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We consider a one-dimensional lattice system of unbounded, real-valued spins with arbitrary strong, quadratic, finite-range interaction. We show the equivalence of correlations of the grand canonical (gce) and the canonical ensemble (ce). As a corollary we obtain that the correlations of the ce decay exponentially plus a volume correction term. Then, we use the decay of correlation to verify a conjecture that the infinite-volume Gibbs measure of the ce is unique on a one-dimensional lattice. For the equivalence of correlations, we modify a method that was recently used by the authors to show the equivalence of the ce and the gce on the level of thermodynamic functions. In this article we also show that the equivalence of the ce and the gce holds on the level of observables. One should be able to extend the methods and results to graphs with bounded degree as long as the gce has a sufficient strong decay of correlations., Comment: 28 pages

We find a self-similar analytical solution for the grand-canonical-ensemble (GCE) statistics and thermodynamics in the critical region of Bose–Einstein condensation. It is valid for an arbitrary trap, loaded with an ideal gas, in the thermodynamic limit. We show that for the quantities, changing by a finite amount across the critical region, the exact GCE result differs from the corresponding canonical-ensemble result by a factor on the order of unity even in the thermodynamic limit. Thus, a widely used GCE approach does not describe correctly the critical phenomena at the phase transition for the actual systems with a fixed number of particles and yields only an asymptotics far outside the critical region.

We consider a classical system of $N$ particles confined in a box $\Lambda\subset\mathbb{R}^d$ interacting via a finite range pair potential. Given the validity of the cluster expansion in the canonical ensemble we compute the error between the finite and the infinite volume free energy and estimate it to be bounded by the area of the surface of the box's boundary over its volume. We also compute the truncated two-point correlation function and find that the contribution from the ideal gas case is of the order $1/N$ while the contribution of the interactions is of the order $1/|\Lambda|$ plus an exponentially small error with the distance.

We revisit the expansion recently proposed by Pulvirenti and Tsagkarogiannis for a system of $N$ continuous particles in the canonical ensemble. Under the sole assumption that the particles interact via a tempered and stable pair potential and are subjected to the usual free boundary conditions, we show the analyticity of the Helmholtz free energy at low densities and, using the Penrose tree graph identity, we establish a lower bound for the convergence radius which happens to be identical to the lower bound of the convergence radius of the virial series in the grand canonical ensemble established by Lebowitz and Penrose in 1964. We also show that the (Helmholtz) free energy can be written as a series in power of the density whose $k$ order coefficient coincides, modulo terms $o(N)/N$, with the $k$-order virial coefficient divided by $k+1$, according to its expression in terms of the $m$-order (with $m\le k+1$) simply connected cluster integrals first given by Mayer in 1942. We finally give an upper bound for the $k$-order virial coefficient which slightly improves, at high temperatures, the bound obtained by Lebowitz and Penrose.

The distinct deviations from the Fermi-Dirac statistics ascertained recently at low temperatures for a one-dimensional, spinless fermionic discrete lattice gas with conserved number of noninteracting particles hopping on the non-degenerated, well-separated single-particle energy levels are studied in numerical and theoretical terms. The generalized distribution is derived in the formn(h)={Yhexp [(ɛh−μ)β]+1{su−1 valid even in the thermodynamic limit, when the discreteness of the energy levels is kept. This distribution demonstrates good agreement with the data obtained numerically both by the canonical partition function technique and by Monte Carlo simulation.

We present a new technique to simulate the contact of a molecular dynamics system with a thermal wall. A canonical ensemble is obtained, and its statistical and thermodynamic fluctuations are studied. The values of the specific heat found by simulation agree with the experimental data. By means of thermal walls at different temperatures, thermal gradients are obtained. The values of the thermal conductivity are consistent with the experimental data.

We apply a technique to simulate the canonical ensemble, mixing molecular dynamics and Monte Carlo techniques, in which particles suffer virtual hard shocks. In the limit of infinite time the system approaches a Boltzmann distribution. A good approximation to the Boltzmann distribution is achieved in computationally accessible time for some model systems including the one-dimensional jellium.

