/3, V_perp_1/2/3, E_perp_1/2/3	Perpendicular vector components	\(\mathbf{A}_\perp = \mathbf{A} - A_\parallel\hat{b}\)
\|J_perp\|, \|V_perp\|, \|E_perp\|	Perpendicular magnitude5	\(\sqrt{\|\mathbf{A}\|^2 - A_\parallel^2}\)
V_s{N}_par, V_s{N}_perp_{1,2,3}, \|V_s{N}_perp\|	Per-species velocity decomposition	same form on V_s{N}_{1,2,3}
E_prime_par	Field-aligned non-ideal residual6	\((\mathbf{E} + \mathbf{V}\times\mathbf{B}) \cdot \hat{b}\)
\|E_prime_perp\|	Perpendicular non-ideal residual magnitude	\(\sqrt{\|\mathbf{E}'\|^2 - E'^2_\parallel}\)
E_ideal_par, E_Hall_par	Field-aligned ideal-MHD / Hall field7	\(\equiv 0\) (cross product \(\perp\) \(\mathbf{B}\))
E_ideal_perp_{1,2,3}, E_Hall_perp_{1,2,3}	Perpendicular ideal-MHD / Hall field	equal the full vectors up to roundoff
\|E_ideal_perp\|, \|E_Hall_perp\|	Perpendicular magnitudes	\(= \|\mathbf{E}^{\mathrm{ideal}}\|\), \(\|\mathbf{E}^{\mathrm{Hall}}\|\) up to roundoff

5. Characteristic Scales

The Tier-3 <field>_s<N> form is the canonical recipe ID; the NRL Plasma Formulary spelling (omega_pe, v_th_e, lambda_D, ...) remains a registered alias and the human-friendly form throughout this document.¹⁰

Canonical	NRL alias	Description	Normalized	SI
d_s0	d_e	Electron skin depth	\(c / \omega_{pe}\)	--
d_s1	d_i	Ion skin depth	\(c / \omega_{pi}\)	--
r_s0	r_e	Electron thermal gyroradius	\(v_{th,e} / \omega_{ce}\)	--
r_s1	r_i	Ion thermal gyroradius	\(v_{th,i} / \omega_{ci}\)	--
omega_p_s0	omega_pe	Electron plasma frequency	\(\sqrt{n_e q_e^2 / m_e}\)	\(\sqrt{n_e e^2 / (\epsilon_0 m_e)}\)
omega_p_s1	omega_pi	Ion plasma frequency	\(\sqrt{n_i q_i^2 / m_i}\)	\(\sqrt{n_i Z^2 e^2 / (\epsilon_0 m_i)}\)
omega_c_s0	omega_ce	Electron cyclotron freq.11	\(\|q_e\| B / m_e\)	\(\|e\| B / m_e\)
omega_c_s1	omega_ci	Ion cyclotron freq.11	$	q_i
lambda_D_s0	lambda_D	Electron Debye length12	\(\sqrt{T_e / (n_e q_e^2)}\)	\(\sqrt{\epsilon_0 T_e / (n_e e^2)}\)
v_A	—	Alfvén speed 37	\(B / \sqrt{\rho_m}\)	\(B / \sqrt{\mu_0 \rho_m}\)
v_th_s0	v_th_e	Electron thermal speed13	\(\sqrt{T_e / m_e}\)	--
v_th_s1	v_th_i	Ion thermal speed13	\(\sqrt{T_i / m_i}\)	--
c_s	—	Sound speed14	\(\sqrt{\gamma P / \rho_m}\)	--
c_ia	—	Ion acoustic speed14	\(\sqrt{(\gamma_e T_e + \gamma_i T_i) / m_i}\)	--
v_ms	—	Fast magnetosonic speed15	\(\sqrt{v_A^2 + c_s^2}\)	--
M_A	—	Alfvén Mach number	\(V / v_A\)	--
M_ms	—	Magnetosonic Mach number	\(V / v_{ms}\)	--
beta	—	Plasma beta	\(2P / B^2\)	\(2\mu_0 P / B^2\)
beta_s0	beta_e1	Electron beta	\(2P_e / B^2\)	\(2\mu_0 P_e / B^2\)
beta_s1	beta_i	Ion beta	\(2P_i / B^2\)	\(2\mu_0 P_i / B^2\)

