Statistical mechanics of quenched inhomogeneous systems with random disorder

We propose a method to treat quenched disordered systems within the frame of an inhomogeneous ensemble of configurations. To describe a definite configuration, we make use of occupation numbers. By choosing an appropriate representation of these occupation numbers, we are able to deal with an inhomogeneous ensemble by elementary techniques as for instance used for uncorrelated occupation numbers in the frame of a translation invariant ensemble.     

Classification of Phase Transitions and Ensemble Inequivalence, in Systems with Long Range Interactions

Systems with long range interactions in general are not additive, which can lead to an inequivalence of the microcanonical and canonical ensembles. The microcanonical ensemble may show richer behavior than the canonical one, including negative specific heats and other non-common behaviors. We propose a classification of microcanonical phase transitions, of their link to canonical ones, and of the possible situations of ensemble inequivalence. We discuss previously observed phase transitions and inequivalence in self-gravitating, two-dimensional fluid dynamics and non-neutral plasmas. We note a number of generic situations that have not yet been observed in such systems.     

Extended gaussian ensemble solution and tricritical points of a system with long-range interactions

The gaussian ensemble and its extended version theoretically play the important role of interpolating ensembles between the microcanonical and the canonical ensembles. Here, the thermodynamic properties yielded by the extended gaussian ensemble (EGE) for the Blume-Capel (BC) model with infinite-range interactions are analyzed. This model presents different predictions for the first-order phase transition line according to the microcanonical and canonical ensembles. From the EGE approach, we explicitly work out the analytical microcanonical solution. Moreover, the general EGE solution allows one to illustrate in details how the stable microcanonical states are continuously recovered as the gaussian parameter γ is increased. We found out that it is not necessary to take the theoretically expected limit γ → ∞ to recover the microcanonical states in the region between the canonical and microcanonical tricritical points of the phase diagram. By analyzing the entropy as a function of the magnetization we realize the existence of unaccessible magnetic states as the energy is lowered, leading to a breaking of ergodicity.     

The Generalized Canonical Ensemble and Its Universal Equivalence with the Microcanonical Ensemble

This paper shows for a general class of statistical mechanical models that when the microcanonical and canonical ensembles are nonequivalent on a subset of values of the energy, there often exists a generalized canonical ensemble that satisfies a strong form of equivalence with the microcanonical ensemble that we call universal equivalence. The generalized canonical ensemble that we consider is obtained from the standard canonical ensemble by adding an exponential factor involving a continuous function g of the Hamiltonian. For example, if the microcanonical entropy is C², then universal equivalence of ensembles holds with g taken from a class of quadratic functions, giving rise to a generalized canonical ensemble known in the literature as the Gaussian ensemble. This use of functions g to obtain ensemble equivalence is a counterpart to the use of penalty functions and augmented Lagrangians in global optimization. linebreak Generalizing the paper by Ellis et al. [J. Stat. Phys. 101:999–1064 (2000)], we analyze the equivalence of the microcanonical and generalized canonical ensembles both at the level of equilibrium macrostates and at the thermodynamic level. A neat but not quite precise statement of one of our main results is that the microcanonical and generalized canonical ensembles are equivalent at the level of equilibrium macrostates if and only if they are equivalent at the thermodynamic level, which is the case if and only if the generalized microcanonical entropy s–g is concave. This generalizes the work of Ellis et al., who basically proved that the microcanonical and canonical ensembles are equivalent at the level of equilibrium macrostates if and only if they are equivalent at the thermodynamic level, which is the case if and only if the microcanonical entropy s is concave.     

Microcanonical ensemble study of a classical fluid of hard rods under gravity

The study of a system of hard rods in a box of finite length in the presence of a uniform gravitational field is made by means of the microcanonical ensemble. Explicit expressions are derived for the phase volume and the density of states, the primary functions of this ensemble. Related statistical quantities are reported, such as the entropy, the temperature, the heat capacity and the forces exerted on the fluid by the bottom and top walls. The microcanonical number density and higher order molecular distribution functions are also derived. Received: 7 April 1998 / Received in final form: 20 July 1998 / Accepted: 28 July 1998     

Complete Equivalence of the Gibbs Ensembles for One-Dimensional Markov Systems

We show the equivalence of the Gibbs ensembles at the level of measures for one-dimensional Markov-Systems with arbitrary boundary conditions. That is, the limit of the microcanonical Gibbs ensemble is a Gibbs measure with an interaction depending on the microcanonical constraint. In fact the usual microcanonical condition is replaced by the sharper constraint that all type frequencies of neighboring spins (including the boundary spins) are fixed. When conditioning on a set of different frequencies of neighboring spins compatible with physical quantities like energy density we get the usual microcanonical ensemble. We show that the limit is a Gibbs measure for a nearest neighbor potential depending on the pair measure which maximizes the entropy on the given set of pair measures. For this we show the large deviation property of the pair empirical measure for arbitrary boundary conditions. We establish analogous results for finite range potentials.     

