On the equilibrium statistical mechanics of isothermal classical self-gravitating matter

The canonical ensemble is investigated for classical self-gravitating matter in a finite container ^{[d] d} ,d=3 and 2. Starting with modified gravitational interactions (smoothed-out singularity), it is proven by explicit construction that, in thew ^*-topology, the canonical equilibrium measure converges to a superposition of Dirac measures when the limit of exact Newtonian gravitational interactions between classical point particles is taken. The consequences of this result for more realistic classical systems are evaluated, and the existence of a gravitational phase transition is proven. The results are discussed with view toward applications in astrophysics and space science. Some attention is paid also to the problem of founding thermodynamics by means of statistical mechanics.     

Statistical equilibrium states and long-time dynamics for a transport equation

Statistical equilibrium states for a linear transport equation were defined in a previous work. We consider here the two-dimensional case: we show that under some mild assumptions these equilibrium states actually describe the long-time dynamics of the system.     

Statistical mechanics of the nonlinear Schrödinger equation. II. Mean field approximation

We investigate a mean field approximation to the statistical mechanics of complex fields with dynamics governed by the nonlinear Schrödinger equation. Such fields, whose Hamiltonian is unbounded below, may model plasmas, lasers, and other physical systems. Restricting ourselves to one-dimensional systems with periodic boundary conditions, we find in the mean field approximation a phase transition from a uniform regime to a regime in which the system is dominated by solitons. We compute explicitly, as a function of temperature and density (L ² norm), the transition point at which the uniform configuration becomes unstable to local perturbations; static and dynamic mean field approximations yield the same result.     

A complementary thermodynamic limit for classical Coulomb matter

The canonical equilibrium measure of classical two-component Coulomb matter with regularized interactions is analyzed in a finite volume. It is shown that, in the mean-field regime, the one-particle density is inhomogeneous on a new characteristic length scale _inh. For a system ofN positive andN negative particles, _inh and the characteristic length scale of correlations _corr (=Debye screening length) are related via _inh=(2N)^1/2 _corr. The major conceptual conclusion that is drawn from this is that one needs two nontrivial complementary thermodynamic limits to define the equilibrium thermodynamics of two-component Coulomb systems. One of them is the standard thermodynamic limit (infinite volume), where one takesN, _corr fixed. Its complementary limit is characterized byN, _inh fixed, and is a finite-volume inhomogeneous mean-field limit. The most prominent new feature in the mean-field thermodynamic limit, which is absent in the standard thermodynamic limit, is an anomalous first-order phase transition where the Coulomb system explodes or implodes, respectively. The phase transition is connected with the existence of a metastable plasma phase far below the ionization temperature.     

Statistical mechanics and dynamics of solvable models with long-range interactions

For systems with long-range interactions, the two-body potential decays at large distances as V(r)1/r^α, with α≤d, where d is the space dimension. Examples are: gravitational systems, two-dimensional hydrodynamics, two-dimensional elasticity, charged and dipolar systems. Although such systems can be made extensive, they are intrinsically non additive: the sum of the energies of macroscopic subsystems is not equal to the energy of the whole system. Moreover, the space of accessible macroscopic thermodynamic parameters might be non convex. The violation of these two basic properties of the thermodynamics of short-range systems is at the origin of ensemble inequivalence. In turn, this inequivalence implies that specific heat can be negative in the microcanonical ensemble, and temperature jumps can appear at microcanonical first order phase transitions. The lack of convexity allows us to easily spot regions of parameter space where ergodicity may be broken. Historically, negative specific heat had been found for gravitational systems and was thought to be a specific property of a system for which the existence of standard equilibrium statistical mechanics itself was doubted. Realizing that such properties may be present for a wider class of systems has renewed the interest in long-range interactions. Here, we present a comprehensive review of the recent advances on the statistical mechanics and out-of-equilibrium dynamics of solvable systems with long-range interactions. The core of the review consists in the detailed presentation of the concept of ensemble inequivalence, as exemplified by the exact solution, in the microcanonical and canonical ensembles, of mean-field type models. Remarkably, the entropy of all these models can be obtained using the method of large deviations. Long-range interacting systems display an extremely slow relaxation towards thermodynamic equilibrium and, what is more striking, the convergence towards quasi-stationary states. The understanding of such unusual relaxation process is obtained by the introduction of an appropriate kinetic theory based on the Vlasov equation. A statistical approach, founded on a variational principle introduced by Lynden-Bell, is shown to explain qualitatively and quantitatively some features of quasi-stationary states. Generalizations to models with both short and long-range interactions, and to models with weakly decaying interactions, show the robustness of the effects obtained for mean-field models.     

