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Nonlinear optical Galton board: Thermalization and continuous limit.

Giuseppe Di Molfetta1, Fabrice Debbasch1, Marc Brachet2

  • 1LERMA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, UPMC Univ. Paris 6, UMR 8112, F-75014 Paris, France.

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PubMed
Summary
This summary is machine-generated.

The nonlinear optical Galton board (NLOGB) and its continuous limit, the nonlinear Dirac equation (NLDE), spontaneously thermalize. Both systems evolve towards complex, long-time thermalized states, demonstrating a novel form of spontaneous thermalization.

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Area of Science:

  • Quantum mechanics
  • Nonlinear dynamics
  • Statistical physics

Background:

  • The nonlinear optical Galton board (NLOGB) is a discrete-time quantum automaton exhibiting complex dynamics.
  • Understanding thermalization in nonlinear systems is crucial for statistical physics and quantum chaos.

Purpose of the Study:

  • To investigate the long-time evolution and thermalization properties of the NLOGB.
  • To derive and analyze the continuous limit of the NLOGB, identified as a nonlinear Dirac equation (NLDE).
  • To demonstrate spontaneous thermalization in both the NLOGB and the NLDE.

Main Methods:

  • Derivation of the continuous limit of the NLOGB to obtain the NLDE.
  • Galerkin truncation applied to the NLDE for numerical evolution.
  • Derivation of conserved quantities for the NLDE.
  • Construction of a stochastic differential equation from conserved quantities.

Main Results:

  • The NLOGB exhibits complex evolution leading to long-time thermalized states.
  • The continuous limit of the NLOGB is shown to be an NLDE.
  • Galerkin-truncated NLDE evolution thermalizes to states similar to the NLOGB.
  • Conserved quantities of the NLDE were used to construct a stochastic differential equation that converges to grand canonical distributions.
  • These distributions reproduce the microcanonical thermalized statistics of the NLDE.

Conclusions:

  • Both the NLOGB and the Galerkin-truncated NLDE demonstrate spontaneous thermalization.
  • The study provides a theoretical framework connecting discrete nonlinear quantum systems to continuous nonlinear field equations and their thermalization properties.