Abstract:
Symmetry properties of the microcanonical entropy surface as a function of the energy and the order parameter are deduced from the invariance group of the Hamiltonian of the physical system. The consequences of these symmetries for the microcanonical order parameter in the high energy and in the low energy phases are investigated. In particular the breaking of the symmetry of the microcanonical entropy in the low energy regime is considered. The general statements are corroborated by investigations of various examples of classical spin systems.

Abstract:
Several approximations are made to study the microcanonical formalism that are valid in the thermodynamics limit. Usually it is assumed that: 1)Stirling approximation can be used to evaluate the number of microstates; 2) the surface entropy can be replace by the volumen entropy; and 3)derivatives can be used even if the energy is not a continuous variable. It is also assumed that the results obtained from the microcanonical formalism agree with those from the canonical one. However, it is not clear if these assumptions are right for very small systems (10-100 particles). To answer this questions, two systems with exact solutions (the Einstein model of a solid and the two-level system)have been solve with and without these approximations.

Abstract:
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 singularities in microcanonical quantities which announce a continuous phase transition in the infinite system are characterised by classical critical exponents. Estimates of the non-classical exponents which emerge only in the thermodynamic limit can nevertheless be obtained by analyzing effective exponents or by applying a microcanonical finite-size scaling theory. This is explicitly demonstrated for two- and three-dimensional Ising systems.

Abstract:
A microcanonical finite-size scaling ansatz is discussed. It exploits the existence of a well-defined transition point for systems of finite size in the microcanonical ensemble. The best data collapse obtained for small systems yields values for the critical exponents in good agreement with other approaches. The exact location of the infinite system critical point is not needed when extracting critical exponents from the microcanonical finite-size scaling theory.

Abstract:
We derive the microcanonical ensemble from the Maximum Entropy Principle (MEP) using the phase space volume entropy of P. Hertz. Maximizing this entropy with respect to the probability distribution with the constraints of normalization and average energy, we obtain the condition of constant energy. This approach is complementary to the traditional derivation of the microcanonical ensemble from the MEP using Shannon entropy and assuming a priori that the energy is constant which results in equal probabilities.

Abstract:
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 their extensive (Shannon-Boltzmann-Gibbs) form.

Abstract:
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 concept of negative temperatures, which by the Gibbs definition, cannot exist. In this paper, we call into question the fundamental assumption that the microcanonical ensemble should be used to define the entropy. Our argument is based on a recently proposed canonical definition of the entropy as a function of energy. We investigate the predictions of the Boltzmann, Gibbs, and canonical definitions for a variety of classical and quantum models, including models which exhibit a first-order phase transition. Our results support the validity of the concept of negative temperature, but not for all models with a decreasing density of states. We find that only the canonical entropy consistently predicts the correct thermodynamic properties, while microcanonical definitions of entropy, including those of Boltzmann and Gibbs, are correct only for a limited set of simple models.

Abstract:
The microcanonical entropy $S=\ln[W(E)]$ is the {\em geometrical measure} of the microscopic redundancy or ignorance about the N-body system. {\em Even for astronomical large systems} is the microcanonical entropy everywhere {\em single valued and multiply differentiable}. Also the microcanonical temperature is at all energies single valued and differentiable. It is further shown that the recently introduced singularities of the {\em microcanonical} entropy like "microcanonical phase transitions", and exotic patterns of the {\em microcanonical} caloric curve $T(E)$ like multi-valuednes or the appearance of "dinosaur's necks" are inconsistent with Boltzmann's fundamental definition of entropy.

Abstract:
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: $\delta Q/T$ 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 $\Omega$. The second pillar singles out the logarithmic function among all possible functions. Hence the construction leads uniquely to the expression $S= k_B \ln \Omega$, 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.

Abstract:
It has been suggested recently that the microcanonical entropy of a system may be accurately reproduced by including a logarithmic correction to the canonical entropy. In this paper we test this claim both analytically and numerically by considering three simple thermodynamic models whose energy spectrum may be defined in terms of one quantum number only, as in a non-rotating black hole. The first two pertain to collections of noninteracting bosons, with logarithmic and power-law spectra. The last is an area ensemble for a black hole with equi-spaced area spectrum. In this case, the many-body degeneracy factor can be obtained analytically in a closed form. We also show that in this model, the leading term in the entropy is proportional to the horizon area A, and the next term is ln A with a negative coefficient.