Quantum statistical hierarchy equation in nonequilibrium systems

An extension is given for the Fourier expansion method with the contraction technique, which was introduced by Balescu for quantum statistical systems. This is attained by introducing a diagrammatic method with a concept of moving contraction. Then the hierarchy equation for the Contracted Fourier coefficient of the Wigner distribution function is obtained. As an application, a generalized master equation involvingn-body collision effects and quantum statistical effects is also derived.     

Direct numerical simulation of the interaction of isotropic turbulence with a shock wave using shock-fitting

The interaction of three-dimensional isotropic turbulence with a plane shock at Mach numbers of $M=2.0$ and $M=3.0$ is investigated via direct numerical simulation. The numerical scheme is based on a characteristic-type formulation of the Navier–Stokes equations and uses fifth-order upwind schemes in space, a fourth order Runge Kutta scheme in time and a shock-fitting as inlet condition. The isotropic turbulence was generated in a separate computation based on a prescribed energy spectrum. This turbulent flow is considered as frozen, and is convected through the shock with a prescribed average shock speed. An FFT interpolation is used to obtain the upstream values at the instantaneous shock location. Turbulence enhancement is observed, and the evolution of velocity fluctuations as well as turbulence microscales are in good agreement with the behaviour observed using shock-capturing. To cite this article: J. Sesterhenn et al., C. R. Mecanique 333 (2005).     

Informative barycentres in statistics

Barycentres of a discrete probability measure on a dually flat statistical manifold are introduced. They are shown to be unique and to behave as barycentres in Euclidean space. The estimation of these barycentres is studied. Potential applicative usefulness of informative barycentres include the problem of interpolating a statistical manifold valued map and the problem of model merging, which consists in merging several statistical models into a unique one. The results are illustrated on the exponential family, for which a projection theorem is proved.     

Spectral analysis of a nonlinear oscillator driven by random and periodic forces. I. Linearized theory

We have combined the techniques of statistical and harmonic linearization to develop a linearized approximation theory for the calculation of the second-order statistics (i.e., autocorrelation functions and spectral densities) of nonlinear systems driven by both random and periodic forces. For the special case of a Duffing oscillator (a damped anharmonic oscillator with a cubic nonlinearity) driven by Gaussian white noise and by a sinusoidal force, explicit expressions for the renormalized (linearized) frequency, the autocorrelation function, and the spectral density of the oscillator displacement in terms of all the system parameters have been derived. We have determined the region of the parameter space in which the applied periodic force has a significant influence on the second-order statistics of the oscillator.This research was supported by the Office of Naval Research, by the National Science Foundation under grant No. CHE78-21460 and by a grant from Charles and Reneé Taubman.     

The translation-rotation paradox in the nucleation theory

The factor of 10¹⁷ proposed by Lothe and Pound in the theory of nucleation of droplets from the vapor phase is studied using classical statistical mechanics. The controversial factor is derived from partition functions for an isolatedn-mer and for ann-molecular cluster imbedded in the bulk liquid phase. The rotational degrees of freedom have no place in, in agreement with Reiss, Katz, and Cohen's treatment. It is proved that the surface free energy of a cluster is proportional to the surface area. The estimate of what Lothe and Pound called the replacement term is different from those of previous authors. It is concluded that is written as a ratio = _g/_l, where _g and _l , are the volumes per molecule in the gas phase and in the liquid phase, respectively. For water at 300K, is approximately equal to 10₄.     

Dynamic properties of the Monte Carlo method in statistical mechanics

By means of the Monte Carlo sampling technique the equilibrium thermodynamics of fluids and magnets can be calculated numerically. We show that the questions of convergence and accuracy of this method can be understood in terms of the dynamics of the appropriate stochastic model. Also, we discuss to what extent various choices of transition probabilities lead to different dynamic properties of the system. As examples of applications, we consider Ising and Heisenberg spin systems. The numerical results about the dynamic correlation functions are compared to simple approximations taken from the theory of the kinetic Ising model.     

The dirac-schwinger covariance condition in classical field theory

A straightforward derivation of the Dirac-Schwinger covariance condition is given within the framework of classical field theory. The crucial role of the energy continuity equation in the derivation is pointed out. The origin of higher order derivatives of delta function is traced to the presence of higher order derivatives of canonical coordinates and momenta in the energy density functional.     

Statistical thermodynamic formalism in the solution of information theory problems

This is the first of several papers dealing with the application of statistical thermodynamic methodology to the solution of coding and communication theory problems. Emphasis is placed on the various ensemble techniques of statistical mechanics, the words or samples of a message taking the place of molecules in the prototype physical system. Analogs of temperature, internal energy, pressure, chemical potential, volume, entropy, etc., are developed. The isomorphism with thermodynamics is complete and these quantities transform (for example, by partial differentiation) in exactly the same way as the prototype physical quantities. The methods are nicely applicable to coding cases involving sources with memory, in which case, correlation can be discussed in terms of analog coupling energies between signals or words so that the store of many-body-problem techniques can be used. In addition, the manipulative freedom stemming from the possibility of choosing from a multiplicity of ensembles constrained by intensive parameters proves a distinct advantage. A concrete example dealing with the choice of a compact code for a nonextended source with memory is presented. The compact code is derived, and some discussion is given concerning the breadth of its power spectrum. In a following paper, its autocorrelation function within the framework of pulse code modulation is derived and transformed by Wiener theory so that the power spectrum is directly exhibited (along with the spectra for several other cases).Research supported under AFOSR Grant No. 70-1877. The present work is contribution No. 2643 of the Department of Chemistry, University of California-Los Angeles.     

The Boltzmann equation for a polyatomic gas

A formulation of the kinetic theory of dilute, classical polyatomic gases is given which parallels the Waldmann development for structureless molecules. In the first section the Boltzmann equation is written in terms of the specific rates of inelastic collision processes and then the properties of these rates and those of the corresponding collision cross sections are examined. The dependence of the distribution function on the dynamical variables is discussed and the equations of change for the gas are derived. Finally, a study is made of the properties of the linearized Boltzmann collision operation. In the second section the Boltzmann equation is deduced from a rigorous statistical-mechanical point of view and discussed in terms of the basic ideas of Bogoliubov. The computationally important special case of impulsive interactions is then considered.This research was supported in part by a grant from the National Science Foundation and in part by the Ames Laboratory of the U. S. Atomic Energy Commission. Contribution No. 2554.     

Thermodynamical proof of the Gibbs formula for elementary quantum systems

An elementary derivation is given of the formula for the thermal equilibrium states of quantum systems that can be described in finite-dimensional Hilbert spaces. The three assumptions made, Passivity, Structural Stability, and Consistency, have phenomenological interpretations. Except at zero temperature, Structural Stability follows already from Passivity and a weak form of Consistency.