Mean entropy of states in classical statistical mechanics

The equilibrium states for an infinite system of classical mechanics may be represented by states over AbelianC* algebras. We consider here continuous and lattice systems and define a mean entropy for their states. The properties of this mean entropy are investigated: linearity, upper semi-continuity, integral representations. In the lattice case, it is found that our mean entropy coincides with theKolmogorov-Sinai invariant of ergodic theory.     

Ergodicity Breaking and Localization of the Nicolai Supersymmetric Fermion Lattice Model

We investigate dynamics of a supersymmetric fermion lattice model introduced by Nicolai (J Phys A 9:1497–1505, 1976). We show that the Nicolai model has infinitely many local constants of motion for its Heisenberg time evolution, and therefore ergodicity (with respect to thermal equilibrium states) breaks. It has infinitely many degenerated classical ground states. This phenomena is considered as localization at zero temperature. From a viewpoint of perturbation theory, we explain why delocalization is suppressed at zero temperature despite its disorder-free translation-invariant quantum interaction.     

Spin waves,vortices, and the structure of equilibrium states in the classicalXY model

We prove that, for spin systems with a continuous symmetry group on lattices of arbitrary dimension, the surface tension vanishes at all temperatures. For the classicalXY model in zero magnetic field, this result is shown to imply absence of interfaces in the thermodynamic limit, at arbitrary temperature. We show that, at values of the temperature at which the free energy of that model is continuously differentiable, i.e. at all except possibly countably many temperatures, there iseither aunique translation-invariant equilibrium state, or all such states are labelled by the elements of the symmetry group, SO(2). Moreover, there areno non-translation-invariant, but periodic equilibrium states. We also reconsider the representation of theXY model as a gas of spin waves and vortices and discuss the possibility that, in four or more dimensions, translation invariance may be broken by imposing boundary conditions which force an (open) vortex sheet through the system. Among our main tools are new correlation inequalities.     

Microscopic nuclear collective motion studied by classical equations of motion

The time-dependent variation principle is used to obtain generally non-canonical equations of motion from any class of quantum states which are parameterized by a set of continuous complex quantities. A class of states is presented whose associated classical dynamics is described by the five collective quadrupole degrees of freedom. Information about the classical dynamics of the system can be obtained from the non-canonical equations by finding physically interesting quantities which are coordinate independent and which characterize the low-energy collective motion. Approximate collective hamiltonians, of either a Bohr-Mottelson or an IBM type, can be found by insisting that the interesting physical quantities which describe the low-energy classical behavior of the many-body system are the same as those describing the classical behavior of the system given by the collective hamiltonian. The method is applied to two simple schematic models, one vibrational and one rotational, and IBM hamiltonians are obtained.     

Generic triviality of phase diagrams in spaces of long-range interactions

We show that interactions with multiple translation-invariant equilibrium states form a very thin set in spaces of long-range interactions of classical or quantum lattice systems. For example, generic finite-dimensional subspaces do not intersect this set. This constitutes a severe violation of the Gibbs Phase Rule.Research supported by NSERC grant A-4015     

Time-dependent correlations for a one-component plasma in a uniform magnetic field

We derive various sum rules for the time-displaced structure function of a classical one-component plasma subjected to an external uniform magnetic field. When the plasma has some translational invariance (i.e., homogeneous or translation-invariant along the field), we find that there are long-wavelength oscillations with well-defined frequencies. The results are obtained from linear response and macroscopic electrodynamics, as well as from the microscopic equations of motion (BBGKY hierarchy). In the presence of the magnetic field, the time-displaced structure function has a polynomial decay at large distances, even in the homogeneous case. When the plasma has no translational invariance, examples show a more complicated temporal behaviour in the long-length-scale limit, involving a superposition of oscillations over a continuous range of frequencies.     

Classical Representation of a Quantum System at Equilibrium

A quantum system at equilibrium is represented by a corresponding classical system, chosen to reproduce the thermodynamic and structural properties. The objective is to develop a means for exploiting strong coupling classical methods (e.g., MD, integral equations, DFT) to describe quantum systems. The classical system has an effective temperature, local chemical potential, and pair interaction that are defined by requiring equivalence of the grand potential and its functional derivatives with respect to the external and pair potentials for the classical and quantum systems. Practical inversion of this mapping for the classical properties is effected via the hypernetted chain approximation, leading to representations as functionals of the quantum pair correlation function. As an illustration, the parameters of the classical system are determined approximately such that ideal gas and weak coupling RPA limits are preserved (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Renormalized Kinetic Theory of Classical Fluids in and out of Equilibrium

We present a theory for the construction of renormalized kinetic equations to describe the dynamics of classical systems of particles in or out of equilibrium. A closed, self-consistent set of evolution equations is derived for the single-particle phase-space distribution function f, the correlation function C=〈δfδf〉, the retarded and advanced density response functions χ ^R,A =δf/δφ to an external potential φ, and the associated memory functions Σ ^R,A,C . The basis of the theory is an effective action functional Ω of external potentials φ that contains all information about the dynamical properties of the system. In particular, its functional derivatives generate successively the single-particle phase-space density f and all the correlation and density response functions, which are coupled through an infinite hierarchy of evolution equations. Traditional renormalization techniques (involving Legendre transform and vertex functions) are then used to perform the closure of the hierarchy through memory functions. The latter satisfy functional equations that can be used to devise systematic approximations that automatically imply the conservation laws of mass, momentum and energy. The present formulation can be equally regarded as (i) a generalization to dynamical problems of the density functional theory of fluids in equilibrium and (ii) as the classical mechanical counterpart of the theory of non-equilibrium Green’s functions in quantum field theory. It unifies and encompasses previous results for classical Hamiltonian systems with any initial conditions. For equilibrium states, the theory reduces to the equilibrium memory function approach used in the kinetic theory of fluids in thermal equilibrium. For non-equilibrium fluids, popular closures of the BBGKY hierarchy (e.g. Landau, Boltzmann, Lenard-Balescu-Guernsey) are simply recovered and we discuss the correspondence with the seminal approaches of Martin-Siggia-Rose and of Rose and we discuss the correspondence with the seminal approaches of Martin-Siggia-Rose and of Rose.     