6. Magnitudes and Differential Operators

Name	Description	Normalized
\|B\|	Magnetic field magnitude	\(\sqrt{B_1^2 + B_2^2 + B_3^2}\)
\|E\|	Electric field magnitude	\(\sqrt{E_1^2 + E_2^2 + E_3^2}\)
\|J\|	Current density magnitude	\(\sqrt{J_1^2 + J_2^2 + J_3^2}\)
\|V\|	Bulk velocity magnitude	\(\sqrt{V_1^2 + V_2^2 + V_3^2}\)
div_B	Divergence of B	\(\nabla \cdot \mathbf{B}\) (should be ~0)
div_E	Divergence of E	\(\nabla \cdot \mathbf{E}\) (\(= \rho_c\) normalized; \(= \rho_c / \epsilon_0\) SI)
curl_B	Curl of B	\(\nabla \times \mathbf{B}\) (\(\propto \mathbf{J}\) in MHD)
vort	Fluid vorticity	\(\nabla \times \mathbf{V}\)
\|vort\|	Vorticity magnitude	\(\sqrt{\mathrm{vort}_1^2 + \mathrm{vort}_2^2 + \mathrm{vort}_3^2}\)
grad	Scalar field gradient	\((\nabla f)_i = \partial f / \partial x_i\)

Differential operators use second-order central finite differences (Cartesian)⁹. Spherical and cylindrical geometries include metric factors.

7. Diagnostics and Validation

Name	Description	Equation
l2_relative_error	Discrete relative L2 norm16	\(\varepsilon_{L_2} = \sqrt{\sum_i (a_i - b_i)^2} / \sqrt{\sum_i b_i^2}\)
linf_error	Absolute max-norm error	\(\varepsilon_{L_\infty} = \max_i \|a_i - b_i\|\)
field_difference	Pointwise signed difference	\(\Delta f_i = a_i - b_i\)
field_energy	Volume integral of a scalar field	\(E = \sum_{i,j,k} f_{i,j,k} \cdot \Delta V\) where \(\Delta V = \prod_k \Delta x_k\)
div_b	Divergence of B	\(\nabla \cdot \mathbf{B}\) (second-order central differences)
max_div_b	Maximum absolute div B	\(\max \|\nabla \cdot \mathbf{B}\|\)
div_e	Divergence of E	\(\nabla \cdot \mathbf{E}\) (second-order central differences)
spatial_mean	Unweighted spatial mean	\(\langle f \rangle = \frac{1}{N}\sum_i f_i\) (uniform Cartesian grids)
spatial_rms	Root-mean-square amplitude	\(f_{rms} = \sqrt{\langle f^2 \rangle}\)
field_extrema	Field min/max	\((\min_i f_i,\; \max_i f_i)\) ignoring NaN

8. Relativistic Corrections

Relativistic corrections¹⁷ become important in magnetically dominated plasmas (\(\sigma \gg 1\)), relativistic reconnection (pair plasmas, pulsar wind nebulae), and relativistic jets. The existing non-relativistic formulas (Sections 3, 5) are recovered when \(\gamma \to 1\) and \(\sigma \to 0\). See conventions.md § Relativistic Conventions for velocity variable choices and numerical considerations.

8.1 Bulk-Flow Quantities

Name	Description	Normalized	Notes
gamma_L	Lorentz factor (from 3-velocity)	\(\gamma = 1/\sqrt{1 - v^2/c^2}\)	bounded \([1, \infty)\)
gamma_L	Lorentz factor (from 4-velocity)	\(\gamma = \sqrt{1 + u^2/c^2}\)	numerically preferred near \(v \approx c\)
sigma	Magnetization 41	\(B^2 / (\rho_m c^2)\)	\(\sigma \gg 1\): magnetically dominated
e_k (rel.)	Rel. kinetic energy density	\((\gamma - 1)\rho_m c^2\)	recovers \(\frac{1}{2}\rho_m v^2\) for \(v \ll c\)
v_A (rel.)	Rel. Alfvén speed 42	\(c\sqrt{\sigma/(1+\sigma)}\)	\(\to c\) as \(\sigma \to \infty\); \(\to B/\sqrt{\rho_m}\) for \(\sigma \ll 1\)
c_s (rel.)	Rel. sound speed 30	\(c\sqrt{\gamma_{eos} P / (\rho_m h_{rel})}\)	uses relativistic enthalpy \(h_{rel}\)
v_ms (rel.)	Rel. magnetosonic speed 41	\(\sqrt{v_A^2 + c_s^2 - v_A^2 c_s^2/c^2}\)	perp. propagation; always \(< c\)

In normalized units where \(c = 1\): \(\sigma = B^2/\rho_m\), \(v_A = \sqrt{\sigma/(1+\sigma)}\), and the magnetosonic composition simplifies to \(v_{ms}^2 = v_A^2 + c_s^2 - v_A^2 c_s^2\).

