A question that is currently highly debated is whether the microcanonical entropy should be expressed as the logarithm of the phase volume (volume entropy, also known as the Gibbs entropy) or as the logarithm of the density of states (surface entropy, also known as the Boltzmann entropy). Rather than postulating them and investigating the consequence of each definition, as is customary, here we adopt a bottom-up approach and construct the entropy expression within the microcanonical formalism upon two fundamental thermodynamic pillars: (i) The second law of thermodynamics as formulated for quasi-static processes: $δQ/T$ is an exact differential, and (ii) the law of ideal gases: $PV=k_B NT$. The first pillar implies that entropy must be some function of the phase volume $Ω$. The second pillar singles out the logarithmic function among all possible functions. Hence the construction leads uniquely to the expression $S= k_B \ln Ω$, that is the volume entropy. As a consequence any entropy expression other than that of Gibbs, e.g., the Boltzmann entropy, can lead to inconsistencies with the two thermodynamic pillars. We illustrate this with the prototypical example of a macroscopic collection of non-interacting spins in a magnetic field, and show that the Boltzmann entropy severely fails to predict the magnetization, even in the thermodynamic limit. The uniqueness of the Gibbs entropy, as well as the demonstrated potential harm of the Boltzmann entropy, provide compelling reasons for discarding the latter at once.
We study the energy and entropies for $N$ independent harmonic oscillators in the microcanonical and the canonical ensembles in the Tsallis classical and the Tsallis quantum statistics of entropic parameter $q$, where $N$ is the number of the oscillators and the value of $q$ is larger than one. The energy and entropies are represented with the physical temperature, and the well-known expressions are obtained for the energy and Rényi entropy. The difference between the microcanonical and the canonical ensembles is the existence of the condition for $N$ and $q$ in the canonical ensemble: $N(q-1)…
We illustrate a novel characterization of nonequivalent statistical mechanical ensembles using the mean-field Blume-Emery-Griffiths (BEG) model as a test model. The novel characterization takes effect at the level of the microcanonical and canonical equilibrium distributions of states. For this reason it may be viewed as a statistical characterization of nonequivalent ensembles which extends and complements the common thermodynamic characterization of nonequivalent ensembles based on nonconcave anomalies of the microcanonical entropy. By computing numerically both the microcanonical and canoni…
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 effect…
In this article, using a known method, a computation is performed of the derivatives of the microcanonical entropy, with respect to the energy up to the 4-th order, using a Laplace transform technique, and adapted it to the case where the total momentum is conserved. The outcome of this computation answers a theoretical question concerning the description of thermodynamics associated with a Hamiltonian flow in presence of an additional conserved quantity besides energy. This is also of practical interest in numerical simulations of the microcanonical thermodynamics associated to classical Hami…
It is generally assumed that the thermodynamic stability of equilibrium state is reflected by the concavity of entropy. We inquire, in the microcanonical picture, on the validity of this statement for systems described by the bi-parametric entropy $S_{_{κ, r}}$ of Sharma-Taneja-Mittal. We analyze the ``composability'' rule for two statistically independent systems, A and B, described by the entropy $S_{_{κ, r}}$ with the same set of the deformed parameters. It is shown that, in spite of the concavity of the entropy, the ``composability'' rule modifies the thermodynamic stability conditions of …
Microcanonical thermodynamics (MCTh) is contrasted to canonical thermodynamics (CTh). At phase transitions of 1.order the two ensembles are NOT equivalent even in the thermodynamic limit . Energy fluctuations do not vanish and phase separations are suppressed in CTh. A proper treatment of fluctuations is neccessary. MCTh allows to address even isolated small systems where phase transitions can be clearly classified into first order and continuous ones. The microcanonical caloric curve T(E) determines the transition temperature, latent heat AND surface entropy/tension. For systems of ca.1000 Na…
We propose a topological approach suitable to establish a connection between thermodynamics and topology in the microcanonical ensemble. Indeed, we report on results that point to the possibility of describing {\it interacting classical spin systems} in the thermodynamic limit, including the occurrence of a phase transition, using topology arguments only. Our approach relies on Morse theory, through the determination of the critical points of the potential energy, which is the proper Morse function. Our main finding is to show that, in the context of the studied classical models, the Euler cha…
Microcanonical equations for several thermodynamic properties of a system, suitable for molecular dynamics simulations, are derived from the nonextensive Tsallis entropy functional. Two possible definitions of temperature, the usual one and a ``physical'' modification which satisfies the zeroth law of thermodynamics, are considered, and the results from both choices are compared. Results for the ideal gas using the first definition of temperature are provided and discussed in relation with the canonical results reported in the literature. The second choice leaves most formulae unchanged from t…