Tiling problems and undecidability in the cluster variation method

In cluster approximations for lattice systems the thermodynamic behavior of the infinite system is inferred from that of a relatively small, finite subsystem (cluster), approximations being made for the influence of the surrounding system. In this context we study, for translation-invariant classical lattice systems, the conditions under which a state for a cluster admits an extension to a global translation-invariant state. This extension problem is related to undecidable tiling problems. The implication is that restrictions of global translation-invariant states cannot be characterized purely locally in general. This means that there is an unavoidable element of uncertainty in the application of a cluster approximation.     

Low-density expanison for unstable interactions and a model of crystallization

For a class of unstable pair interactions in classical continuous systems of identical particles the high-temperature thermodynamic behavior is shown to be normal by extending low-density theorems for the correlation functions. In an example we prove a transition between a translation-invariant phase at high temperatures and low densities and solid with long-range oder at low temperatures. The transition is catastropic in the sense that it is accompanied by the divergence of thermodynamic quantities. We also exhibit counterexamples of unstable interactions in any dimension which do not give rise to a low-temperature catastrophe.     

Invariance identities associated with finite gauge transformations and the uniqueness of the equations of motion of a particle in a classical gauge field

A certain class of geometric objects is considered against the background of a classical gauge field associated with an arbitrary structural Lie group. It is assumed that the components of these objects depend on the gauge potentials and their first derivatives, and also on certain gauge-dependent parameters whose properties are suggested by the interaction of an isotopic spin particle with a classical Yang-Mills field. It is shown that the necessary and sufficient conditions for the invariance of the given objects under a finite gauge transformation are embodied in a set of three relations involving the derivatives of their components. As a special case these so-called invariance identities indicate that there cannot exist a gauge-invariant Lagrangian that depends on the gauge potentials, the interaction parameters, and the4-velocity components of a test particle. However, the requirement that the equations of motion that result from such a Lagrangian be gauge-invariant, uniquely determines the structure of these equations.     

The Canonical Path Integral Quantization of CP1 Model

The Hamilton-Jacobi method of quantizing singular systems is discussed. The equations of motion are obtained as total differential equations in many variables. It is shown that if the system is integrable, then one can obtain the canonical phase space coordinates and the set of the canonical Hamilton-Jacobi partial differential equations without any need to introduce unphysical auxiliary fields. As an example we quantize the CP¹ model using the canonical path integral quantization formalism to obtain the path integral as an integration over the canonical phase-space coordinates.     

Equilibrium states for a classical lattice gas

Various definitions of thermodynamic equilibrium states for a classical lattice gas are given and are proved to be equivalent. In all cases, a set of equations is given, the solutions of which are by definition equilibrium states. Examples are the condition of Lanford and Ruelle, and the KMS boundary condition. In connection with this, it is shown that the time translation for classical interactions exists as an automorphism of the quantum algebra of observables, under conditions which are weaker than those found for quantum interactions.     

Exact variational methods and cluster-variation approximations

It is shown that for classical,d-dimensional lattice models with finite-range interactions the translation-invariant equilibrium states have the property that their mean entropy is completely determined by their restriction to a subset of the lattice that is infinite in (d–1) dimensions and has a width equal to the range of the interaction in the dth dimension. This property is used to show proper convergence toward the exact result for a hierarchy of approximations of the cluster-variation method that uses one-dimensionally increasing basis clusters in a two-dimensional lattice.     

Free energy in a Markovian model of a lattice spin system

A Markov process which may be thought of as a classical lattice spin system is considered. States of the system are probability measures on the configuration space, and we study the evolution of the free energy of these states with time. It is proved that for all initial states the free energy is nonincreasing and that it strictly decreases from any initial state which is shift invariant but not an equilibrium state. Finally we show that the state of the system converges weakly to the set of Gibbsian Distributions for the given interaction, and that all shift invariant equilibrium states are Gibbsian Distributions.This work was done while the author was a postdoctoral fellow in the Adolph C. and Mary Sprague Miller Institute for Basic Research in Science.     

Equilibrium states and ergodic properties of infinite systems

We discuss the ergodic theoretic structure of infinite classical systems and present results on the ergodic properties of some simple model systems, e.g., ideal gas, Lorentz gas, Harmonic crystal. (The ergodic properties of the latter system are shown to be related in a simple way to the spectrum of the force matrix; when the spectrum is absolutely continuous, as in the translation-invariant crystal, the flow is Bernoulli.) We argue that ergodic properties, suitably refined by the inclusion of space translations, and other structure, are important for an understanding of nonequilibrium properties of macroscopic systems [1–5]. Possible additional structures include requirements of stability for the stationary state. We shall present results on the classical analog of the work by Haag, Kastler, and Trych-Pohlmeyer [6], Araki [7], and others [8]. The existence of a time evolution and equilibrium states for various anharmonic crystal systems will also be discussed [9].Supported in part by AFOSR Grant No. 73-2430B.