Statistical mechanics of classical systems

An algebra of thermodynamic operators of the fluctuations of physical quantities in classical statistical mechanics is found and its properties studied. A method is proposed for obtaining equations that describe the equilibrium and nonequilibrium statistical ensembles of classical systems.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 6–11, May, 1980.     

Statistical mechanics of rotating systems

A microcanonical distribution function depending on the total energyE and thez-componentM of the total angular momentum of a rotating system is examined. ForM=0 the generalized microcanonical ensemble is found to give the same entropy as the usual microcanonical ensemble. The moment of inertia of a rotating gas is calculated, and the kinetic energy of rotation is given as a power series in the small parameterM ²/2I ₀E_int, whereI ₀ is the moment of inertia of the gas at rest andE _int the internal energy.     

Statistical mechanics of lattice systems

We study the thermodynamic limit for a classical system of particles on a lattice and prove the existence of infinite volume correlation functions for a large set of potentials and temperatures.On partial leave from the University of Aix-Marseille.     

Statistical mechanics of quantum spin systems

The thermodynamic limit of a quantum spin system is considered. It is demonstrated that for a large class of interactions and a wide range of the thermodynamic parameters the equilibrium state of the system is describable by an extremalZ ^v -invariant state (a single phase state) over aC* algebra of local observables. It is further shown that the equilibrium state may be obtained as the solution of a variational problem involving the mean entropy. These results extend results previously obtained for classical spin systems byGallavotti, Miracle-Sole andRuelle.     

Statistical mechanics of systems of unbounded spins

We develop the statistical mechanics of unboundedn-component spin systems on the latticeZ ^v interacting via potentials which are superstable and strongly tempered. We prove the existence and uniqueness of the infinite volume free energy density for a wide class of boundary conditions. The uniqueness of the equilibrium state (whose existence is established in general) is then proven for one component ferromagnetic spins whose free energy is differentiable with respect to the magnetic field.     

Statistical mechanics of confined systems with rotational excitations

New experimental and numerical investigations of confined systems of particles demonstrate the existence of rotational excitations. We develop here a statistical theory of finite systems, including rotational modes, by introducing the angular momentum into the formalism and constructing the relevant distributions. As special applications we study systems driven to a prescribed kinetic energy by negative friction or special isokinetic thermostats. Several distribution functions which are solutions of the Liouville or Fokker–Planck equations are given. The theory is applied to Coulomb clusters confined by parabolic forces.     

Statistical mechanics of quantum spin systems. III

In the algebraic formulation the thermodynamic pressure, or free energy, of a spin system is a convex continuous functionP defined on a Banach space of translationally invariant interactions. We prove that each tangent functional to the graph ofP defines a set of translationally invariant thermodynamic expectation values. More precisely each tangent functional defines a translationally invariant state over a suitably chosen algebra of observables, i. e., an equilibrium state. Properties of the set of equilibrium states are analysed and it is shown that they form a dense set in the set of all invariant states over . With suitable restrictions on the interactions, each equilibrium state is invariant under time-translations and satisfies the Kubo-Martin-Schwinger boundary condition. Finally we demonstrate that the mean entropy is invariant under time-translations.     

Statistical mechanics of quantum spin systems. II

In the first part of this paper we continue the general analysis of quantum spin systems. It is demonstrated, for a large class of interactions, that time-translations form a group of automorphisms of theC*-algebra of quasi-local observables and that the thermodynamic equilibrium states are invariant under this group. Further it is shown that the equilibrium states possess the Kubo-Martin-Schwinger analyticity and boundary condition properties. In the second part of the paper we give a general analysis of states which are invariant under space and time translations and also satisfy the KMS boundary condition. A discussion of these latter conditions and their connection with the decomposition of invariant states into ergodic states is given. Various properties pertinent to this discussion are derived.Supported in part by the Office of Naval Research Contract No. Nonr 1866 (5).     

Statistical mechanics of systems nonlinearly displaced from equilibrium I

A graphical approach is used to extend the response theory expression for nonequilibrium averages to any arbitrry order in deviations from equilibrium. Various ways in which the perturbation series about equilibrium can be resummed into gradient or Chapman-Enskog like expansions are presented. As a matter of illustration, examples from the hydrodynamics of simple fluids and the motion of a Brownian particle are considered. Specifically the normal stresses, shear dependent viscosity and velocity dependent friction constant are examined.     

Statistical mechanics of quantum lattice systems without translation invariance

The well-known results concerning the equilibrium of a translation invariant quantum lattice system — existence of the pressure and of the time automorphisms, variational principle for the pressure — are generalized to a large class of quantum lattice systems with potentials not exhibiting covariance under the group of lattice translations.     

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.     

Statistical mechanics of systems with heterogeneous agents: minority games

We study analytically a simple game theoretical model of heterogeneous interacting agents. We show that the stationary state of the system is described by the ground state of a disordered spin model which is exactly solvable within the simple replica symmetric ansatz. Such a stationary state differs from the Nash equilibrium where each agent maximizes her own utility. The latter turns out to be characterized by a replica symmetry broken structure. Numerical results fully agree with our analytical findings.     

Conductivity of quasi-one-dimensional metal systems

A new technique is constructed for treatment of one-dimensional and quasione-dimensional metal systems with random impurities. The technique is applied for calculation of conductivity in several systems (see contents). The influence of impurity scattering on electron pairing is discussed (§ 3).     

Statistical mechanics of viscoelasticity

A unified treatment of viscoelasticity is developed in the framework of non-equilibrium classical statistical mechanics. An exact correspondence between Mori's continued-fractions formalism and rheological circuits is shown.     

Statistical mechanics of money

In a closed economic system, money is conserved. Thus, by analogy with energy, the equilibrium probability distribution of money must follow the exponential Boltzmann-Gibbs law characterized by an effective temperature equal to the average amount of money per economic agent. We demonstrate how the Boltzmann-Gibbs distribution emerges in computer simulations of economic models. Then we consider a thermal machine, in which the difference of temperatures allows one to extract a monetary profit. We also discuss the role of debt, and models with broken time-reversal symmetry for which the Boltzmann-Gibbs law does not hold. The instantaneous distribution of money among the agents of a system should not be confused with the distribution of wealth. The latter also includes material wealth, which is not conserved, and thus may have a different (e.g. power-law) distribution. Received 22 June 2000     

Statistical mechanics of interfaces

On the basis of van der Waals theory for interfaces we evaluate explicitly the small-wavevector behavior of the pair correlation function along an interface. A correction to the density profile is also found. The results obtained are in full accord with capillary wave theory when wave amplitudes are regarded as small.