An analytical method to compute thermodynamic properties of a given Hamiltonian system is proposed. This method combines ideas of both dynamical systems and ensemble approaches to thermodynamics, providing de facto a possible alternative to traditional Ensemble methods. Thermodynamic properties are extracted from effective motion equations. These equations are obtained by introducing a general variational principle applied to an action averaged over a statistical ensemble of paths defined on the constant energy surface. The method is applied first to the one dimensional (β)-FPU chain and to th…
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 sim…
The development of reliable methods for estimating microcanonical averages constitutes an important issue in statistical mechanics. One possibility consists of calculating a given microcanonical quantity by means of typical relations in the grand-canonical ensemble. But given that distinct ensembles are equivalent only at the thermodynamic limit, a natural question is if finite size effects would prevent such procedure. In this work we investigate thoroughly this query in different systems yielding first and second order phase transitions. Our study is carried out from the direct comparison wi…
In this series of papers we shall carry out a reconsideration of the thermodynamical behavior of the called HMF model, a paradigmatic ferromagnetic toy model exhibiting many features of the real long-range interacting systems. This first work is devoted to perform the microcanonical description of this model system: the calculation of microcanonical entropy and some fundamental thermodynamic observables, the distribution and correlation functions, as well as the analysis of the thermodynamical stability and the relevant thermodynamic limit.
European Molecular Biology Laboratory（欧洲分子生物学实验室）；Heidelberg University（海德堡大学）；Utrecht University（乌得勒支大学）；Politecnico di Torino（都灵理工大学）；INFN, Sezione di Torino（意大利国家核物理研究所都灵分部）
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…
In the general case of a many-body Hamiltonian system, described by an autonomous Hamiltonian $H$, and with $K\geq 0$ independent conserved quantities, we derive the microcanonical thermodynamics. By a simple approach, based on the differential geometry, we derive the microcanonical entropy and the derivatives of the entropy with respect to the conserved quantities. In such a way, we show that all the thermodynamical quantities, as the temperature, the chemical potential or the specific heat, are measured as a microcanonical average of the appropriate microscopic dynamical functions that we ha…
The microscopic foundation of the generalized equilibrium statistical mechanics based on the Tsallis entropy is given by using the Gibbs idea of statistical ensembles of the classical and quantum mechanics. The equilibrium distribution functions are derived by the thermodynamic method based upon the use of the fundamental equation of thermodynamics and the statistical definition of the functions of the state of the system. It is shown that if the entropic index $ξ=1/(q-1)$ in the microcanonical ensemble is an extensive variable of the state of the system, then in the thermodynamic limit $\tild…
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 Thermodynam…
The positivity of the heat capacity is the hallmark of thermal stability of systems in thermodynamic equilibrium. We show that this property remains valid for systems with negative derivative of energy with respect to temperature, as happens to some system described by the microcanonical ensemble. The demonstration rests on considering a trajectory on the Gibbs equilibrium surface, and its projection on the entropy-energy plane. The Gibbs equilibrium surface has the convexity property, but the projection might lack this property, leading to a negative derivative of energy with respect to tempe…
For the spherical model with nearest-neighbour interactions, the microcanonical entropy s(e,m) is computed analytically in the thermodynamic limit for all accessible values of the energy e and the magnetization m per spin. The entropy function is found to be concave (albeit not strictly concave), implying that the microcanonical and the canonical ensembles are equivalent, despite the long-range nature of the spherical constraint the spins have to obey. Two transition lines are identified in the (e,m)-plane, separating a paramagnetic phase from a ferromagnetic and an antiferromagnetic one. The …
When studying the thermodynamic properties of mesoscopic systems the most appropriate microcanonical entropy is the volume entropy, i.e. the logarithm of the volume of phase space enclosed by the hypersurface of constant energy. For systems with broken ergodicity, the volume entropy has discontinuous jumps at values of energy that correspond to separatrix trajectories. Simultaneously there is a convex intruder in the entropy function and a region of negative specific heat below such critical energies. We illustrate this with a simple model composed of a chain of 3 particles which interact via …
For more than 100 years, one of the central concepts in statistical mechanics has been the microcanonical ensemble, which provides a way of calculating the thermodynamic entropy for a specified energy. A controversy has recently emerged between two distinct definitions of the entropy based on the microcanonical ensemble: (1) The Boltzmann entropy, defined by the density of states at a specified energy, and (2) The Gibbs entropy, defined by the sum or integral of the density of states below a specified energy. A critical difference between the consequences of these definitions pertains to the c…
We derive the global phase diagram of a self-gravitating $N$-body system enclosed in a finite three-dimensional spherical volume $V$ as a function of total energy and angular momentum, employing a microcanonical mean-field approach. At low angular momenta (i.e. for slowly rotating systems) the known collapse from a gas cloud to a single dense cluster is recovered. At high angular momenta, instead, rotational symmetry can be spontaneously broken and rotationally asymmetric structures (double clusters) appear.
