We show that the configurational probability distribution of a classical gas always belongs to the q-exponential family. Hence, the configurational subsystem is non-extensive in the sense of Tsallis. One of the consequences of this observation is that the thermodynamics of the configurational subsystem is uniquely determined up to a scaling function. As an example we consider a system of non-interacting harmonic oscillators. In this example, the scaling function can be determined from the requirement that in the limit of large systems the microcanonical temperature of the configurational subsystem should coincide with that of the canonical ensemble. The result suggests that Renyi's entropy function is the relevant one rather than that of Tsallis.
We introduce a simple model of hard spheres with gravitational interactions, for which we study a suitable scaling limit. Usual extensive properties are maintained notwithstanding the long range of gravitational interaction. We show that a local thermalization spontaneously emerges within a microcanonical description of the stationary state. In the considered scaling limit, the resulting density profile can be determined in a hydrostatic approach.
In the present work, we present a detailed discussion of a Riemannian metric structure originally introduced in [Gori et al., \textit{J. Stat. Mech.}, \textbf{9} 093204 (2018)] on the configuration space and on phase space allowing us to interpret the derivatives of the configurational microcanonical entropy and of the canonical entropy in terms of integrals of extrinsic geometrical quantities associated with the equipotential level sets.
In contrast to the canonical case, microcanonical thermodynamic functions can show nonanalyticities also for finite systems. In this paper we contribute to the understanding of these nonanalyticities by working out the relation between nonanalyticities of the microcanonical entropy and its configurational counterpart. If the configurational microcanonical entropy $ω_N^c(v)$ has a nonanalyticity at $v=v_c$, then the microcanonical entropy $ω_N(ε)$ has a nonanalyticity at the same value $ε=v_c$ of its argument for any finite value of the number of degrees of freedom $N$. The presence of the kine…
The phenomenon of partial equivalence of statistical ensembles is illustrated by discussing two examples, the mean-field XY and the mean-field spherical model. The configurational parts of these systems exhibit partial equivalence of the microcanonical and the canonical ensemble. Furthermore, the configurational microcanonical entropy is a smooth function, whereas a nonanalytic point of the configurational free energy indicates the presence of a phase transition in the canonical ensemble. In the presence of a standard kinetic energy contribution, partial equivalence is removed and a nonanalyti…
The entropy definition in the microcanonical ensemble is revisited. We propose a novel definition for the microcanonical entropy that resolve the debate on the correct definition of the microcanonical entropy. In particular we show that this entropy definition fixes the problem inherent the exact extensivity of the caloric equation. Furthermore, this entropy reproduces results which are in agreement with the ones predicted with standard Boltzmann entropy when applied to macroscopic systems. On the contrary, the predictions obtained with the standard Boltzmann entropy and with the entropy we pr…
Two recently proposed expressions for the computation of the entropy in the microcanonical ensemble are compared, and their equivalence is proved. These expressions are valid for a certain class of statistical mechanics systems, that can be called mean-field-like systems. Among these, this work considers only the systems with the most usual hamiltonian structure, given by a kinetic energy term plus interaction terms depending only on the configurational coordinates.
The thermodynamic properties of systems with long-range interactions is still an ongoing challenge, both from the point of view of theory as well as computer simulation. In this work we study a model system, a Coulomb gas confined inside a sphere, by using the Wang-Landau algorithm. We have computed the configurational density of states (CDOS), the thermodynamic entropy and the caloric curve, and compared with microcanonical Metropolis simulations, while showing how concepts such as the configurational inverse temperature can be used to understand some aspects of thermodynamic behavior. A dyna…
Monte Carlo simulations have boosted the numerical study of several different physical systems and in particular, the canonical ensemble has been especially useful because of the existence of easy and efficient relaxation algorithms required to minimize the energy, the relevant extensive thermodynamic variable appearing in the probability distribution, which drives the system after a thermalization process to equilibrium. Nevertheless, the nature does not know about statistical ensembles and therefore it is desirable and a theoretical challenge to show how to perform efficient numerical simula…
We investigate the limiting cases of Tsallis statistics. The viewpoint adopted is not the standard information-theoretic one, where one derives the distribution from a given measure of information. Instead the mechanical approach recently proposed in [M. Campisi, G.B. Bagci, Phys. Lett. A (2006), doi:10.1016/j.physleta.2006.09.081], is adopted, where the distribution is given and one looks for the associated physical entropy. We show that, not only the canonical ensemble is recovered in the limit of $q$ tending to one, as one expects, but also the microcanonical ensemble is recovered in the li…
