Statistical mechanics of an integrable system

Abstract

We provide here an explicit example of Khinchin's idea that the validity of equilibrium statistical mechanics in high dimensional systems does not depend on the details of the dynamics. This point of view is supported by extensive numerical simulation of the one-dimensional Toda chain, an integrable non-linear Hamiltonian system where all Lyapunov exponents are zero by definition. We study the relaxation to equilibrium starting from very atypical initial conditions and focusing on energy equipartion among Fourier modes, as done in the original Fermi-Pasta-Ulam-Tsingou numerical experiment. We find evidence that in the general case, i.e., not in the perturbative regime where Toda and Fourier modes are close to each other, there is a fast reaching of thermal equilibrium in terms of a single temperature. We also find that equilibrium fluctuations, in particular the behaviour of specific heat as function of temperature, are in agreement with analytic predictions drawn from the ordinary Gibbs ensemble, still having no conflict with the established validity of the Generalized Gibbs Ensemble for the Toda model. Our results suggest thus that even an integrable Hamiltonian system reaches thermalization on the constant energy hypersurface, provided that the considered observables do not strongly depend on one or few of the conserved quantities. This suggests that dynamical chaos is irrelevant for thermalization in the large-$N$ limit, where any macroscopic observable reads of as a collective variable with respect to the coordinate which diagonalize the Hamiltonian. The possibility for our results to be relevant for the problem of thermalization in generic quantum systems, i.e., non-integrable ones, is commented at the end.

6 pages, 4 figures + Appendix

References in corpus (8)

• Thermalization and its mechanism for generic isolated quantum systems
• Quantum Quench in the Transverse Field Ising Chain
• Proof of the Ergodic Theorem and the H-Theorem in Quantum Mechanics
• Effective thermal dynamics following a quantum quench in a spin chain
• Long-Time Behavior of Macroscopic Quantum Systems: Commentary Accompanying the English Translation of John von Neumann's 1929 Article on the Quantum Ergodic Theorem
• Domain wall dynamics in the Landau--Lifshitz magnet and the classical-quantum correspondence for spin transport
• (Non equilibrium) Thermodynamics of Integrable models: The Generalized Gibbs Ensemble description of the classical Neumann Model
• Stationarization and Multithermalization in spin glasses

Cited by in corpus (24)

• Lyapunov spectrum scaling for classical many-body dynamics close to integrability
• Entropy growth during free expansion of an ideal gas
• Thermalization slowing down in multidimensional Josephson junction networks
• Ergodic observables in non-ergodic systems: the example of the harmonic chain
• Dynamics and transport in the boundary-driven dissipative Klein-Gordon chain
• Thermalization without chaos in harmonic systems
• Quantum and classical contributions to entropy production in fermionic and bosonic Gaussian systems
• Quantum master equation from the eigenstate thermalization hypothesis
• On the foundations of statistical mechanics
• Long-time relaxation of a finite spin bath linearly coupled to a qubit
• Microscopic contributions to the entropy production at all times: From nonequilibrium steady states to global thermalization
• Thermalization of the Ablowitz-Ladik lattice in the presence of non-integrable perturbations
• Open-system eigenstate thermalization in a noninteracting integrable model
• Thermalization of isolated harmonic networks under conservative noise
• Quantum Enhancement of Thermalization
• Prethermalization in Fermi-Pasta-Ulam-Tsingou chains
• Integrable hydrodynamics of Toda chain: case of small systems
• Statistical Features of High-Dimensional Hamiltonian Systems
• Thermalization is typical in large classical and quantum harmonic systems
• Large-deviations approach to thermalization: the case of harmonic chains with conservative noise
• Trotter transition in BCS pairing dynamics
• Thermalization dynamics of macroscopic weakly nonintegrable maps
• Generalised Gibbs Ensemble for spherically constrained harmonic models
• Lax dynamics