Statistical Mechanics of an Integrable System (original) (raw)

Nonequilibrium statistical mechanics of open classical systems

XIVth International Congress on Mathematical Physics, 2006

We describe the ergodic and thermodynamical properties of chains of anharmonic oscillators coupled, at the boundaries, to heat reservoirs at positive and different temperatures. We discuss existence and uniqueness of stationary states, rate of convergence to stationarity, heat flows and entropy production, Kubo formula and Gallavotti-Cohen fluctuation theorem.

Statistical Physics and Dynamical Systems: Models of Phase Transitions

2007

This paper explores the connection between dynamical system properties and statistical phyics of ensembles of such systems. Simple models are used to give novel phase transitions; particularly for finite N particle systems with many physically interesting examples. 1. INTRODUCTION The developments in dynamical systems and in Statistical Mechanics have occured for over a century. The ergodic hypothesis of J W Gibbs and Boltzmann's H theorem were re-investigated as integrable and chaotic dynamical systems were found. Poincare, Birkhoff, Krylov formulated these problems. Kolmogorov, Arnold and Moser theorem gave a detailed description of phase space as 'islands of integrable and sea of chaotic regions'. From chaotic billiards to integrable Toda lattices the range of the systems includes mixed type of systems. Work of Ruelle, Benettin, Galgani, Casati, Galavotti and others clarified some properties of classical and quantum systems. Recent work of E.G.D.Cohen, L Casetti and others has given a rigourous basis for Riemannian space based description of dynamics of Hamiltonian systems and K entropy. However the Statistical mechanics of dynamical systems remains to be formulated. Ensembles with model Hamiltonians showing chaos transitions and topological transitions have become significant as simulation and empirical work has grown. A partition function inclusive of the Kolmogorov entropy and the Euler characteristic has been defined. Nano clusters of particles show properties dependent on the boundaries, number, energy and interaction parameters. Strongly interacting systems in condensed matter , nucleons and quarks are also possible applications. This requires that any dynamical system model such as with maps, or differential equations for the 'units' in the physical system, will have consequences for the statistical mechanics of their ensemble. It leads to novel phase transitions and new interpretations of thermodynamic phase transitions. This has been shown in this paper for some simple models. The Baker transform is used to model kinetics of melting in two dimensions. The Henon Heiles like models are used to model ergodic channels with correlated charge densities in superconductors. The gas of molecules with Henon Heiles Hamiltonian is shown to have a 'phase' transition dependent on the chaotic transition in the molecule. Quantum chaos also creates a transition in the ensemble of such systems and the Poisson, GOE, GUE and Husimi distributions are an example of Wigner distributions on phase space. These ideas could be generalised to more complex model maps or Hamiltonians, that are used in physical systems. Hence a statistical physics of finite (any N) number of particles is expected to have definite properties dependent on

Exponential Convergence to Non-Equilibrium Stationary States in Classical Statistical Mechanics

Communications in Mathematical Physics, 2002

We continue the study of a model for heat conduction consisting of a chain of non-linear oscillators coupled to two Hamiltonian heat reservoirs at different temperatures. We establish existence of a Liapunov function for the chain dynamics and use it to show exponentially fast convergence of the dynamics to a unique stationary state. Ingredients of the proof are the reduction of the infinite dimensional dynamics to a finite-dimensional stochastic process as well as a bound on the propagation of energy in chains of anharmonic oscillators. 1

Dynamical chaos in the integrable Toda chain induced by time discretization

arXiv (Cornell University), 2023

We use the Toda chain model to demonstrate that numerical simulation of integrable Hamiltonian dynamics using time discretization destroys integrability and induces dynamical chaos. Specifically, we integrate this model with various symplectic integrators parametrized by the time step τ and measure the Lyapunov time T Λ (inverse of the largest Lyapunov exponent Λ). A key observation is that T Λ is finite whenever τ is finite but diverges when τ → 0. We compare the Toda chain results with the nonitegrable Fermi-Pasta-Ulam-Tsingou chain dynamics. In addition, we observe a breakdown of the simulations at times T B ≫ T Λ due to certain positions and momenta becoming extremely large ("Not a Number"). This phenomenon originates from the periodic driving introduced by symplectic integrators and we also identify the concrete mechanism of the breakdown in the case of the Toda chain.

New trends in nonequilibrium statistical mechanics: classical and quantum systems

Journal of Statistical Mechanics: Theory and Experiment, 2020

The nonlinear relaxation process in many condensed matter systems proceeds through metastable states, giving rise to long-lived states. Stochastic manybody systems, classical and quantum, often display a complex and slow relaxation towards a stationary state. A common phenomenon in the dynamics of out of equilibrium systems is the metastability, and the problem of the lifetime of metastable states involves fundamental aspects of nonequilibrium statistical mechanics. In spite of such ubiquity, the microscopic understanding of metastability and related out of equilibrium dynamics still raise fundamental questions. The aim of this meeting is to bring together scientists interested in the challenging problems connected with dynamics of out of equilibrium classical and quantum physical systems from both theoretical and experimental point of view, within an interdisciplinary context. Specifically, three main areas of outof-equilibrium statistical mechanics will be covered: long range interactions and multistability, anomalous diffusion, and quantum systems. Moreover, the conference will be a discussion forum to promote new ideas in this fertile research field, and in particular new trends such as quantum thermodynamics and novel types of quantum phase transitions occurring in non-equilibrium steady states, and topological phase transitions.