Phase transitions in small systems: Microcanonical vs. canonical ensembles

We compare phase transition(-like) phenomena in small model systems for both microcanonical and canonical ensembles. The model systems correspond to a few classical (non-quantum) point particles confined in a one-dimensional box and interacting via Lennard-Jones-type pair potentials. By means of these simple examples it can be shown already that the microcanonical thermodynamic functions of a small system may exhibit rich oscillatory behavior and, in particular, singularities (non-analyticities) separating different microscopic phases. These microscopic phases may be identified as different microphysical dissociation states of the small system. The microscopic oscillations of microcanonical thermodynamic quantities (e.g., temperature, heat capacity, or pressure) should in principle be observable in suitably designed evaporation/dissociation experiments (which must realize the physical preconditions of the microcanonical ensemble). By contrast, singular phase transitions cannot occur, if a small system is embedded into an infinite heat bath (thermostat), corresponding to the canonical ensemble. For the simple model systems under consideration, it is nevertheless possible to identify a smooth canonical phase transition by studying the distribution of complex zeros of the canonical partition function.     

Finite-size scaling in a microcanonical ensemble

The finite-size scaling technique is extended to a microcanonical ensemble. As an application, equilibrium magnetic properties of anL×L square lattice Ising model are computed using the microcanonical ensemble simulation technique of Creutz, and the results are analyzed using the microcanonical ensemble finite-size scaling. The computations were done on the multitransputer system of the Condensed Matter Theory Group at the University of Mainz.     

Microcanonical simulation of Ising systems

Numerical simulations of the microcanonical ensemble for Ising systems are described. We explain how to write very fast algorithms for such simulations, relate correlations measured in the microcanonical ensemble to those in the canonical ensemble and discuss criteria for convergence and ergodicity.     

Large Deviation Principles and Complete Equivalence and Nonequivalence Results for Pure and Mixed Ensembles

We consider a general class of statistical mechanical models of coherent structures in turbulence, which includes models of two-dimensional fluid motion, quasi-geostrophic flows, and dispersive waves. First, large deviation principles are proved for the canonical ensemble and the microcanonical ensemble. For each ensemble the set of equilibrium macrostates is defined as the set on which the corresponding rate function attains its minimum of 0. We then present complete equivalence and nonequivalence results at the level of equilibrium macrostates for the two ensembles. Microcanonical equilibrium macrostates are characterized as the solutions of a certain constrained minimization problem, while canonical equilibrium macrostates are characterized as the solutions of an unconstrained minimization problem in which the constraint in the first problem is replaced by a Lagrange multiplier. The analysis of equivalence and nonequivalence of ensembles reduces to the following question in global optimization. What are the relationships between the set of solutions of the constrained minimization problem that characterizes microcanonical equilibrium macrostates and the set of solutions of the unconstrained minimization problem that characterizes canonical equilibrium macrostates? In general terms, our main result is that a necessary and sufficient condition for equivalence of ensembles to hold at the level of equilibrium macrostates is that it holds at the level of thermodynamic functions, which is the case if and only if the microcanonical entropy is concave. The necessity of this condition is new and has the following striking formulation. If the microcanonical entropy is not concave at some value of its argument, then the ensembles are nonequivalent in the sense that the corresponding set of microcanonical equilibrium macrostates is disjoint from any set of canonical equilibrium macrostates. We point out a number of models of physical interest in which nonconcave microcanonical entropies arise. We also introduce a new class of ensembles called mixed ensembles, obtained by treating a subset of the dynamical invariants canonically and the complementary set microcanonically. Such ensembles arise naturally in applications where there are several independent dynamical invariants, including models of dispersive waves for the nonlinear Schrödinger equation. Complete equivalence and nonequivalence results are presented at the level of equilibrium macrostates for the pure canonical, the pure microcanonical, and the mixed ensembles.     

Remarks on the use of a microcanonical ensemble to study phase transitions in lattice gauge theory

The phase transitions of the Z₂ and U₁ lattice gauge theories are described by sampling a microcanonical ensemble, in which the gauge field is in thermal equilibrium with a system of auxiliary variables called demons. It is shown that the microcanonical ensemble yields a more complete picture of a first-order transition than that obtained by sampling the Boltzmann distribution.     

Direct calculation of fluctuation formulae in the microcanonical ensemble

We present simple derivations of the classical microcanonical ensemble formulae for fluctuations which involve the kinetic energy and microscopic pressure function. The new derivations, which confirm earlier results of Lebowitz et al. [1] and Cheung [2], proceed in a direct way from basic microcanonical ensemble theory without recourse to fluctuation expressions of other ensemble theories. The method developed is applicable to shell ensembles in general.     

