Adiabatic Elimination in Relativistic Stochastic Mechanics

TL;DR

This work develops a covariant framework for relativistic diffusion by applying adiabatic elimination to fast momentum variables in a $(1+1)$-dimensional relativistic stochastic setting. It compares equation-of-motion and distribution-level reductions, derives a spacetime diffusion equation with relativistic corrections, and introduces a new dimensionless scale to assess the validity of the elimination. A complementary path-integral coarse-graining is presented to achieve higher accuracy at the cost of computational effort, highlighting trade-offs between covariance preservation and practical modeling. Numerical simulations corroborate the analytic reductions, illustrating how relativistic effects slow diffusion relative to the Newtonian case and how coarse-grained observables relate to the full phase-space dynamics. The results provide a systematic route to relativistic diffusion descriptions relevant for plasmas, cosmology, and nucleosynthesis, and point to future work on covariant coarse-graining and higher-dimensional extensions.

Abstract

We investigate the adiabatic elimination of fast variables in relativistic stochastic mechanics, which is analyzed in the equation of motion and in the distribution function, with relativistic corrections explicitly derived. A new dimensionless parameter is introduced to characterize the timescale. The adiabatic elimination is also compared with the path integral coarse graining, which is more general yet computationally demanding.

Figures (6)

• Figure 1: Mean coordinate value $t$ as a function of different evolution parameters.
• Figure 2: Mean squared momentum and displacement with respect to the evolution parameter.
• Figure 3: Mean squared displacement over $\left\langle t\right\rangle$ and $t$
• Figure 4: Comparison between theoretical curve and simulation data at $t=100$.
• Figure 5: Time evolution of the distribution function with $m=1, \kappa=1, \beta=1$.