Eur. Phys. J. B 23, 487-496 (2001)
https://doi.org/10.1007/s100510170040

Covariant Kolmogorov equation and entropy current for the relativistic Ornstein-Uhlenbeck process

¹ Laboratoire de Radioastronomie, ENS, 24 rue Lhomond, 75231 Paris Cedex 05, France
² CNRS, Laboratoire G.D. Cassini, Observatoire de Nice, 06304 Nice Cedex 04, France

Corresponding author: ^a rivet@obs-nice.fr

Received: 30 May 2001
Published online: 15 October 2001

Abstract

The relativistic Ornstein-Uhlenbeck Process (ROUP), which is a toy-model of relativistic irreversible phenomena, is studied statistically in an explicitly covariant manner. An 8-dimensional phase space is introduced (four dimensions for space-time coordinates, and four dimensions for the 4-momentum coordinates), on which `extended' probability distributions are defined (the usual probability distribution is recovered as their restriction to the mass shell). An explicitly covariant Kolmogorov equation is derived for these `extended' probability distributions. The whole formalism is used to introduce a 4-current of conditional entropy and prove that the 4-divergence of this 4-current is always positive. This constitutes an H-theorem for the ROUP.