Canonical typicality

It is well known that a system weakly coupled to a heat bath is described by the canonical ensemble when the composite S + B is described by the microcanonical ensemble corresponding to a suitable energy shell. This is true for both classical distributions on the phase space and quantum density matrices. Here we show that a much stronger statement holds for quantum systems. Even if the state of the composite corresponds to a single wave function rather than a mixture, the reduced density matrix of the system is canonical, for the overwhelming majority of wave functions in the subspace corresponding to the energy interval encompassed by the microcanonical ensemble. This clarifies, expands, and justifies remarks made by Schr?dinger in 1952.     

Crossover from Quantum to Boltzmann Statistics for Free Particles in a Single Harmonic Trap

With invoking analytical formulae in number theory and numerical calculations, we calculate the number of microstates in microcanonical ensemble for free particles in a single harmonic trap which in whole space defines a thermodynamic system but not a spatially homogeneous one. Once the number of excitation quanta m is larger than the square of the particle number N ² as m⪰O(N ²) when N≫1, the number of microcanonical microstates for an ideal, harmonically trapped Bose or Fermi gas gradually converge to the Boltzmann microcanonical microstates for the classical particles with a proper consideration of the indistinguishability.     

近独立子系系统的统计规律(二)——微正则分布函数

从微正则系统理论出发,导出了任意数目粒子构成的近独立子系系统的分布函数,在粒子数N趋于无限大的极限情况下,得到了麦克斯韦分布律.     

Microcanonical equilibrium properties of chiral liquid crystals—a Monte Carlo study

Chiral liquid crystals have been investigated by means of a multicanonical Monte Carlo approach in order to characterize their phase behaviour by microcanonical equilibrium properties. The liquid crystals were described by three-dimensional lattice systems with intermolecular interactions given by the chiral Lebwohl-Lasher potential. Self-determined boundary conditions have been applied in order to enable the formation of chiral phases with equilibrium pitch. Selected thermodynamic properties, e.g. microcanonical entropy, temperature, heat capacity and a set of order parameters have been determined with dependence on microcanonical total energy. A cholesteric phase with temperature-induced helix inversion could be proven where the helical superstructure of the single component system studied changed its handedness through an infinite-pitch system. The thermodynamical behaviour in the microcanonical ensemble was found to be very similar to the behaviour in the canonical ensemble. The study of microcanonical equilibrium properties by means of multicanonical Monte Carlo simulations was shown to be a powerful tool for the study of the phase behaviour of model liquid crystals.     

Microcanonical ensemble study of a gas column under gravity

The study of a classical ideal gas column of finite height H in a uniform gravitational field g is made by the microcanonical ensemble at energy E. The primary functions of this ensemble, the phase volume and the density of states, are derived. Related statistical quantities, such as the entropy, the temperature and the heat capacity, are also reported. The equivalence in the thermodynamic limit between the calculated microcanonical expressions and those obtained from the canonical ensemble is shown numerically. The expression for the temperature is used to analyze the temperature change when the gas is permitted to expand into an evacuated region increasing the height of the column from H ₁ to H ₂. The microcanonical single-particle momentum and height distributions are also reported.     

Partial equivalence of statistical ensembles and kinetic energy

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 nonanalyticity arises also microcanonically. Hence in contrast to the common belief, kinetic energy, even though a quadratic form in the momenta, has a nontrivial effect on the thermodynamic behaviour. As a by-product we present the microcanonical solution of the mean-field spherical model with kinetic energy for finite and infinite system sizes.     

Microcanonical Finite-Size Scaling

In the microcanonical ensemble, suitably defined observables show nonanalyticities and power-law behavior even for finite systems. For these observables, a microcanonical finite-size scaling theory is established and combined with the experimentally observed power-law behavior. Scaling laws are obtained which relate exponents of the finite system and critical exponents of the infinite system to the system-size dependence of the affiliated microcanonical observables.     

Statistical hadronization and hadronic microcanonical ensemble I

We present a full treatment of the microcanonical ensemble of the ideal hadron-resonance gas starting from a quantum-mechanical formulation which is appropriate for the statistical model of hadronization. By using a suitable transition operator for hadronization we are able to recover the results of the statistical theory, particularly the expressions of the rates of different channels. Explicit formulae are obtained for the phase space volume or density of states of the ideal relativistic gas in quantum statistics as a cluster decomposition, generalizing previous ones in the literature. The problem of the computation of averages in the hadron gas microcanonical ensemble and the comparison with canonical ones will be the main subject of a forthcoming second paper.Received: 8 July 2003, Revised: 17 October 2003, Published online: 5 May 2004