Relativistic Langevin Equation derived from a particle-bath Lagrangian

TL;DR

This paper derives a relativistic Langevin equation from a Lorentz-covariant particle-bath Lagrangian, providing a rigorous foundation for modeling relativistic Brownian motion and exploring its non-Markovian nature.

Contribution

It introduces a first-principles derivation of the relativistic Langevin equation from a covariant Lagrangian, extending previous non-relativistic models to fully relativistic regimes.

Findings

Derivation of a Lorentz-covariant Langevin equation from a particle-bath Lagrangian

Identification of the relativistic Langevin equation with models used in high-speed statistical mechanics

Discussion of non-Markovian effects and symmetry considerations in relativistic stochastic dynamics

Abstract

We show how a relativistic Langevin equation can be derived from a Lorentz-covariant version of the Caldeira-Leggett particle-bath Lagrangian. In one of its limits, we identify the obtained equation with the Langevin equation used in contemporary extensions of statistical mechanics to the near-light-speed motion of a Brownian particle in non-relativistic dissipative fluids. The proposed framework provides a more rigorous and first-principles form of the Langevin equation often quoted or postulated as ansatz in previous works. We then refine the aforementioned results by considering more terms in the particle-bath coupling, which improves the precision of the approximation for fully relativistic settings where not only the tagged particle but also the thermal bath motion is relativistic. We discuss the implications of the apparent breaking of space-time translation and parity invariance, showing that these effects are not necessarily in contradiction with the assumptions of statistical mechanics. The intrinsically non-Markovian character of the fully relativistic generalized Langevin equation derived here, and of the associated fluctuation-dissipation theorem, is also discussed.