Nonextensive Statistical Mechanics: Construction and Physical Interpretation

Abstract

Statistical mechanics is clearly mechanics (classical, quantum, special or general relativistic, or any other) plus the theory of probabilities, as is well known. It is our understanding, however, that it is more than that. It is also the adoption of a specific entropic functional, which will, in some sense, adequately shortcut the vast, and for most practical purposes useless, detailed microscopic mechanical information on the system. It is, in particular, through this functional that the connection with thermodynamics and its macroscopic laws will be established. This particular functional is determined by the specific type (or geometry) of occupation of the phase space (or Hilbert space or analogous space). This geometrical structure depends in turn not only on the microscopic dynamics that the system obeys, but also on the initial conditions at which the system is placed at t = 0. In colloquial terms, we could say that the microscopic dynamics determine where the system is allowed to live, whereas the initial conditions determin where it likes to live within the allowed region. This viewpoint is consistent with Einstein's perspective on classical statistical mechanics, and especially with his criticism [82, 92] of the celebrated Boltzmann principle However, the problem is that, up to now, no systematic manner exists for univocally determining the entropic functional to be used, given the dynamics and the initial conditions. The optimization of this entropy under the physically appropriate constraints is expected to provide the correct probability distribution for the microscopic states of the macroscopic stationary state of the system. Boltzmann, then complemented by Gibbs, proposed the celebrated form which is the foundation of standard statistical mechanics.