Determination of the vapor–liquid transition of square-well particles using a novel generalized-canonical-ensemble-based method

The square-well(SW) potential is one of the simplest pair potential models and its phase behavior has been clearly revealed, therefore it has become a benchmark for checking new theories or numerical methods. We introduce the generalized canonical ensemble(GCE) into the isobaric replica exchange Monte Carlo(REMC) algorithm to form a novel isobaric GCE-REMC method, and apply it to the study of vapor–liquid transition of SW particles. It is validated that this method can reproduce the vapor–liquid diagram of SW particles by comparing the estimated vapor–liquid binodals and the critical point with those from the literature. The notable advantage of this method is that the unstable vapor–liquid coexisting states,which cannot be detected using conventional sampling techniques, are accessed with a high sampling efficiency. Besides,the isobaric GCE-REMC method can visit all the possible states, including stable, metastable or unstable states during the phase transition over a wide pressure range, providing an effective pathway to understand complex phase transitions during the nucleation or crystallization process in physical or biological systems.     

Statistical mechanics of a nonlinear stochastic model

A multivariable Fokker-Planck equation (FPE) is used to investigate the equilibrium and dynamical properties of a nonlinear stochastic model. The model displays a phase transition. The equilibrium distributions are found to be non-Gaussian; the deviation from Gaussian is especially significant near the transition point. To study the nonequilibrium behavior of the model, a self-consistent dynamic mean field (SCDMF) theory is derived and used to transform the FPE to a systematic hierarchy of equations for the cumulant moments of the time-dependent distribution function. These equations are numerically solved for a variety of initial conditions. During the time evolution of the system from an initial unstable equilibrium state to the final equilibrium state, three distinct time stages are found.Supported by a grant from the National Research Council of Canada (to RCD) and by the Sherman Fairchild Foundation (to RZ).Also Sherman Fairchild Distinguished Scholar, 1974–75, at the California Institute of Technology, where the early part of this research was done.     

Nonequilibrium statistical mechanics of systems with long-range interactions

Systems with long-range (LR) forces, for which the interaction potential decays with the interparticle distance with an exponent smaller than the dimensionality of the embedding space, remain an outstanding challenge to statistical physics. The internal energy of such systems lacks extensivity and additivity. Although the extensivity can be restored by scaling the interaction potential with the number of particles, the non-additivity still remains. Lack of additivity leads to inequivalence of statistical ensembles. Before relaxing to thermodynamic equilibrium, isolated systems with LR forces become trapped in out-of-equilibrium quasi-stationary states (qSSs), the lifetime of which diverges with the number of particles. Therefore, in the thermodynamic limit LR systems will not relax to equilibrium. The qSSs are attained through the process of collisionless relaxation. Density oscillations lead to particle–wave interactions and excitation of parametric resonances. The resonant particles escape from the main cluster to form a tenuous halo. Simultaneously, this cools down the core of the distribution and dampens out the oscillations. When all the oscillations die out the ergodicity is broken and a qSS is born. In this report, we will review a theory which allows us to quantitatively predict the particle distribution in the qSS. The theory is applied to various LR interacting systems, ranging from plasmas to self-gravitating clusters and kinetic spin models.     

On the equation of state of classical one-component systems with long-range forces

Several definitions of the pressure are introduced for one-component systems and shown to be nonequivalent in the presence of a rigid neutralizing background. Relations between these pressures are derived for finite and infinite systems; these relations depend on the asymptotic behavior of the force at infinity, with the Coulomb force at the borderline between different properties. It is argued that only one of those definitions is physically acceptable and its properties are discussed in relation to the asymptotic behavior of the force. It is seen in particular that a knowledge of the state of the infinite system is not sufficient to determine its thermodynamic properties. The results are illustrated by some typical examples.For example, for two-dimensional systems with three-dimensional Coulomb interaction see refs. 2–4.     

An alternative approach to calculate the density of states in nonextensive statistical mechanics

A relation between the generalized partition function (Tsallis) and density of states is established by using the method of integral transform which enables reducing some integral equations into the algebraic equations. Inverse Mellin transformation of this equation gives the density of states. Similar relation is also hold the for standard partition function (Boltzmann-Gibbs) and the density of states. Using these relations, we recover the density of states for the classical ideal gas within both statistics.     

哈密顿正则方程在生物膜与胶体粒子相互作用研究中的应用

介绍经典分析力学中的哈密顿正则方程在生物膜与胶体粒子相互作用研究中的一个具体应用.由Helfrich理论模型得到体系的哈密顿,用正则方程给出一组常微分方程,并用打靶法对其进行求解得到体系的稳定构型随膜参数变化的规律.     