8.2 Thermal Corrections

Relevant when the thermal energy approaches the rest mass energy (\(T \gtrsim mc^2\)), as in relativistic pair plasma reconnection or hot accretion flows.

Name	Rel. form	Notes
omega_c	\(\|q\|B / (\gamma m)\)	\(\gamma\) = thermal Lorentz factor; particles gyrate slower
omega_p	\(\omega_p / \sqrt{\langle\gamma\rangle}\)	mean thermal \(\langle\gamma\rangle\); reduces effective plasma frequency
v_th	\(v_{th}/\sqrt{1 + v_{th}^2/c^2}\)	practical cap at \(c\); proper treatment uses Juttner distribution

The thermal corrections use the mean thermal Lorentz factor \(\langle\gamma\rangle\), which for a relativistic Maxwellian (Juttner distribution) depends on the dimensionless temperature \(\Theta = T/(mc^2)\). For \(\Theta \ll 1\) the classical results are recovered; for \(\Theta \gg 1\) the ultra-relativistic limit applies.

8.3 Which Gamma?

Three distinct Lorentz factors arise in plasma analysis:

• \(\gamma_{bulk}\) — from the fluid (bulk) velocity \(\mathbf{V}\). This is what lorentz_factor() computes from the moment velocity fields V_1/V_2/V_3 or u_1/u_2/u_3. Used in relativistic kinetic energy, Alfvén speed, and Mach numbers.

• \(\langle\gamma\rangle_{thermal}\) — the mean Lorentz factor of the thermal distribution. Relevant for cyclotron and plasma frequency corrections (Section 8.2). Not directly available from fluid moments; approximated from temperature as \(\langle\gamma\rangle \approx 1 + (5/2)\Theta\) for \(\Theta \ll 1\).

• \(\gamma_{particle}\) — the per-particle Lorentz factor from the full distribution function. Only available in full particle data, not from the moment-based fields that pypic analyzes. Relevant for particle-level diagnostics (energy spectra, acceleration studies) but outside the scope of fluid/moment derived quantities.

9. Reconnection and Anisotropy Diagnostics

Name	Description	Normalized	SI
J_dot_E	Energy conversion rate18	\(\mathbf{J} \cdot \mathbf{E}\)	\(\mathbf{J} \cdot \mathbf{E}\) [W/m³]
D_e	Electron-frame dissipation24	\(\gamma_e\bigl[\mathbf{J}\cdot(\mathbf{E}+\mathbf{V}_e\times\mathbf{B}) - \rho_c\,\mathbf{V}_e\cdot\mathbf{E}\bigr]\)	same [W/m³]
R_recon	Local dimensionless reconnection rate25	\(\lvert\mathbf{E}+\mathbf{V}\times\mathbf{B}\rvert / (v_A\,\lvert\mathbf{B}\rvert)\)	dimensionless
E_prime	Non-ideal electric field19	\(\mathbf{E} + \mathbf{V} \times \mathbf{B}\)	same
E_ideal	Ideal (convective) E field19	\(-\mathbf{V} \times \mathbf{B}\)	same
E_Hall	Hall electric field19	$\mathbf{J} \times \mathbf{B} / (n_e	q_e
D_ng	Aunai nongyrotropy26	\(2\,\|\mathbf{N}\|_F / \mathrm{Tr}(\mathbf{P})\)	dimensionless
A_phi	Scudder agyrotropy27	\(\lvert\lambda_1^\perp - \lambda_2^\perp\rvert / (\lambda_1^\perp + \lambda_2^\perp)\)	dimensionless
psi	Magnetic flux function (2D)20	\(-\int B_2\, dx\)	--
firehose	Firehose parameter21	\((P_\parallel - P_\perp)/(B^2/2) - 1\)	--
mirror	Mirror parameter21	\(P_\perp/P_\parallel - 1 - 1/\beta_\perp\)	--
theta_shear	Magnetic shear angle23	$\theta = \arccos(\mathbf{B}_a \cdot \mathbf{B}_b /	\mathbf{B}_a
find_saddle_points	X-point detection22	\(\det(H) = \psi_{xx}\psi_{yy} - \psi_{xy}^2 < 0\)	--
reconnection_rate	Reconnection rate (2D)22	\(R = \partial\psi/\partial t\|_X\)	--
schindler_xi	3D reconnection criterion28	\(\Xi(\mathbf{x}_0) = \int_{\mathcal{L}} E_\parallel\,d\ell\)	--