Microcanonical thermodynamics studies the operations that can be performed on systems with well-defined energy. So far, this approach has been applied to classical and quantum systems. Here we extend it to arbitrary physical theories, proposing two requirements for the development of a general microcanonical framework. We then formulate three resource theories, corresponding to three different sets of basic operations: i) random reversible operations, resulting from reversible dynamics with fluctuating parameters, ii) noisy operations, generated by the interaction with ancillas in the microcan…
The recent experimental realization of exotic matter states in isolated quantum systems and the ensuing controversy about the existence of negative absolute temperatures demand a careful analysis of the conceptual foundations underlying microcanonical thermostatistics. Here, we provide a detailed comparison of the most commonly considered microcanonical entropy definitions, focussing specifically on whether they satisfy or violate the zeroth, first and second law of thermodynamics. Our analysis shows that, for a broad class of systems that includes all standard classical Hamiltonian systems, o…
The equivalence of thermodynamic results in the canonical and the microcanonical ensembles has been questioned in some calculations for spin models with long-range interactions. We show that these claims of inequivalence are related to an inadequate definition of the independent (density) variables in the microcanonical ensemble. We illustrate this point with the example of a simple spin-1 ideal paramagnet, and then revisit the original calculations of Barré, Mukamel, and Ruffo, for a mean-field spin-1 Blume-Capel model. If the microcanonical ensemble is defined in terms of adequate density va…
We present general and rigorous results showing that the microcanonical and canonical ensembles are equivalent at all three levels of description considered in statistical mechanics - namely, thermodynamics, equilibrium macrostates, and microstate measures - whenever the microcanonical entropy is concave as a function of the energy density in the thermodynamic limit. This is proved for any classical many-particle systems for which thermodynamic functions and equilibrium macrostates exist and are defined via large deviation principles, generalizing many previous results obtained for specific cl…
Thermodynamics allows the application of Statistical Mechanics to finite and even small systems. As surface effects cannot be scaled away, one has to be careful with the standard arguments of splitting a system into two or bringing two systems into thermal contact with energy or particle exchange: Not only the volume part of the entropy must be considered. The addition of any other macroscopic constraint like a dividing surface, or the enforcement of gradients of the energy/particle reduce the entropy. As will be shown here, when removing such constraint in regions of a negative heat capacity,…
The dynamical justifications which lie at the basis of an effective Statistical Mechanics for self gravitating systems are formulated, analyzing some among the well known obstacles thought to prevent a rigorous Statistical treatment. It is shown that N-body gravitational systems satisfy a strong chaos criterion, so supporting the assumption of an increasingly uniform spreading of orbits over the constant energy surface, i.e., the asymptotic evolution towards a microcanonical distribution. We then focus on the necessary conditions for the equivalence of statistical ensembles and remark that thi…
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 subsy…
A microcanonical first order transition, connecting a clustered to a homogeneous phase, is studied from both the thermodynamic and dynamical point of view for a N-body Hamiltonian system with infinite-range couplings. In the microcanonical ensemble specific heat can be negative, but besides that, a microcanonical first order transition displays a temperature discontinuity as the energy is varied continuously (a dual phenomenon to the latent heat in the canonical ensemble). In the transition region, the entropy per particle exhibits, as a function of the order parameter, two relative maxima sep…
We derive a new formulation to calculate the excess chemical potential of a fraction of $N_1$ particles interacting with $N_2$ particles of a different species. The excess chemical potential is calculated numerically from first principles by coupling molecular dynamics and Thomas-Fermi density functional theory to take into account the contribution arising from the quantum electrons on the forces acting on the ions. The choice of this simple functional is motivated by the fact that the present paper is devoted to the derivation and the validation of the method but more complicated functionals …
Depending on the exact experimental conditions, the thermodynamic properties of physical systems can be related to one or more thermostatistical ensembles. Here, we survey the notion of thermodynamic temperature in different statistical ensembles, focusing in particular on subtleties that arise when ensembles become non-equivalent. The 'mother' of all ensembles, the microcanonical ensemble, uses entropy and internal energy (the most fundamental, dynamically conserved quantity) to derive temperature as a secondary thermodynamic variable. Over the past century, some confusion has been caused by …