The stationary state and the heat equation for a variant of Davies' model of heat conduction

We study a variant of Davies' model of heat conduction, consisting of a chain of (classical or quantum) harmonic oscillators, whose ends are coupled to thermal reservoirs at different temperatures, and where neighboring oscillators interact via intermediate reservoirs. In the weak coupling limit, we show that a unique stationary state exists, and that a discretized heat equation holds. We give an explicit expression of the stationary state in the case of two classical oscillators. The heat equation is obtained in the hydrodynamic limit, and it is proved that it completely describes the macroscopic behavior of the model.     

Statistical mechanical theory of the great red spot of jupiter

In a previous study a permanent isolated vortex like the Great Red Spot of Jupiter was obtained as a statistical equilibrium for the classical quasigeostrophic model of atmospheric motion on rapidly rotating planets. We provide here a theoretical basis for this work and relate it to a previous model of the spot (Rossby soliton).     

Statistical mechanics of holonomic systems as a Brownian motion on smooth manifolds

The statistical mechanics of arbitrary holonomic scleronomous systems subjected to arbitrary external forces is described by specializing the Lagrange and Hamilton equations of motion to those of the Brownian motion on a manifold. In this context, the Klein‐Kramers and Smoluchowski equations are derived in covariant form, and it is demonstrated that these equations have equilibrium solutions corresponding to the Gibbs distribution, in agreement with standard thermodynamics. At last, the Langevin dynamics corresponding to the Smoluchowski limit is found to exactly correspond to the Brownian motion on a smooth manifold. These results find significant applications in the study of several statistical properties of constrained molecular assemblies (e.g. polymers) of interest in chemistry, physics and biology.     

On the solutions and the steady states of a master equation

A complete characterization of the time behavior of the means and variance of a stochastic process which is generated by a finite number of independent systems is presented based on the master equation for the conditional probability. It is found that the means and variance relax to a steady state and that the steady state will be independent of the initial state if and only if a matrix related to the transition matrix is nonsingular. Finally, the result that the variance approaches its steady-state form at twice the rate of the means is shown to depend on the nonsingularity of the same matrix.     

Mapping of the position-dependent mass Schroedinger equation under point canonical transformation

In this paper, the three-dimensional radial position-dependent mass Schroedinger equation is exactly solved through mapping this wave equation into the constant mass Schroedinger equation with Coulomb potential by means of point canonical transformation. The wavefunctions here can be given in terms of confluent hypergeometric functions.     

Examples of Bosonic de Finetti States over Finite Dimensional Hilbert Spaces

According to the Quantum de Finetti Theorem, locally normal infinite particle states with Bose–Einstein symmetry can be represented as mixtures of infinite tensor powers of vector states. This note presents examples of infinite-particle states with Bose–Einstein symmetry that arise as limits of Gibbs ensembles on finite dimensional spaces, and displays their de Finetti representations. We consider Gibbs ensembles for systems of bosons in a finite dimensional setting and discover limits as the number of particles tends to infinity, provided the temperature is scaled in proportion to particle number     

Mean field kinetic theory of a classical electron gas in a periodic potential. II. Qualitative analysis of the mean-field solution in one dimension

A qualitative analysis is made of the static and dynamic behavior of a one-dimensional classical electron gas in a periodic potential in the framework of a mean-field kinetic theory. The mean-field equations have been formally solved elsewhere in terms of the trajectories of one electron in the mean-field equilibrium potential, which determines the local electronic density. Taking advantage of the relative simplicity of the mean-field expressions in one dimension, we study the effects of the temperature upon the local electronic density, the static structure factor, and the spectrum of the fluctuations in the long-wavelength limit. At high temperatures, the system tends to behave like a homogeneous electron gas; however, the collective plasmon mode at zero wavenumber is damped and shifted below the plasma frequency. At low temperatures, the system behaves as an ensemble of independent electrons strongly localized in the neighborhood of the fixed ions that create the periodic potential; the plasmon mode then vanishes. We consider the physical relevance of these predictions. They turn out to be quite reasonable, despite the failure of meanfield theory to predict the phase of the model.     

广义经典力学中Hamilton-Tabarrok-Leech正则方程的对称性理论

提出广义Hamilton-Tabarrok-Leech正则方程的对称性理论.列写系统的运动方程.研究系统的Noether对称性、形式不变性和Lie对称性，并求出相应的守恒量.举例说明结果的应用. 关键词： 广义经典力学 H-T-L 正则方程 对称性 守恒量