10. Spectral Analysis

Name	Description	Equation
power_spectrum_1d	1D power spectral density29	\(P(k) = \|\hat{f}(k)\|^2 \Delta x / (2\pi N)\)
power_spectrum_2d	Radially averaged 2D PSD29	\(P(k) = \langle \|\hat{f}\|^2 \rangle_{\|k\|=k}\)
power_spectrum_3d	Spherically averaged 3D PSD29	\(P(k) = \langle \|\hat{f}\|^2 \rangle_{\|k\|=k}\)

---

1. pypic library-side convenience aliases bound to the canonical _s{N} names at FieldDataset construction time (Pe ↔ P_s0, Pi ↔ P_s1, Ve ↔ V_s0, …) for the common two-species electron/ion case. They are not part of the cross-tool v1.0 contract — non-pypic consumers (Rust, JS) read only the numbered canonical names from disk. See Aliases for the full registration list. ↩↩↩↩↩

2. The fluid form uses mass density (\(\rho_m\), single-fluid equation of state); the per-species form uses number density (\(n_s\), kinetic moments per particle), with \(\gamma = 5/3\) (3D) by default. The gyrotropic exponent of 5 is independent of \(\gamma\) — it arises from the CGL double-adiabatic invariants, not from the equation of state. See conventions.md § γ Convention. ↩

3. h_rel uses the constant-\(\Gamma\) (Synge-type) approximation; see ³⁰ for the variable-\(\Gamma\) treatment. ↩

4. iPIC3D outputs EF per species as a deposited moment. In ideal MHD (\(\mathbf{q} = 0\)), the total energy flux equals \(\mathbf{KEF} + \mathbf{EHF}\). The conductive heat flux \(\mathbf{q}\) captures non-Maxwellian and non-adiabatic transport — it is the physically interesting residual in reconnection exhausts, shocks, and turbulence. All quantities exist as both total (MHD: EHF_1, EHF_2, EHF_3) and per-species (PIC: KEF_s0_1, HFi¹, q_i). Precision notes: HF = EF - KEF is exact (no closure assumption). EHF uses the scalar (isotropic) pressure; for anisotropic plasmas the exact enthalpy flux involves the full pressure tensor (\(EHF_i = P_{ij} V_j + \tfrac{1}{2} P_{jj} V_i\)). The isotropic form is a useful reference — discrepancies between HF and EHF reflect both anisotropy and heat conduction effects. ↩

5. Pythagorean identity holds exactly: \(\|\mathbf{A}\|^2 = A_\parallel^2 + \|\mathbf{A}_\perp\|^2\). The perpendicular magnitude uses this directly — half the temporaries of materializing the three perpendicular components. ↩

6. \(E'_\parallel\) quantifies frozen-in flux violation: it is zero in ideal MHD and non-zero only where the non-ideal terms in Ohm's law (resistivity, electron inertia, pressure divergence) break the frozen-in condition. In 2D it equals \(\partial\psi/\partial t\) at the X-point — the standard reconnection-rate quantity. This identity assumes a finite out-of-plane guide field so \(\hat{b}\) at the null points out of plane and \(E'_\parallel = E_z\); in the strict anti-parallel limit \(\hat{b}\) is undefined at the magnetic null and \(E'_\parallel\) is NaN (consistent with the \(|\mathbf{B}| = 0\) handling). Cross-reference §9 footnote 13. ³¹³². ↩

7. \(\mathbf{E}^{\mathrm{ideal}} = -\mathbf{V}\times\mathbf{B}\) and \(\mathbf{E}^{\mathrm{Hall}} \propto \mathbf{J}\times\mathbf{B}\) are cross products with \(\mathbf{B}\), hence orthogonal to \(\mathbf{B}\) by the scalar triple product identity. E_ideal_par and E_Hall_par therefore evaluate to zero up to floating-point roundoff. Kept in the registry for diagnostic symmetry with E_par / E_prime_par and as a numerical-precision check on the cross-product implementation in the destaggered co-located grid. The underlying scalar projection \(A_\parallel = \mathbf{A}\cdot\hat{b}\) and rejection \(\mathbf{A}_\perp = \mathbf{A} - A_\parallel\hat{b}\) are introduced as part of the single-particle gyromotion decomposition ³³. ↩

