[Paper Review] On a relativistic Fokker-Planck equation in kinetic theory

José Antonio Alcántara Felix, Simone Calogero|arXiv (Cornell University)|Nov 24, 2010

Statistical Mechanics and Entropy21 references3 citations

TL;DR

This paper introduces and rigorously analyzes a relativistic Fokker-Planck equation that preserves Lorentz invariance in the absence of friction, ensuring finite signal propagation speed—key for relativistic consistency. It establishes existence of steady states for all masses in both plasma and gravitational mean-field models, improving on non-relativistic counterparts where existence is only known for small masses.

ABSTRACT

A relativistic kinetic Fokker-Planck equation that has been recently proposed in the physical literature is studied. It is shown that, in contrast to other existing relativistic models, the one considered in this paper is invariant under Lorentz transformations in the absence of friction. A similar property (invariance by Galilean transformations in the absence of friction) is verified in the non-relativistic case. In the first part of the paper some fundamental mathematical properties of the relativistic Fokker-Planck equation are established. In particular, it is proved that the model is compatible with the finite propagation speed of particles in relativity. In the second part of the paper, two non-linear relativistic mean-field models are introduced. One is obtained by coupling the relativistic Fokker-Planck equation to the Maxwell equations of electrodynamics, and is therefore of interest in plasma physics. The other mean-field model couples the Fokker-Planck dynamics to a relativistic scalar theory of gravity (the Nordström theory) and is therefore of interest in gravitational physics. In both cases the existence of steady states for all possible prescribed values of the mass is established. In the gravitational case this result is better than for the corresponding non-relativistic model, the Vlasov-Poisson-Fokker-Planck system, for which existence of steady states is known only for small mass.

Motivation & Objective

• To establish a relativistic Fokker-Planck equation that maintains Lorentz invariance when friction is absent, mirroring Galilean invariance in the non-relativistic case.
• To ensure the model respects the finite propagation speed of particles, incompatible with classical Fokker-Planck equations.
• To extend the model to non-linear mean-field systems coupling to Maxwell electrodynamics (plasma physics) and Nordström gravity (gravitational physics).
• To prove existence of steady-state solutions for all prescribed mass values in both relativistic mean-field models.
• To demonstrate that the relativistic model overcomes limitations of the non-relativistic Vlasov-Poisson-Fokker-Planck system, which only guarantees steady states for small masses.

Proposed method

• Adopts the relativistic Fokker-Planck equation with relativistic velocity $\hat{p} = p / \sqrt{1 + |p|^2}$ and a relativistic diffusion matrix $D = (I + p \otimes p)/\sqrt{1 + |p|^2}$, ensuring physical consistency with special relativity.
• Proves Lorentz invariance of the equation under zero-friction conditions using transformation properties of the relativistic momentum and velocity.
• Applies a generalized uniqueness argument based on Fritz John’s method to establish finite propagation speed, by showing that if initial data are compactly supported, the solution remains zero outside the relativistic light cone.
• Introduces two non-linear mean-field models: one coupling the Fokker-Planck equation to Maxwell’s equations for plasma physics, and another to Nordström’s scalar theory of gravity for gravitational physics.
• Uses functional analytic techniques, including weighted $L^2$ estimates and spectral theory, to control the dynamics and prove existence of steady states.
• Employs a cut-off function $\phi_k$ and Young’s inequality to control error terms in the energy estimates, leading to the conclusion that the solution must vanish under appropriate limits.

Experimental results

Research questions

• RQ1 Does the proposed relativistic Fokker-Planck equation preserve Lorentz invariance in the absence of friction, analogous to Galilean invariance in the non-relativistic case?
• RQ2 Can the model be shown to respect the finite propagation speed of particles, avoiding the instantaneous diffusion problem of classical Fokker-Planck equations?
• RQ3 Does the relativistic mean-field model coupled to Maxwell’s equations admit steady-state solutions for all prescribed mass values in plasma physics?
• RQ4 Does the relativistic mean-field model coupled to Nordström’s scalar gravity admit steady-state solutions for all prescribed mass values in gravitational physics?
• RQ5 How does the existence theory for steady states in this relativistic model compare to the non-relativistic Vlasov-Poisson-Fokker-Planck system, particularly regarding mass constraints?

Key findings

• The relativistic Fokker-Planck equation is Lorentz invariant when friction is absent, confirming its consistency with special relativity.
• Solutions to the equation exhibit finite propagation speed: if initial data are compactly supported in position, the solution remains zero outside the relativistic light cone.
• The non-linear relativistic mean-field model coupled to Maxwell’s equations admits steady-state solutions for all prescribed mass values.
• The non-linear relativistic mean-field model coupled to Nordström’s scalar gravity admits steady-state solutions for all prescribed mass values, a stronger result than the non-relativistic Vlasov-Poisson-Fokker-Planck system, which only guarantees existence for small masses.
• The proof of existence of steady states relies on a novel energy estimate using cut-off functions and Young’s inequality, ultimately showing that the solution must vanish under appropriate limits.
• The model avoids the unphysical infinite-speed diffusion of classical Fokker-Planck equations by construction, ensuring compatibility with relativistic causality.