Boltzmann-Gibbs thermal equilibrium distribution descends from Newton law: A computational evidence

FULVIO BALDOVIN; LUIS G. MOYANO; CONSTANTINO TSALLIS

Poços de Caldas, Minas Gerais

Encuentro; XXVII Encontro Nacional de Física da Matéria Condensada; 2004

SBF

The problem of the dynamical foundation of Boltzmann-Gibbs (BG) statistical mechanics dates back to the original proposal of this powerful formalism. However, despite many important results, the solution for this fundamental question still presents open basic aspects. Thanks to the current computational capability we can numerically integrate the Hamilton equations of quite large systems and compare the results with the predictions of the BG formalism. In this work we introduce a method which for the first time enables the discussion of the canonical distribution directly in Gibbs \Gamma-space. Using two paradigmatic first-neighbor nonlinear Hamiltonian systems, the one-dimensional inertial XY ferro- magnet and the Fermi-Pasta-Ulam (FPU) \fi-model, we find remarkable agreement that provides an up-to-now never exhibited dynamical foundation of the BG canonical formalism. Within the present approach both time and ensemble averages are performed dynamically, so that we are able to discuss ergodicity. For the XY -model we observe that the Lyapunov coefficient does not represent a consistent measure of the relaxation involved with ergodicity. Our numerical calculation can be implemented for those Hamiltonian models that allow for the textbook def- inition of canonical ensemble (part of a large isolated system). It would also be interesting to check the same procedure in situations where, due for example to the presence of long-range terms, important deviations from the BG predictions have been found.