Attempts to establish microcanonical entropy as an adiabatic invariant date back to works of Gibbs and Hertz. More recently, a consistency relation based on adiabatic invariance has been used to argue for the validity of Gibbs (volume) entropy over Boltzmann (surface) entropy. Such consistency relation equates derivatives of thermodynamic entropy to ensemble average of the corresponding quantity in micro-state space (phase space or Hilbert space). In this work we propose to re-examine such a consistency relation when the number of particles ($N$) is considered as the independent thermodynamic …
The basic quantity for the description of the statistical properties of physical systems is the density of states or equivalently the microcanonical entropy. Macroscopic quantities of a system in equilibrium can be computed directly from the entropy. Response functions such as the susceptibility are for example related to the curvature of the entropy surface. Interestingly, physical quantities in the microcanonical ensemble show characteristic properties of phase transitions already in finite systems. In this paper we investigate these characteristics for finite Ising systems. The singularitie…
According to the recently obtained thermodynamic uncertainty relation, the microcanonical regions with a negative heat capacity can be accessed within a canonical-like description by using a thermostat with a fluctuating inverse temperature. This far-reaching conclusion is used in this Letter for enhancing the potentialities of the well-known Swendsen-Wang cluster algorithm in order to access to the anomalous microcanonical states of the q=10 state Potts model on a square lattice, which exhibits a first-order phase transition in its thermodynamical description.
Conventional thermo-statistics address infinite homogeneous systems within the canonical ensemble. However, some 150 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 canonical thermo-statistics. Then it is not astonishing, that a phase transition of first order is signaled canonically by a Yang-Lee singularity. Thus it is only treated correctly by microcanonical Boltzmann-Planck statisti…
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…
Conventional thermo-statistics address infinite homogeneous systems within the canonical 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 singularity. Thus it is only treated correctly by microcanonical Boltzmann-Planck stati…
Boltzmann's principle is used to select the "most probable" realization (macrostate) of an isolated or closed thermodynamic system, containing a small number of particles ($N \llsp \infty$), for both classical and quantum statistics. The inferred probability distributions provide the means to define intensive variables and construct thermodynamic relationships for small microcanonical systems, which do not satisfy the thermodynamic limit. This is of critical importance to nanoscience and quantum technology.
This short paper presents a nontechnical introduction to the problem of nonequivalent microcanonical and canonical ensembles. Both the thermodynamic and the macrostate levels of definition of nonequivalent ensembles are introduced. The many relationships that exist between these two levels are also explained in simple physical terms.
In the present work, we discuss how the functional form of thermodynamic observables can be deduced from the geometric properties of subsets of phase space. The geometric quantities taken into account are mainly extrinsic curvatures of the energy level sets of the Hamiltonian of a system under investigation. In particular, it turns out that peculiar behaviours of thermodynamic observables at a phase transition point are rooted in more fundamental changes of the geometry of the energy level sets in phase space. More specifically, we discuss how microcanonical and geometrical descriptions of pha…
The notion of negative absolute temperature emerges naturally from Boltzmann's definition of "surface" microcanonical entropy in isolated systems with a bounded energy density. Recently, the well-posedness of such construct has been challenged, on account that only the Gibbs "volume" entropy ---and the strictly positive temperature thereof--- would give rise to a consistent thermodynamics. Here we present analytical and numerical evidence that Boltzmann microcanonical entropy provides a consistent thermometry for both signs of the temperature. In particular, we show that Boltzmann (negative) t…
Boltzmann's microcanonical entropy is the link between statistical physics and thermodynamics, forasmuch as the behavior of any thermodynamic quantity is directly related to the number of microscopic configurations. Accordingly, in this work, we investigate the behavior of the logarithm of the density of states of the three-state Potts model with an external field applied to one of the states using joint entropic simulations based on the Wang-Landau algorithm. Our analysis reveals that the microcanonical entropy curve is detachable, and each resulting path is related to the formation of cluste…
