Linearized Relativistic Boltzmann Equation

• The linearized relativistic Boltzmann equation describes small perturbations of a dilute relativistic gas from local equilibrium using a linearized collision operator.
• It features a self-adjoint, nonpositive operator with a five-dimensional null space that ensures conservation of particle number and energy-momentum.
• Spectral analysis reveals discrete fluid modes and exponential decay, providing a rigorous basis for deriving relativistic hydrodynamics and transport coefficients.

The linearized relativistic Boltzmann equation governs small perturbations of a dilute, relativistic gas from local equilibrium. It plays a central role in the derivation of hydrodynamics, the analysis of relaxation rates, and the paper of transport phenomena in kinetic theory under special or general relativistic conditions. Its structure, spectral properties, and analytic behavior are essential for understanding both mathematical well-posedness and the foundational aspects of relativistic dissipative fluid dynamics.

1. Formulation and Linearization

The relativistic Boltzmann equation for the one-particle distribution function $f(x^\mu, p^\nu)$ on the mass shell ($p^\mu p_\mu = m^2c^2$, $p^0>0$) is

$p^\mu \frac{\partial f}{\partial x^\mu} = C[f,f],$

with the collision integral $C[f,f]$ expressing binary, Lorentz-invariant scattering. In the presence of background fields or curved spacetime, additional streaming terms (e.g., Christoffel or Lorentz forces) appear (Kremer, 2014).

Linearization is performed by expanding $f = f^{\mathrm{eq}} + \delta f$ around a local equilibrium—typically the Jüttner distribution,

$$f^{\mathrm{eq}}_p(x) = \frac{1}{e^{(p\cdot u-\mu)/T} - a},$$

where $a$ encodes Bose/Fermi or Maxwell statistics, $T$ the temperature, $\mu$ the chemical potential, and $u^\mu$ the hydrodynamic 4-velocity (Tsumura et al., 2015).

Deviations $\delta f$ are assumed small, and the collision operator is linearized: $$C[f^{\mathrm{eq}} + \delta f] \approx C[f^{\mathrm{eq}}] + \hat{L}[\delta f],$$ where $C[f^{\mathrm{eq}}] = 0$, and $\hat L$ is the linearized collision operator.

In component form, the linearized relativistic Boltzmann equation is

$$p^\mu \partial_\mu \delta f_p(x) = \hat L[\delta f]_p(x).$$

2. Structure and Properties of the Linearized Collision Operator

The linearized operator $\hat L$ has the explicit structure

$$[\hat L \delta f]_p = \frac{1}{p \cdot u} \sum_q \left.\frac{\delta C[f]_p}{\delta f_q}\right|_{f = f^{\mathrm{eq}}} \delta f_q,$$

which, upon integration, can be expressed as a sum of "gain" and "loss" terms reflecting post- and pre-collisional distributions (Zhao et al., 2022, Jang, 2016).

Key properties are:

• Self-adjointness and seminegativity: $\hat L$ is self-adjoint and nonpositive with respect to the weighted inner product

$$\langle \psi, \chi \rangle = \int dp\, (p \cdot u) f^{\mathrm{eq}}_p [1+a f^{\mathrm{eq}}_p]\, \psi_p \chi_p.$$

• Null space (collision invariants): The kernel of $\hat L$ is five-dimensional, spanned by $\{1, p^\mu\}$ (conservation of particle number and energy-momentum) (Tsumura et al., 2015, Zhao et al., 2022).
• Spectral gap: On the orthogonal complement of the null space, $\hat L$ admits a strictly negative spectrum, with precise, coercive lower bounds for hard potentials and appropriate angular singularities (Jang, 2016, Jang et al., 2021).

In the presence of non-cutoff kernels, $\hat L$ exhibits fractional Laplacian-type regularization, and the main quantitative norm is a geometric fractional Sobolev norm with a velocity weight (Jang, 2016).

3. Spectral Analysis and Fluid-Mode Decomposition

For homogeneous or (spatially) periodic problems, one studies the Cauchy problem

$$\partial_t f + \hat v \cdot \nabla_x f - Lf = \Gamma(f, f),$$

where $L$ is the linearized operator and $\Gamma$ a quadratic nonlinearity. In Fourier space, the generator

$B(k) = L - i\hat v \cdot k$

has the following spectral decomposition (Zhao et al., 2022):

• Essential spectrum: Bounded away from the imaginary axis, i.e., $\Re \lambda \le -c_0 < 0$ for nonzero frequencies.
• Discrete fluid modes: Five eigenvalues analytic in $k$ near $k=0$ correspond to the hydrodynamic variables; these modes govern long-wavelength dissipative dynamics and decay algebraically in time $\sim (1+t)^{-3/2}$ in three dimensions, reflecting sound and diffusion (Zhao et al., 2022).
• Microscopic modes: The remainder of the spectrum is strictly negative, ensuring exponential decay of non-fluid perturbations.

