First order phase transitions in the canonical and the microcanonical ensemble

In the canonical ensemble any singularity of a thermodynamic function at a temperatureT _c is smeared over a temperature range of orderT _T /N. Therefore it is rather difficult to distinguish between a discontinuous and a continuous phase transition on the basis of numerical data obtained for finite systems in the canonical ensemble. It is demonstrated for four model systems that this problem cannot be circumvented by considering higher cumulants of the energy distribution or cumulant ratios. On the other hand, the distinction between first and a second order phase transition is rather direct if based on the microcanonical density of states which is readily obtainable in the dynamical ensemble.     

A new simulation method for the determination of phase equilibria in mixtures in the grand canonical ensemble

A grand canonical Monte Carlo (GCMC) simulation method is presented for the determination of the phase equilibria of mixtures. The coexistence is derived by expanding the pressure into a Taylor series as a function of the temperature and the chemical potentials that are the independent intensive variables of the grand canonical ensemble. The coefficients of the Taylor series can be calculated from ensemble averages and fluctuation formulae that are obtained from GCMC simulations in both phases. The method is able to produce the equilibrium data in a certain domain of the (T, p) plane from two GCMC simulations. The vapour-liquid equilibrium results obtained for a Lennard-Jones mixture agree well with the corresponding Gibbs ensemble Monte Carlo data.     

Electron-hole droplet condensation — A phase transition in an open system

We derive from hydrodynamical equations a generalized Ginzburg-Landau potential for the amplitude of the u stable mode, which describes a first order phase transition in the electron-hole plasma.     

The pressure ensemble as a tool for describing the hadron-quark phase transition

We apply the technique of Gaussian regularization to the grand canonical pressure partition function to show that under physically meaningful boundary conditions (non-existence of external confining vessels, i.e., no fixed volumes) the energy density and similar intensive quantities have, in a statistical bootstrap model of extended hadrons (Van der Waals type volume corrections), indeed the singularity claimed in previous papers. Earlier results obtained with an entirely different technique (which had been criticized) are recovered and shown to be correct. The technique used here is useful in all cases where the volume is not imposed from the outside, but results from the internal dynamics of the system as is generally the case in high energy physics and astrophysics.     

A molecular dynamics method for simulations in the canonical ensemble

A molecular dynamics simulation method which can generate configurations belonging to the canonical (T, V, N) ensemble or the constant temperature constant pressure (T, P, N) ensemble, is proposed. The physical system of interest consists of N particles (f degrees of freedom), to which an external, macroscopic variable and its conjugate momentum are added. This device allows the total energy of the physical system to fluctuate. The equilibrium distribution of the energy coincides with the canonical distribution both in momentum and in coordinate space. The method is tested for an atomic fluid (Ar) and works well.     

On the meaning of the canonical ensemble

Thermal equilibrium between (quantum) systems is taken to mean stability for the combined system. Necessary and sufficient conditions for such stability are found and used to show that any system in equilibrium with suitably complex second system (heat bath) will be characterized by a canonical ensemble. Thus the notion of temperature is derived directly from that of equilibrium, without, for example, recourse to microcanonical ensembles or information theory. Discussed briefly are the generalization of these results to grand canonical ensembles and their application to the equilibrium between a black hole and the surrounding radiation field.     

Condensation of eigen microstate in statistical ensemble and phase transition

In a statistical ensemble with M microstates, we introduce an M × M correlation matrix with correlations among microstates as its elements. Eigen microstates of ensemble can be defined using eigenvectors of the correlation matrix. The eigenvalue normalized by M represents weight factor in the ensemble of the corresponding eigen microstate. In the limit M →∞, weight factors drop to zero in the ensemble without localization of the microstate. The finite limit of the weight factor when M →∞ indicates a condensation of the corresponding eigen microstate. This finding indicates a transition into a new phase characterized by the condensed eigen microstate. We propose a finite-size scaling relation of weight factors near critical point, which can be used to identify the phase transition and its universality class of general complex systems. The condensation of eigen microstate and the finite-size scaling relation of weight factors are confirmed using Monte Carlo data of one-dimensional and two-dimensional Ising models.     

