title: Nonlinear optical Galton board: Thermalization and continuous limit.
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.
authors.affiliation: LERMA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, UPMC Univ. Paris 6, UMR 8112, F-75014 Paris, France.
authors.forename: Giuseppe
authors.initials: G
authors.lastname: Di Molfetta
authors.affiliation: LERMA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, UPMC Univ. Paris 6, UMR 8112, F-75014 Paris, France.
authors.forename: Fabrice
authors.initials: F
authors.lastname: Debbasch
authors.affiliation: 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.
authors.forename: Marc
authors.initials: M
authors.lastname: Brachet