8. Swisdak's gyrotropy measure ³⁴ (Eq. A8), built from the first two invariants of the full pressure tensor: \(I_1 = \mathrm{Tr}(\mathbf{P}) = P_{11} + P_{22} + P_{33}\) and \(I_2 = P_{11}P_{22} + P_{11}P_{33} + P_{22}P_{33} - P_{12}^2 - P_{13}^2 - P_{23}^2\). Bounded \(Q \in [0, 1]\): \(Q = 0\) for a perfectly gyrotropic plasma, \(Q \to 1\) at maximal agyrotropy. Frame-invariant (the invariants and \(P_\parallel = \hat{b} \cdot \mathbf{P}\cdot\hat{b}\) are both rotation-invariant under joint rotation of \(\mathbf{P}\) and \(\hat{b}\)). Swisdak plots \(\sqrt{Q}\) in figures to share a linear scale with \(A_\phi\) and \(D_{ng}\), but the definition (and what compute("agyrotropy") returns) is \(Q\) — not \(\sqrt{Q}\). Alternatives in the literature: Scudder's \(A_\phi\) ³⁵ and Aunai's \(D_{ng}\) ³⁶ — pypic standardizes on \(Q\) because, unlike \(A_\phi\), it senses off-axis (\(\hat{b}\)-coupling) components of \(\mathbf{P}\), and unlike \(D_{ng}\) it has the closed-form denominator above and an interpretable upper bound. Both alternatives ship as A_phi and D_ng (per-species A_phi_s{N} / D_ng_s{N}, with _e / _i aliases). See §9. ↩

9. Richardson convergence testing. For a \(p\)-th order scheme with grid spacing \(h\), the truncation error scales as \(O(h^p)\). Halving \(h\) reduces the error by a factor of \(2^p\): a ratio of ~2 indicates first order, ~4 second order, ~8 fourth order. This is the standard method for verifying stencil order ³⁹⁴⁰. The convergence tests in test_operators.py use this approach with sinusoidal fields at three resolutions, checking interior points only to avoid the lower-order one-sided boundary stencils. ↩↩

10. Both spellings resolve to the same array: compute("omega_pe") and compute("omega_p_s0") are interchangeable. Multi-species runs (omega_p_s2, lambda_D_s3, ...) synthesize via SPECIES_TEMPLATES on demand. ↩

11. Cyclotron frequencies are positive by convention (magnitudes). See conventions.md § Cyclotron Frequency. ³⁷³³ ↩↩

12. lambda_D is specifically the electron Debye length. See conventions.md § Debye Length. ³⁷ ↩

13. NRL convention: \(v_{th} = \sqrt{T/m}\) (1D Maxwellian standard deviation). The \(\sqrt{2T/m}\) convention (most probable speed) differs by \(\sqrt{2}\) and propagates into gyroradius. See conventions.md § Thermal Speed. ³⁷ ↩↩

14. MHD sound speed; differs from ion acoustic speed \(c_{ia}\). See conventions.md § Sound Speed vs Ion Acoustic. ³³ ↩↩

15. Maximum fast-mode phase speed (perpendicular propagation, \(\theta = 90°\)). The general dispersion relation is angle-dependent. See conventions.md § Fast Magnetosonic Speed. ³⁸ ↩

16. L2 norm is discrete and unweighted — volume factors cancel on the same grid (LeVeque convention). L∞ is absolute, not relative, to avoid division near field nulls. See conventions.md § Error Norms and Divergence. ↩

17. Several characteristic scales in Section 5 have relativistic generalizations listed here. When physics.relativistic = true in the simulation config, compute() uses these forms automatically. The non-relativistic limit is always recovered by setting \(\gamma_L = 1\) (or equivalently \(v \ll c\), \(\sigma \ll 1\)). ↩

18. Positive \(\mathbf{J} \cdot \mathbf{E} > 0\) means particles gain energy from fields (electromagnetic-to-kinetic energy conversion). Per-species decomposition \(\mathbf{J}_s \cdot \mathbf{E}\) identifies which species is energized. The energy conversion rate is the source term in the Poynting theorem: \(\partial u_{EM}/\partial t + \nabla \cdot \mathbf{S} = -\mathbf{J} \cdot \mathbf{E}\). ⁴³ §6.8, ⁴⁴. ↩