For a random graph subject to a topological constraint, the microcanonical ensemble requires the constraint to be met by every realisation of the graph (`hard constraint'), while the canonical ensemble requires the constraint to be met only on average (`soft constraint'). It is known that breaking of ensemble equivalence may occur when the size of the graph tends to infinity, signalled by a non-zero specific relative entropy of the two ensembles. In this paper we analyse a formula for the relative entropy of generic discrete random structures recently conjectured by Squartini and Garlaschelli. We consider the case of a random graph with a given degree sequence (configuration model), and show that in the dense regime this formula correctly predicts that the specific relative entropy is determined by the scaling of the determinant of the matrix of canonical covariances of the constraints. The formula also correctly predicts that an extra correction term is required in the sparse regime and in the ultra-dense regime. We further show that the different expressions correspond to the degrees in the canonical ensemble being asymptotically Gaussian in the dense regime and asymptotically Poisson in the sparse regime (the latter confirms what we found in earlier work), and the dual degrees in the canonical ensemble being asymptotically Poisson in the ultra-dense regime. In general, we show that the degrees follow a multivariate version of the Poisson-Binomial distribution in the canonical ensemble., Journal of Statistical Physics, (), 1-19

We examine the minimization of information entropy for measures on the phase space of bounded domains, subject to constraints that are averages of grand canonical distributions. We describe the set of all such constraints and show that it equals the set of averages of all probability measures absolutely continuous with respect to the standard measure on the phase space (with the exception of the measure concentrated on the empty configuration). We also investigate how the set of constrains relates to the domain of the microcanonical thermodynamic limit entropy. We then show that, for fixed constraints, the parameters of the corresponding grand canonical distribution converge, as volume increases, to the corresponding parameters (derivatives, when they exist) of the thermodynamic limit entropy. The results hold when the energy is the sum of any stable, tempered interaction potential that satisfies the Gibbs variational principle (e.g.~Lennard-Jones) and the kinetic energy. The same tools and the strict convexity of the thermodynamic limit pressure for continuous systems (valid whenever the Gibbs variational principle holds) give solid foundation to the folklore local homeomorphism between thermodynamic and macroscopic quantities., Comment: Section 2 revised and improved. The rest of the article has been rewritten accordingly. 22 pages, 2 figures

International audience; We study the equivalence of microcanonical and canonical ensembles in continuous systems, in the sense of the convergence of the corresponding Gibbs measures and the first order corrections. We are particularly interested in extensive observables, like the total kinetic energy. This result is obtained by proving an Edgeworth expansion for the local central limit theorem for the energy in the canonical measure, and a corresponding local large deviations expansion. As an application we prove a formula due to Lebowitz-Percus-Verlet that express the asymptotic microcanonical variance of the kinetic energy in terms of the heat capacity.

Motivated by recent experiments with two-component Bose-Einstein condensates, we study fully-connected spin models subject to an additional constraint. The constraint is responsible for the Hilbert space dimension to scale only linearly with the system size. We discuss the unconventional statistical physical and thermodynamic properties of such a system, in particular the absence of concentration of the underlying probability distributions. As a consequence, expectation values are less suitable to characterize such systems, and full distribution functions are required instead. Sharp signatures of phase transitions do not occur in such a setting, but transitions from singly peaked to doubly peaked distribution functions of an "order parameter" may be present., 15 pages, 6 figures

In general nonequilibrium steady states, directly replacing the canonical ensemble by the nonequilibrium invariant distribution yields a free energy function that is insufficient in characterizing the dynamical landscape. We address the problem by defining the free action, which is like a free energy on path space. Through a representative example, we demonstrate the conceptual and practical usefulness of the free action for quantifying the dynamics of nonequilibrium steady states, including those exhibiting phase transitions.

