Thermo-statistics or Topology of the Microcanonical Entropy Surface

Abstract

Boltzmann's principle S(E, N, V · · · ) = ln W (E, N, V · · · ) allows the interpretation of Statistical Mechanics of a closed system as Pseudo-Riemannian geometry in the space of the conserved parameters E, N, V · · · (the conserved mechanical parameters in the language of Ruppeiner [1]) without invoking the thermodynamic limit. The topology is controlled by the curvature of S(E, N, V · · · ). The most interesting region is the region of (wrong) positive maximum curvature, the region of phase-separation. This is demonstrated among others for the equilibrium of a typical non-extensive system, a self-gravitating and rotating cloud in a spherical container at various energies and angular-momenta. A rich variety of realistic configurations, as single stars, multistar systems, rings and finally gas, are obtained as equilibrium microcanonical phases. The global phase diagram, the topology of the curvature, as function of energy and angular-momentum is presented. No exotic form of thermodynamics like Tsallis [2,3] non-extensive one is necessary. It is further shown that a finite (even mesoscopic) system approaches equilibrium with a change of its entropy ∆S ≥ 0 (Second Law) even when its Poincarré recurrence time is not large.

Statistical mechanics and dynamics of solvable models with long-range interactions

“…Gross has focused his attention on the theoretical treatment of "Small" systems [165,166,167]. His approach privileged the use of the microcanonical ensemble.…”

“…To get a better understanding of the microcanonical phase diagram and also in order to compare our results with those obtained for self-gravitating systems [94,97] and for finite systems [113,114,165,166], we consider the temperature-energy relation T (ε) (also called in the literature "caloric curve"). Also this curve has two branches: a high energy branch (92) corresponding to m = 0, and a low energy branch obtained from (53) their first derivatives at the crossing point can be different, resulting in a jump in the temperature, i.e.…”

Second Law in Classical Non-Extensive Systems

“…They fail to describe the equilibrium of small systems like quantum systems, as well the largest systems possible like self-gravitating ones, or the thermodynamical most important situations like phase-separation with their negative heat-capacity [1].…”

“…self-gravitating systems and small quantum systems) [1,2,3]. A new derivation of the Second Law is presented that respects these fundamental complications.…”

Bimodalities: A survey of experimental data and models

“…Anomaly of thermodynamical potentials A first order phase transition is characterized by an inverted curvature of the relevant thermodynamical potential (entropy, free energy) [7,8]. This feature is also equivalent to a bimodality in the event probability of the given order parameter X as displayed in the left part of fig.…”