Nonlinear diffusion in relativistic kinetic theory
Simone Calogero
TL;DR
The paper addresses the need for a relativistic diffusion model that preserves particle number, energy, and momentum while remaining Lorentz invariant and compatible with gravity. It develops a nonlinear diffusion operator with a diffusion scalar $\Phi_f$ and drift $K_f$ that covariantly transform under Lorentz transformations, recovers the Newtonian limit as $c\to\infty$, and yields a special-relativistic diffusion with a non-Jüttner equilibrium that tends to Maxwellian in the Newtonian limit. The spatially homogeneous relativistic equilibria are explicit and show nonlinearity persists in the homogeneous setting, with finite-energy conditions requiring $\lambda>2$. The framework extends to general relativity by formulating diffusion on Lorentzian manifolds, preserving $\nabla_\alpha N^\alpha=0$ and $\nabla_\beta T^{\alpha\beta}=0$, and enabling direct coupling to Einstein equations via the contracted Bianchi identities, with potential cosmological applications. Overall, the work provides a consistent, conservation-law–satisfying diffusion model across Newtonian, special-relativistic, and general-relativistic regimes.
Abstract
A nonlinear Lorentz invariant kinetic diffusion equation is introduced, which is consistent with the conservation laws of particles number, energy and momentum. The equilibrium solution converges to the Maxwellian density in the Newtonian limit, but it is not given by the Jüttner distribution commonly employed in relativistic kinetic theory. The nonlinear kinetic diffusion equation on a general Lorentzian manifold is consistent with the contracted Bianchi identities and therefore can be coupled to the Einstein equations of general relativity.