Symmetry-resolved properties of the trace distance in thermalizing SU(2) systems
National Laboratory of Solid State Microstructures and School of Physics, Nanjing University（南京大学固体微结构物理国家重点实验室和物理学院）；Collaborative Innovation Center of Advanced Microstructures, Nanjing University（南京大学先进微结构协同创新中心）；School of Physics and Astronomy, Shanghai Jiao Tong University（上海交通大学物理与天文学院）；School of Science, Key Laboratory of High Performance Scientific Computation, Xihua University（西华大学理学院高性能科学计算重点实验室）；School of Physics, Zhejiang University（浙江大学物理学院）
The foundations for a thermo-statistical description of the called non extensive Hamiltonian systems are reconsidered. The relevance of the parametric resonance as a fundamental mechanism of the Hamiltonian chaoticity in those systems with bound motions in the configurational space is discussed. The universality of this mechanism suggests us the possibility of performing a thermo-statistical description with microcanonical basis in the context of the long-range interacting Hamiltonian systems. The concept of selfsimilarity is proposed as an appropriate generalization of the well-known extensiv…
We show that within classical statistical mechanics without taking the thermodynamic limit, the most general Boltzmann factor for the canonical ensemble is a q-exponential function. The only assumption here is that microcanonical distributions have to be separable from of the total system energy, which is the prerequisite for any sensible measurement. We derive that all separable distributions are parametrized by a mathematical separation constant Q which can be related to the non-extensivity q-parameter in Tsallis distributions. We further demonstrate that nature fixes the separation constant…
Microcanonical Thermodynamics allows the application of Statistical Mechanics on one hand to closed finite and even small systems and on the other to the largest,self-gravitating ones. However, one has to reconsider the fundamental principles of Statistical Mechanics especially its key quantity, entropy. Whereas in conventional Thermostatistics the homogeneity and extensivity of the system and the concavity of its entropy S(E) are central conditions, these fail for the systems considered here. E.g. at phase separation the entropy S(E) is necessarily convex to make e^{S(E)-E/T} bimodal in E (th…
The statistical mechanics of a cloud of particles interacting via their gravitational potentials is an old problem which encounters some issues when the traditional Boltzmann-Gibbs statistics is applied. In this article, we consider the generalized statistics of Tsallis and analyze the statistical and thermodynamical implications for a self-gravitating gas, obtaining analytical and convergent expressions for the equation of state and specific heat in the canonical as well as microcanonical ensembles. Although our results are comparable in both ensembles, it turns out that only in the canonical…
Ensemble inequivalence has been previously displayed only for long-range interacting systems with non-extensive energy. In order to perform the thermodynamic limit, such systems require an unphysical, so-called, Kac rescaling of the coupling constant. We here study models defined on long-range random networks, which avoid such a rescaling. The proposed models have an extensive energy, which is however non-additive. For such long-range random networks, pairs of sites are coupled with a probability decaying with the distance $r$ as $1/r^δ$. In one dimension and with $0 \leq δ<1$, surface energy …
Conventional thermo-statistics address infinite homogeneous systems within the canonical ensemble. (Only in this case this is equivalent to the fundamental microcanonical ensemble.) However, some 170 years ago the original motivation of thermodynamics was the description of steam engines, i.e. boiling water. Its essential physics is the separation of the gas phase from the liquid. Of course, boiling water is inhomogeneous and as such cannot be treated by conventional thermo-statistics. Then it is not astonishing, that a phase transition of first order is signaled canonically by a Yang-Lee sing…
We describe in detail two numerical simulation methods valid to study systems whose thermostatistics is described by generalized entropies, such as Tsallis. The methods are useful for applications to non-trivial interacting systems with a large number of degrees of freedom, and both short-range and long-range interactions. The first method is quite general and it is based on the numerical evaluation of the density of states with a given energy. The second method is more specific for Tsallis thermostatistics and it is based on a standard Monte Carlo Metropolis algorithm along with a numerical i…
In a first part the scope of classical thermodynamics and statistical mechanics is discussed in the broader context of formal dynamical systems, including computer programmes. In this context classical thermodynamics appears as a particular theory suited to a subset of all dynamical systems. A statistical mechanics similar to the one derived with the microcanonical ensemble emerges from dynamical systems provided it contains, 1) a finite non-integrable part of its phase space which is, 2) ergodic at a satisfactory degree after a finite time. The integrable part of phase space provides the cons…