We propose a general formalism to study the static properties of a system composed of particles with nearest neighbor interactions that are located on the sites of a one-dimensional lattice confined by walls (“confined Takahashi lattice gas”). Linear recursion relations for generalized partition functions are derived, from which thermodynamic quantities, as well as density distributions and correlation functions of arbitrary order can be determined in the presence of an external potential. Explicit results for density profiles and pair correlations near a wall are presented for various situations. As a special case of the Takahashi model we consider in particular the hard rod lattice gas, for which a system of nonlinear coupled difference equations for the occupation probabilities has been presented by Robledo and Varea. A solution of these equations is given in terms of the solution of a system of independent linear equations. Moreover, for zero external potential in the hard-rod system we specify various central regions between the confining walls, where the occupation probabilities are constant and the correlation functions are translationally invariant in the canonical ensemble. In the grand canonical ensemble such regions do not exist.

The exponential family of random graphs is one of the most promising class of network models. Dependence between the random edges is defined through certain finite subgraphs, analogous to the use of potential energy to provide dependence between particle states in a grand canonical ensemble of statistical physics. By adjusting the specific values of these subgraph densities, one can analyze the influence of various local features on the global structure of the network. Loosely put, a phase transition occurs when a singularity arises in the limiting free energy density, as it is the generating function for the limiting expectations of all thermodynamic observables. We derive the full phase diagram for a large family of 3-parameter exponential random graph models with attraction and show that they all consist of a first order surface phase transition bordered by a second order critical curve., 14 pages, 8 figures

We explore differences between canonical and grand canonical ensembles of 500–2000 free electrons confined in a spherical well with a radius from 1.2 to 2 nm. The averaged occupation numbers of the electronic energy levels and their variances are calculated. For isolated Ag and Au particles, the sum of the variances of all occupation numbers differs from the corresponding bulk-metal value by a factor of 0.005 to 2.

When described in a grand canonical ensemble, a finite Coulomb system exhibits charge fluctuations. These fluctuations are studied in the case of a classical (i.e., non-quantum) system with no macroscopic average charge. Assuming the validity of macroscopic electrostatics gives, on a three-dimensional finite large conductor of volume V, a mean square charge 〈Q 2〉 which goes as V 1/3. More generally, in a short-circuited capacitor of capacitance C, made of two conductors, the mean square charge on one conductor is 〈Q 2〉=TC, where T is the temperature and C the capacitance of the capacitor. The case of only one conductor in a grand canonical ensemble is obtained by removing the other conductor to infinity. The general formula is checked in the weak-coupling (Debye–Huckel) limit for a spherical capacitor. For two-dimensional Coulomb systems (with logarithmic interactions), there are exactly solvable models which reveal that, in some cases, macroscopic electrostatics is not applicable even for large conductors. This is when the charge fluctuations involve only a small number of particles. The mean square charge on one two-dimensional system alone, in the grand canonical ensemble, is expected to be, at most, one squared elementary charge.

Classical spin systems with nonadditive long-range interactions are studied in the microcanonical ensemble. It is expected that the entropy of such a system is identical to that of the corresponding mean-field model, which is called "exactness of the mean-field theory". It is found out that this expectation is not necessarily true if the microcanonical ensemble is not equivalent to the canonical ensemble in the mean-field model. Moreover, necessary and sufficient conditions for exactness of the mean-field theory are obtained. These conditions are investigated for two concrete models, the \alpha-Potts model with annealed vacancies and the \alpha-Potts model with invisible states., Comment: 23 pages, to appear in J. Stat. Phys

The Arak process is a solvable stochastic process which generates coloured patterns in the plane. Patterns are made up of a variable number of random non-intersecting polygons. We show that the distribution of Arak process states is the Gibbs distribution of its states in thermodynamic equilibrium in the grand canonical ensemble. The sequence of Gibbs distributions forms a new model parameterised by temperature. We prove that there is a phase transition in this model, for some non-zero temperature. We illustrate this conclusion with simulation results. We measure the critical exponents of this off-lattice model and find they are consistent with those of the Ising model in two dimensions.