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Relativistic Lévy processes

Lucas G. B. de Souza1,2,*, M. G. E. da Luz3,†, E. P. Raposo4,‡, Evaldo M. F. Curado5,6,§, and G. M. Viswanathan1,6,∥

  • *Contact author: lucas.gabrielsouza@fisica.ufrn.br
  • †Contact author: luz@fisica.ufpr.br
  • ‡Contact author: ernesto.raposo@ufpe.br
  • §Contact author: evaldo@cbpf.br
  • ∥Contact author: gandhi.viswanathan@gmail.com
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Phys. Rev. E 112, 044125 – Published 14 October, 2025

DOI: https://doi.org/10.1103/wf2s-g7lv

Abstract

We study sums of independent and identically distributed random velocities in special relativity. We show that the resulting one-dimensional velocity distributions are not only stable under relativistic velocity addition, but define a genuinely new class of stochastic processes, namely relativistic Lévy processes. Given a system, this allows identifying distinct relativistic regimes in terms of the distribution's concavity at the origin and the probability of measuring relativistic velocities. These features provide a protocol to assess the relevance of stochastic relativistic effects in actual experiments. As supporting evidence, we find agreement with previous results about heavy-ion diffusion and show that our findings are consistent with the distribution of momentum deviations observed in measurements of antiproton cooling.

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Phys. Rev. E 112, 044125– Published 14 October, 2025
  • Received 3 September 2024
  • Revised 22 July 2025
  • Accepted 3 September 2025
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