19. Generalized Ohm's law: \(\mathbf{E} = -\mathbf{V} \times \mathbf{B} + \frac{\mathbf{J} \times \mathbf{B}}{n_e q_e} - \frac{\nabla P_e}{n_e q_e} + \ldots\) The non-ideal residual \(\mathbf{E}' = \mathbf{E} + \mathbf{V} \times \mathbf{B}\) vanishes in ideal MHD and is non-zero where the frozen-in condition breaks down. The Hall term uses \(|q_e|\) (charge magnitude, always positive) following the standard convention where \(\mathbf{E}_{Hall}\) points in the \(\mathbf{J} \times \mathbf{B}\) direction. Dominant at ion skin depth scales where ion and electron motions decouple. ³¹³². ↩↩↩

20. For 2D geometry (\(\partial/\partial z = 0\)): \(\mathbf{B} = \nabla\psi \times \hat{z} + B_z \hat{z}\) defines the in-plane field via the flux function \(\psi\). Contours of \(\psi\) are in-plane field lines. The reconnected flux is \(\Delta\psi\) between the X-point and O-point. The reconnection rate is \(\partial\psi/\partial t\) at the X-point, equal to the out-of-plane electric field \(E_z\) there. Computed by cumulative integration: \(\psi(x, y) = -\int_0^x B_y(x', y)\, dx'\) (negative sign from \(B_y = -\partial\psi/\partial x\)). ⁴⁵ §3.1. ↩

21. Firehose: unstable when \(P_\parallel - P_\perp > B^2/2\) (parallel pressure excess drives field-line bending). Mirror: unstable when \(P_\perp/P_\parallel > 1 + 1/\beta_\perp\) (perpendicular pressure excess drives density compressions along field lines). Both arise from the CGL double-adiabatic framework and are the primary kinetic instabilities in collisionless plasmas with temperature anisotropy. ⁴⁶⁴⁷⁴⁸. ↩↩

22. X-points (magnetic nulls in 2D) are saddle points of the flux function \(\psi\) where \(\nabla\psi = 0\) and the Hessian determinant is negative. The Hessian \(H_{ij} = \partial^2\psi / \partial x_i \partial x_j\) is computed via second-order central finite differences. A negative determinant indicates saddle (hyperbolic) topology rather than an extremum (elliptic). The reconnection rate \(R = \partial\psi/\partial t\) at the X-point equals the out-of-plane electric field \(E_z\) there, and measures the rate of magnetic flux transfer between topologically distinct regions. ⁴⁵ §3.1, ³¹ Ch. 2. ↩↩

23. The magnetic shear angle measures the rotation of the magnetic field direction across a boundary (e.g. current sheet or magnetopause). \(\theta = 0\) for parallel fields, \(\pi\) for anti-parallel. Used in component reconnection analysis to determine whether reconnection geometry is anti-parallel (\(\theta \approx \pi\)) or component (\(\theta < \pi\)). ⁴⁹. ↩

24. Zenitani's electron-frame dissipation measure: the Lorentz-boosted scalar \(D_e = \gamma_e\bigl[\mathbf{J}\cdot \mathbf{E}' - \rho_c\,(\mathbf{V}_e\cdot\mathbf{E})\bigr]\) with \(\mathbf{E}' = \mathbf{E} + \mathbf{V}_e\times\mathbf{B}\). The first term is the electron-frame Joule heating; the \(\rho_c\,\mathbf{V}_e\cdot\mathbf{E}\) subtraction removes the bulk energy-transfer contribution to which \(\mathbf{J}\cdot\mathbf{E}\) is otherwise sensitive. Positive in the electron diffusion region, vanishes in ideal MHD. The canonical EDR localizer in modern collisionless PIC reconnection analysis. The non-relativistic limit is \(D_e \to \mathbf{J}\cdot\mathbf{E}' - \rho_c\,\mathbf{V}_e\cdot\mathbf{E}\); the \(\gamma_e\) prefactor activates when physics.relativistic is set in the dataset config. ⁵⁰. ↩

25. Dimensionless local reconnection rate normalized by the local Alfvén speed and local field magnitude — a per-cell map of where ideal MHD fails. Distinct from the canonical macroscopic rate \(R_{\mathrm{global}} = E_{\mathrm{rec}}/(v_{A,\mathrm{up}}\, B_{\mathrm{up}})\) measured at the X-point with upstream-asymptotic quantities; both saturate near \(0.1\) for fast collisionless reconnection (the source of frequent conflation), but address different questions. Per-cell scalar built from the same \(\mathbf{E}'\) that drives the 2D reconnection rate \(\partial\psi/\partial t\), divided by \(v_A\,\lvert\mathbf{B}\rvert\) so the result is comparable across runs with different field strengths. ⁵¹. ↩