Spectral analysis justifies the reduction of kinetic theory to effective macroscopic equations in the hydrodynamic regime (Tsumura et al., 2015, Tsumura et al., 2012).

4. Hydrodynamic Limits and Transport Coefficients

The linearized equation provides the foundation for the Chapman–Enskog and the renormalization-group (RG) reduction to dissipative relativistic hydrodynamics (Tsumura et al., 2015, Tsumura et al., 2012). The deviation $\delta f$ is expanded in eigenfunctions,

$\Psi_p = \sum_{X} [\hat L^{-1} \hat X]_p X,$

where $X$ runs over dissipative currents (bulk, heat, shear). Constitutive relations and transport coefficients (shear viscosity $\eta$, bulk viscosity $\zeta$, conductivity $\lambda$) are given by explicit inner products involving $\hat L^{-1}$: $$\eta = -\frac{1}{10T} \langle \hat{\pi}^{\mu\nu}, \hat L^{-1} \hat{\pi}_{\mu\nu} \rangle,\quad \zeta = -\frac{1}{T} \langle \hat{\Pi}, \hat L^{-1} \hat{\Pi} \rangle$$ and so on (Tsumura et al., 2015, Kremer, 2014, Tsumura et al., 2012).

Relaxation times are obtained from higher powers of $\hat L^{-1}$ and have Green–Kubo-like representations in terms of correlation integrals,

$$\tau_{\Pi} = - \frac{\langle \hat \Pi, \hat L^{-2} \hat \Pi \rangle}{\langle \hat \Pi, \hat L^{-1} \hat \Pi \rangle}$$

(Tsumura et al., 2015).

5. Causality, Stability, and Well-Posedness

Second-order hydrodynamic equations reduced from the linearized Boltzmann framework feature relaxation terms guaranteeing causal, stable evolution: $\Pi = -\zeta\, \theta - \tau_\Pi \dot \Pi + \dots$ and analogous relations for the other currents (Tsumura et al., 2015). Linear stability analysis of the full coupled system shows:

• Causality: The characteristic velocities of fluctuations are maximally luminal ($v_{\rm ch} \le 1$) (Tsumura et al., 2015).
• Stability: All nontrivial perturbations decay due to the negative definiteness of the inner product $(\chi, \hat L^{-1} \chi)<0$ for non-zero-modes (Tsumura et al., 2015, Zhao et al., 2022).

Mathematically, the existence and uniqueness of global classical solutions and the optimal large-time decay rates have been established under general scattering kernels, including those without angular cutoff (Jang, 2016, Zhao et al., 2022).

6. Carleman Representation and Non-Cutoff Behavior

For general, non-cutoff kernels (with angular singularity), the Carleman dual representation is employed to express the linearized operator in forms amenable to fractional Sobolev estimates (Jang, 2016). The sharp coercive bounds and spectral gap remain valid due to the precise geometric structure of the Møller relative speed and the Jüttner weight.

Failure of convenience transformations, such as nonrelativistic $p' \to p$ variable substitutions, necessitates the use of Lorentz-invariant reduction strategies and careful spectral analysis (Jang et al., 2021). The consequences include:

• Nonlocal, fractional dissipation in momentum variables.
• Lorentz-covariant modeling of collision frequencies.
• Uniform positivity and spectral gap for the weighted $L^2$ norms.

7. Relaxation Time Approximation and Its Refinement

The Anderson–Witting Relaxation Time Approximation (RTA) provides a simplified model for the linearized collision operator,

$$\hat L_{\text{RTA}}[\phi]_k = -\frac{E_k}{\tau_R} f^{(0)}_k\, \phi(k),$$

but fails to preserve conservation laws if $\tau_R$ is energy dependent. The corrected RTA projects out the five collisional invariants: $$\hat L_{\text{RTA}}^{\text{new}}[\phi](k) = -\frac{E_k}{\tau_R} f^{(0)}_k\,\left[ \phi(k) - A(x) - B_\mu(x) k^\mu \right]$$ with $A$ and $B_\mu$ chosen to annihilate the zero modes and enforce Landau matching, restoring the proper null space and spectral properties (Rocha et al., 2021).

Table: Properties of the Linearized Collision Operator

Property	Description	Reference
Self-adjoint, nonpositive	True in weighted inner product	(Tsumura et al., 2015)
Five-dimensional null space	Spanned by $\{1, p^\mu\}$	(Tsumura et al., 2012)
Spectral gap (coercivity)	Strictly negative on orthogonal complement	(Jang, 2016, Zhao et al., 2022)
Carleman representation	Fractional-Sobolev structure in non-cutoff case	(Jang, 2016)
Nonlocal dissipation	Fractional Laplacian-type for hard/soft potentials	(Jang et al., 2021)
Causality/stability	Causal signal speed and decay of perturbations	(Tsumura et al., 2015)

These structural and analytic results underlie the theoretical and computational modeling of relativistic kinetic systems, demonstrating both the mathematical rigidity and physical consistency of the linearized relativistic Boltzmann equation framework.