Nonlinear optical Galton board: Thermalization and continuous limit

Giuseppe Di Molfetta; Fabrice Debbasch; Marc Brachet

doi:10.1103/PhysRevE.92.042923

Nonlinear optical Galton board: Thermalization and continuous limit

Phys Rev E Stat Nonlin Soft Matter Phys. 2015 Oct;92(4):042923. doi: 10.1103/PhysRevE.92.042923. Epub 2015 Oct 28.

Authors

Giuseppe Di Molfetta¹, Fabrice Debbasch¹, Marc Brachet²

Affiliations

¹ LERMA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, UPMC Univ. Paris 6, UMR 8112, F-75014 Paris, France.
² Laboratoire de Physique Statistique de l'Ecole Normale Supérieure / PSL Research University, associé au CNRS et aux Universités Pierre-et-Marie-Curie Paris 06 et Paris Diderot, and 24 Rue Lhomond, 75231 Paris, France.

PMID: 26565320
DOI: 10.1103/PhysRevE.92.042923

Abstract

The nonlinear optical Galton board (NLOGB), a quantum walk like (but nonlinear) discrete time quantum automaton, is shown to admit a complex evolution leading to long time thermalized states. The continuous limit of the Galton board is derived and shown to be a nonlinear Dirac equation (NLDE). The (Galerkin-truncated) NLDE evolution is shown to thermalize toward states qualitatively similar to those of the NLOGB. The NLDE conserved quantities are derived and used to construct a stochastic differential equation converging to grand canonical distributions that are shown to reproduce the (microcanonical) NLDE thermalized statistics. Both the NLOGB and the Galerkin-truncated NLDE are thus demonstrated to exhibit spontaneous thermalization.