In contrast to the canonical ensemble where thermodynamic functions are smooth for all finite system sizes, the microcanonical entropy can show nonanalytic points also for finite systems, even if the Hamiltonian is smooth. The relation between finite and infinite system nonanalyticities is illustrated by means of a simple classical spin-like model which is exactly solvable for both, finite and infinite system sizes, showing a phase transition in the latter case. The microcanonical entropy is found to have exactly one nonanalytic point in the interior of its domain. For all finite system sizes,…
In a pioneer work, John Nagle has shown that an Ising chain with competing short and long-range interactions displays second and first-order phase transitions separated by a tricritical point. More recently, it has been claimed that Nagle's model provides an example of the inequivalence between canonical and microcanonical calculations. We then revisit Nagle's original solution, as well as the usual formulation of the problem in a canonical ensemble, which lead to the same results. Also, in contrast to recent claims, we show that an alternative formulation in the microcanonical ensemble, with …
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…
We address the thermodynamics (equilibrium density profiles, phase diagram, instability analysis...) and the collapse of a self-gravitating gas of Brownian particles in D dimensions, in both canonical and microcanonical ensembles. In the canonical ensemble, we derive the analytic form of the density scaling profile which decays as f(x)=x^{-α}, with alpha=2. In the microcanonical ensemble, we show that f decays as f(x)=x^{-α_{max}}, where α_{max} is a non-trivial exponent. We derive exact expansions for alpha_{max} and f in the limit of large D. Finally, we solve the problem in D=2, which displ…
The so-called $χ^{2}$-superstatistics of Beck and Cohen (BC) is employed to investigate the infinite-range Blume-Capel model, a well-known representative system displaying inequivalence of canonical and microcanonical phase diagrams. While not being restricted to any of those particular thermodynamic limits, our analytical result can smoothly recover both canonical and microcanonical ensemble solutions as its nonextensive parameter $q$ is properly tuned. Additionally, we compare our findings to ones previously obtained from a generalized canonical framework named Extended Gaussian ensemble (EG…
We calculate the thermodynamic entropy of the mean-field $ϕ^4$ spin model in the microcanonical ensemble as a function of the energy and magnetization of the model. The entropy and its derivative are obtained from the theory of large deviations, as well as from Rugh's microcanonical formalism, which is implemented by computing averages of suitable observables in microcanonical molecular dynamics simulations. Our main finding is that the entropy is a concave function of the energy for all values of the magnetization, but is nonconcave as a function of the magnetization for some values of the en…
In this paper, we consider the volume enclosed by the microcanonical ensemble in phase space as a statistical ensemble. This can be interpreted as an intermediate image between the microcanonical and the canonical pictures. By maintaining the ergodic hypothesis over this ensemble, that is, the equiprobability of all its accessible states, the equivalence of this ensemble in the thermodynamic limit with the microcanonical and the canonical ensembles is suggested by means of geometrical arguments. The Maxwellian and the Boltzmann-Gibbs distributions are obtained from this formalism. In the appen…
The anisotropic quantum Heisenberg model with Curie-Weiss-type interactions is studied analytically in several variants of the microcanonical ensemble. (Non)equivalence of microcanonical and canonical ensembles is investigated by studying the concavity properties of entropies. The microcanonical entropy s(e,m) is obtained as a function of the energy e and the magnetization vector m in the thermodynamic limit. Since, for this model, e is uniquely determined by m, the same information can be encoded either in s(m) or s(e,m1,m2). Although these two entropies correspond to the same physical settin…
We study a quantum spin system on the $d$-dimensional hypercubic lattice $Λ$ with $N=L^d$ sites with periodic boundary conditions. We take an arbitrary translation invariant short-ranged Hamiltonian. For this system, we consider both the canonical ensemble with inverse temperature $β_0$ and the microcanonical ensemble with the corresponding energy $U_N(β_0)$. For an arbitrary self-adjoint operator $\hat{A}$ whose support is contained in a hypercubic block $B$ inside $Λ$, we prove that the expectation values of $\hat{A}$ with respect to these two ensembles are close to each other for large $N$ …
The Renyi statistics in the canonical and microcanonical ensembles is examined in the general case and in particular for the ideal gas. In the microcanonical ensemble the Renyi statistics is equivalent with the Boltzmann-Gibbs statistics. By the exact analytical results for the ideal gas, it is shown that in the canonical ensemble in the thermodynamic limit the Renyi statistics is also equivalent with the Boltzmann-Gibbs statistics. Furthermore it satisfies the requirements of the equilibrium thermodynamics, i.e. the thermodynamical potential of the statistical ensemble is a homogeneous functi…