26. Aunai's degree of nongyrotropy: \(D_{ng} = 2\,\|\mathbf{N}\|_F / \mathrm{Tr}(\mathbf{P})\) where \(\mathbf{N} = \mathbf{P} - P_\parallel\,\hat{b}\hat{b} - P_\perp(\mathbf{I} - \hat{b}\hat{b})\) is the non-gyrotropic part of the pressure tensor and \(\|\cdot\|_F\) is the Frobenius norm. Frame-invariant. Captures both perpendicular anisotropy (eigenvalue spread in the perp 2×2 block) and off-axis (\(\hat{b}\)-coupling) nongyrotropy. Closed-form identity: \(\|\mathbf{N}\|_F^2 = \mathrm{Tr}(\mathbf{P}^2) - P_\parallel^2 - 2 P_\perp^2\). pypic standardizes on Swisdak's \(Q\) (see agyrotropy) for its closed form and bounded range; \(D_{ng}\) ships as a research alternative for literature comparisons. ³⁶³⁴. ↩

27. Scudder's electron agyrotropy: eigenvalue-ratio of the perpendicular \(2\times 2\) block of \(\mathbf{P}\) in the field-aligned frame, \(A_\phi = \lvert\lambda_1^\perp - \lambda_2^\perp\rvert / (\lambda_1^\perp + \lambda_2^\perp)\). Bounded \([0, 1]\), frame-invariant. Only captures perpendicular anisotropy — misses off-axis (\(\hat{b}\)-coupling) nongyrotropy that \(D_{ng}\) and Swisdak's \(Q\) both detect. Closed form: \(A_\phi = \sqrt{\max(0,\;\|\Pi\|_F^2/(2 P_\perp^2) - 1)}\), where \(\Pi\) is the double-projected perpendicular pressure tensor (same object computed inside agyrotropy). ³⁵³⁴. ↩

28. Schindler-Hesse-Birn 3D general-magnetic-reconnection criterion. Reconnection is defined by \(\Xi(\mathbf{x}_0) \neq 0\) along a magnetic field line \(\mathcal{L}(\mathbf{x}_0)\) — no 3D null is required (unlike the 2D X-point definition). Vanishes identically in ideal MHD because \(\mathbf{E}^{\mathrm{ideal}} = -\mathbf{V}\times\mathbf{B} \perp \mathbf{B}\), so non-zero \(\Xi\) isolates the non-ideal contributions in Ohm's law. Implemented as an RK4 field-line trace from each seed (trace_field_line) with \(E_\parallel = \mathbf{E}\cdot\hat{b}\) trapezoid-integrated along arc length. Returns one scalar per seed — not a per-cell field, so dispatched outside the compute() registry as pypic.reconnection.schindler_xi. A runnable worked example on a guide-field Harris sheet — where no 3D null exists yet \(\Xi\) still localizes the X-line — is examples/ex_schindler_xi.py. ⁵². ↩

29. The PSD uses Parseval-consistent normalization: \(\int P(k)\, dk\) equals the variance of the windowed signal. Wavenumbers are in radians per unit length: \(k = 2\pi / \lambda\). Windowing (Hann by default) reduces spectral leakage from finite domain boundaries at the cost of broadening spectral peaks. The one-sided spectrum doubles the power at all frequencies except DC and Nyquist to account for negative-frequency contributions. Common reference slopes: \(k^{-5/3}\) (Kolmogorov inertial range), \(k^{-3}\) (2D enstrophy cascade), \(k^{-7/3}\) (sub-ion Hall-MHD). ⁵³⁵⁴. ↩↩↩

30. A. Mignone and J. C. McKinney. Equation of state in relativistic magnetohydrodynamics: variable versus constant adiabatic index. Monthly Notices of the Royal Astronomical Society, 378:1118–1130, 2007. doi:10.1111/j.1365-2966.2007.11849.x. ↩↩

31. Joachim Birn and Eric R. Priest. Reconnection of Magnetic Fields: Magnetohydrodynamics and Collisionless Theory and Observations. Cambridge University Press, 2007. doi:10.1017/cbo9780511536151. ↩↩↩