Underlying probability distributions of the canonical ensemble

The canonical distributions are chi-square distributions which are derived from parent distributions for nonconjugate fluctuating thermodynamic variables. The probability distributions are generated by discrete random variables which are the number of degrees of freedom and the number of particles. Randomized sampling of the total number of degrees of freedom and total number of particles gives rise, respectively, to fluctuations in the energy and volume.     

Molecular dynamical simulation of the canonical ensemble

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.     

Superfluidity-insulator phase transition in ensemble of Fermi atoms in optical lattice

An ultracold Fermi gas in an optical lattice with a parabolic potential is modeled by the quantum Monte Carlo method. The gas density profile is calculated in the Hubbard model; it is shown that a domain with a density of one atom per site is formed in the trap that corresponds to the Mott insulator state. The insulator phase is surrounded by a superfluid region occupying the center of the trap, as well as its periphery.     

From phenomenological thermodynamics to the canonical ensemble

Given the generic canonical probability in phase φ=exp[β(Ψ-H)], contact is traditionally made with phenomenological thermodynamics by comparing the identity δ〈φ〉=0 with the relationTδS=δU+δW, δ indicating an arbitrary infinitesimal variation of the thermodynamic coordinates and angular brackets ensemble means. This paper is concerned with the inverse problem of finding both the generic form of the phase functionw such thatS=〈w〉 and the explicit form φ=αexp[(F-H)/kT] of the canonical distribution on the basis of the requirement that the consequences of the phenomenological laws must be safeguarded, i.e., the relations between the quantities which are their concomitants must also be satisfied by their statistical representatives. Given the generic statistical formalism and specifically thatU=〈H〉, δW=?〈δH〉, the problem is soluble, granted the following generic assumption: “the statistical representative of the entropy is the ensemble mean of a function which depends upon the phase through φ alone.”     

Density-functional theory of inhomogeneous fluids in the canonical ensemble

We present a density-functional approach for dealing with inhomogeneous fluids in the canonical ensemble. A general relation is proposed between the free-energy functionals in the canonical and the grand canonical ensembles. The minimization of the canonical-ensemble free-energy functional gives rise to Euler-Lagrange equations which involve averaged Ornstein-Zernike equations of second and third order. The theory is especially appropriate for systems with a small, fixed number of particles. As an example of application we obtain accurate results for the density profile of a hard-sphere fluid in a closed spherical cavity that contains only a few particles.     

刚性多原子分子的正则系综分子动力学算法

通过将Nose所定义的扩展系统哈密顿量推广到刚性多原子分子系统中，严格地推导了刚性多原子分子在正则系综下的运动方程.在此基础上，证明了依靠所给出的运动方程能得到平衡态的正则分布，并给出了该方程所对应的可逆而守恒的积分格式. 关键词： 刚性多原子分子 正则系综 分子动力学算法     

Exact correlation functions of the BCS model in the canonical ensemble

We evaluate correlation functions of the BCS model for a finite number of particles. The integrability of the Hamiltonian relates it with the Gaudin algebra G[sl(2)]. Therefore, a theorem that Sklyanin proved for the Gaudin model, can be applied. Several diagonal and off-diagonal correlators are calculated. The finite-size scaling behavior of the pairing correlation function is studied.     

A kinetic model for droplet growth in the transition regime

A variant of the moment expansion method, used in an earlier paper to describe the flow of a gas toward an absorbing sphere, is applied to a more realistic model of a droplet condensing from a supersaturated vapor. In the simplest version a spherical droplet absorbs all incoming vapor molecules, but spontaneously emits molecules with a Maxwellian distribution at the droplet temperature and with the corresponding saturated vapor density. From a solution of the stationary linearized Boltzmann equation with these boundary conditions we obtain expressions for the heat and mass currents toward the sphere as a function of the supersaturation and the temperature difference between the droplet and the vapor at infinity. For small droplet radii the known free flow limit is obtained in a natural way. From the calculated expressions for the heat and mass current we derive evolution equations for the radius and temperature of the droplet. The temperature evolves more rapidly and can thus be eliminated adiabatically; the resulting growth curve for the radius shows a sharp transition from a kinetically controlled regime for small radii to a regime dominated by heat conduction for large radii. The effect of incomplete absorption at the surface is also studied. The actual calculations are carried out for Maxwell molecules, with parameters corresponding to argon at 0.65T _c and 100% supersaturation.