Geometrical aspects in Equilibrium Thermodynamics

We discuss different aspects of the present status of the Statistical Physics focusing the attention on the non-extensive systems, and in particular, on the so called small systems. Multimicrocanonical Distribution and some of its geometric aspects are presented. The same could be a very atractive way to generalize the Thermodynamics. It is suggested that if the Multimicrocanonical Distribution could be equivalent in the Thermodynamic Limit with some generalized Canonical Distribution, then it is possible to estimate the entropic index of the non-extensive thermodynamics of Tsallis without any additional postulates.Comment: 6 pages, RevTeX. Revised Versio

On the dynamical anomalies in numerical simulations of selfgravitating systems

According to self-similarity hypothesis, the thermodynamic limit could be defined from the scaling laws for the system self-similarity by using the microcanonical ensemble. This analysis for selfgravitating systems yields the following thermodynamic limit: send N to infinity, keeping constant E/N^{(7/3)} and LN^{(1/3)}, in which is ensured the extensivity of the Boltzmann entropy S_{B}=lnW(E,N). It is shown how the consideration of this thermodynamic limit allows us to explain the origin of dynamical anomalies in numerical simulations of selfgravitating systems.Comment: RevTex4, 4 pages, no figure

Generalizing the Extensive Postulates

We addressed the problem of generalizing the extensive postulates of the standard thermodynamics in order to extend it to the study of nonextensive systems. We did it in analogy with the traditional analysis, starting from the microcanonical ensemble, but this time, considering its equivalence with some generalized canonical ensemble in the thermodynamic limit by means of the scaling properties of the fundamental physical observables.Comment: 5 pages, RevTeX, no figures, Revised Versio

Some geometrical aspects of the Microcanonical Distribution

In the present work is presented some considerations for a possible generalization of the Microcanonical thermoestatics (M.Th) of D.H.E. Gross.The same reveals a geometric aspect that commonly it has been disregarded so far: the local reparametrization invariance . This new characteristic leads to the needing of generalizing the methods of M.Th to be consequent with this property.Comment: 4pages, RevTeX. Revised versio

Thermo-Statistical description of the Hamiltonian non extensive systems: The reparametrization invariance

In the present paper we continue our reconsideration about the foundations for a thermostatistical description of the called Hamiltonian nonextensive systems (see in cond-mat/0604290). After reviewing the selfsimilarity concept and the necessary conditions for the ensemble equivalence, we introduce the reparametrization invariance of the microcanonical description as an internal symmetry associated with the dynamical origin of this ensemble. Possibility of developing a geometrical formulation of the thermodynamic formalism based on this symmetry is discussed, with a consequent revision about the classification of phase-transitions based on the concavity of the Boltzmann entropy. The relevance of such conceptions are analyzed by considering the called Antonov isothermal model.Comment: RevTex with 10 pages and 2 eps figure

Remarks about the Phase Transitions within the Microcanonical description

According to the reparametrization invariance of the microcanonical ensemble, the only microcanonically relevant phase transitions are those involving an ergodicity breaking in the thermodynamic limit: the zero-order phase transitions and the continuous phase transitions. We suggest that the microcanonically relevant phase transitions are not associated directly with topological changes in the configurational space as the Topological Hypothesis claims, instead, they could be related with topological changes of certain subset A of the configurational space in which the system dynamics is effectively trapped in the thermodynamic limit N→∞.Comment: RevTeX, 4 pages, no figure. Revised versio

Where the Tsallis Statistic is valid?

In the present paper are analysed the conditions for the validity of the Tsallis Statistics. The same have been done following the analogy with the traditional case: starting from the microcanonical description of the systems and analysing the scaling properties of the fundamental macroscopic observables in the Thermodynamic Limit. It is shown that the Generalized Legendre Formalism in the Tsallis Statistic only could be applied for one special class of the bordering systems, those with non exponential growth of the accessible states density in the thermodynamic limit and zero-order divergency behavior for the fundamental macroscopic observables, systems located in the chaos threshold.Comment: 9 pages, RevTe

Remarks about the thermodynamic limit in selfgravitating systems

The present effort addresses the question about the existence of a well-defined thermodynamic limit for the astrophysical systems with the following power law form: to tend the number of particles, N, the total energy, E, and the characteristic linear dimension of the system, L, to infinity, keeping constant E/N^{\Lambda_{E}} and L/N^{\Lambda_{L}}, being \Lambda_{E} and \Lambda_{L} certain scaling exponent constant. This study is carried out for a system constituted by a non-rotating fluid under the influence of its own Newtonian gravitational interaction. The analysis yields that a thermodynamic limit of the above form will only appear when the local pressure depends on the energy density of fluid as $p=\gamma\epsilon$, being $\gamma$ certain constant. Therefore, a thermodynamic limit with a power law form can be only satisfied by a reduced set of models, such as the selfgravitating gas of fermions and the Antonov isothermal model.Comment: RevTex 4, 4 pages. Comments about the de Vega and Sanchez discussion with V. Laliena about the applicability of the diluted limit in selfgravitating system

Remarks about the microcanonical description of astrophysical systems

We reconsider some general aspects about the mean field thermodynamical description of the astrophysical systems based on the microcanonical ensemble. Starting from these basis, we devote a special attention to the analysis of the scaling laws of the thermodynamical variables and potentials in the thermodynamic limit. Geometrical considerations motivate a way by means of which could be carried out a well-defined generalized canonical-like description for this kind of systems, even being nonextensive. This interesting possibility allows us to extend the applicability of the Standard Thermodynamic methods, even in the cases in which the system exhibits a negative specific heat. As example of application, we reconsider the classical Antonov problem of the isothermal spheres.Comment: RevTex4, 10 pages, 6 eps figures. Submitted Version to PR

Basis of a non Riemannian Geometry within the Equilibrium Thermodynamics

Microcanonical description is characterized by the presence of an internal symmetry closely related with the dynamical origin of this ensemble: the reparametrization invariance. Such symmetry possibilities the development of a non Riemannian geometric formulation within the microcanonical description of an isolated system, which leads to an unexpected generalization of the Gibbs canonical ensemble and the classical fluctuation theory for the open systems (where the inverse temperature and the total energy E behave as complementary thermodynamical quantities), the improvement of Monte Carlo simulations based on the canonical ensemble, as well as a reconsideration of any classification scheme of the phase transitions based on the concavity of the microcanonical entropy.Comment: Revtex style. 21 pages and 10 eps figure