32. M. Hesse, N. Aunai, D. Sibeck, and J. Birn. On the electron diffusion region in planar, asymmetric, systems. Geophysical Research Letters, 41:8673–8680, 2014. doi:10.1002/2014GL061586. ↩↩

33. Francis F. Chen. Introduction to Plasma Physics and Controlled Fusion. Springer, 3rd edition, 2016. doi:10.1007/978-3-319-22309-4. ↩↩↩

34. M. Swisdak. Quantifying gyrotropy in magnetic reconnection. Geophysical Research Letters, 43:43–49, 2016. doi:10.1002/2015GL066980. ↩↩↩

35. J. Scudder and W. Daughton. “Illuminating” electron diffusion regions of collisionless magnetic reconnection using electron agyrotropy. Journal of Geophysical Research, 113:A06222, 2008. doi:10.1029/2008JA013035. ↩↩

36. N. Aunai, M. Hesse, and M. Kuznetsova. Electron nongyrotropy in the context of collisionless magnetic reconnection. Physics of Plasmas, 20:092903, 2013. doi:10.1063/1.4820953. ↩↩

37. J. D. Richardson. NRL Plasma Formulary. Naval Research Laboratory, 2019. ↩↩↩↩

38. Richard Fitzpatrick. Plasma Physics: An Introduction. CRC Press, 2014. doi:10.1201/b17263. ↩

39. Lewis Fry Richardson. The approximate arithmetical solution by finite differences of physical problems involving differential equations. 1911. Philosophical Transactions of the Royal Society A, 210:307–357. doi:10.1098/rsta.1911.0009. ↩

40. Randall J. LeVeque. Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM, 2007. doi:10.1137/1.9780898717839. ↩

41. A. Mignone and G. Bodo. An HLLC Riemann solver for relativistic flows – I. Hydrodynamics. Monthly Notices of the Royal Astronomical Society, 364:126–136, 2005. doi:10.1111/j.1365-2966.2005.09546.x. ↩↩

42. Yuri Lyubarsky. On the relativistic magnetic reconnection. Monthly Notices of the Royal Astronomical Society, 358:113–119, 2005. doi:10.1111/j.1365-2966.2005.08767.x. ↩

43. J. D. Jackson. Classical Electrodynamics. Wiley, 3rd edition, 1999. ↩

44. Seiji Zenitani and Masahiro Hoshino. The generation of nonthermal particles in the relativistic magnetic reconnection of pair plasmas. The Astrophysical Journal, 562:L63–L66, 2001. doi:10.1086/337972. ↩

45. Dieter Biskamp. Magnetic Reconnection in Plasmas. Cambridge University Press, 2000. doi:10.1017/cbo9780511599958. ↩↩

46. P. Hellinger, P. Trávníček, J. C. Kasper, and A. J. Lazarus. Solar wind proton temperature anisotropy: linear theory and WIND/SWE observations. Geophysical Research Letters, 33:L09101, 2006. doi:10.1029/2006GL025925. ↩

47. S. Peter Gary. Theory of Space Plasma Microinstabilities. Cambridge University Press, 1993. doi:10.1017/cbo9780511551512. ↩

48. Matthew W. Kunz, Alexander A. Schekochihin, and James M. Stone. Firehose and mirror instabilities in a collisionless shearing plasma. Physical Review Letters, 112:205003, 2014. doi:10.1103/PhysRevLett.112.205003. ↩

49. K. J. Trattner, S. A. Mullen, H. U. Frey, and S. M. Petrinec. Location of the reconnection line at the magnetopause during southward IMF conditions. Journal of Geophysical Research, 112:A08213, 2007. doi:10.1029/2006JA012029. ↩

50. S. Zenitani, M. Hesse, A. Klimas, and M. Kuznetsova. New measure of the dissipation region in collisionless magnetic reconnection. Physical Review Letters, 106:195003, 2011. doi:10.1103/PhysRevLett.106.195003. ↩

51. L. Comisso and A. Bhattacharjee. On the value of the reconnection rate. Journal of Plasma Physics, 82:595820601, 2016. doi:10.1017/S002237781600101X. ↩

52. K. Schindler, M. Hesse, and J. Birn. General magnetic reconnection, parallel electric fields, and helicity. Journal of Geophysical Research, 93:5547–5557, 1988. doi:10.1029/JA093iA06p05547. ↩

53. Uriel Frisch. Turbulence: The Legacy of A. N. Kolmogorov. Cambridge University Press, 1995. doi:10.1017/cbo9781139170666. ↩

54. William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery. Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, 3rd edition, 2007. ↩