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.     

Reflexion of a spherical wave by the plane interface between a perfect fluid and a porous medium

By using the Fourier transform of the system of equations and continuity conditions, it is easy to obtain the Fourier transform of the solution. This last function is decomposed into several terms which are identified as Fourier images of known functions. An exact representation of the scattered sound pressure field is obtained as a combination of the radiation of the image source and layer potentials. Approximations are given when the point spherical source is located on the interface.     

Equilibrium states for a one dimensional lattice gas

The thermodynamic equilibrium state can be defined directly for an infinite system via an equilibrium condition or via the variational principle. Both definitions are used to calculate the equilibrium state for a one dimensional lattice gas with finite range interactions.     

Equilibrium states for classical systems

It is shown that the Dobrushin-Lanford-Ruelle equations for the probability measure μ, and the Kirkwood-Salsburg type equations for the lattice or continuum correlation functions ?, and for the spin correlation functions σ, are essentially equivalent for all ?, σ, and μ satisfying certain boundedness conditions. It is also noted that the lattice equations are identical to the equations for the stationary states of a certain Markoff process. This extends previous results of Ruelle, Brascamp and Holley who proved some of these equivalences for states.     

Spinor model of a perfect fluid

Different characteristics of matter influencing the evolution of the uUniverse haves been simulated by means of a nonlinear spinor field. We have considered two cases where the spinor field nonlinearity occurs either as a result of self-action or due to the interaction with a scalar field.     

Instability criteria for steady flows of a perfect fluid

An instability criterion based on the positivity of a Lyapunov-type exponent is used to study the stability of the Euler equations governing the motion of an inviscid incompressible fluid. It is proved that any flow with exponential stretching of the fluid particles is unstable. In the case of an arbitrary axisymmetric steady integrable flow, a sufficient condition for instability is exhibited in terms of the curvature and the geodesic torsion of a stream line and the helicity of the flow.     

Equilibrium distribution function of a relativistic dilute perfect gas

An alternative to the Jüttner distribution has been recently proposed by several authors. The literature on the topic is reviewed critically. It is found that the Jüttner distribution is correct and that the alternative distribution contradicts quantum field theory, statistical physics and continuum mechanics.     

Isometric motions of a perfect charged fluid

Space-times with perfect charged fluids as sources, that admit groups G_r of isometric motions, are investigated. It is assumed that the velocity vector of the fluid is collinear to the timelike Killing vector ⁱ of group G_r. It is shown that the macroscopic motion of a perfect charged fluid can occur only in the direction of such a Killing vector ⁱ that defines an operator in an invariant subgroup or, in particular, an operator of the center of the group. Parametric representations of the generalized equations of state for the pressure p, the energy density of the fluid, , and the electric charge density are established. All these quantities are functions of the norm of the Killing vector ⁱ and the projection of the 4-potential of the electromagnetic field onto this vector ⁱ. In the approximation of the weak field in the coordinate system where ⁱ= ₄ ⁱ , these functional dependences imply that p, , and are functions of Newtonian and electrostatic potentials.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 25–29, October, 1987.     

Darboux transformation of coherent states for a Lobachevski plane

A Darboux transformation of coherent states is considered for a singular oscillator. A coordinate representation and a holomorphic representation are obtained for the Darboux transformation operator and the coherent states. The Hermitian metric and the Kaler potential of the transformed system are calculated. It is established that a Darboux transformation corresponds to curvature of the phase space of the classical system. Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 29–36, September, 1997.     

The Beltrami spectrum for incompressible fluid flows

Recently V. Yakhot, S. Orszag, and their co-workers have suggested that turbulent flows in various regions of space organize into a coherent hierarchy of weakly interacting superimposed approximate Beltrami flows. A mathematical framework is developed here to study organized Beltrami hierarchies in a systematic fashion. This framework is applied to several important classes of examples with universal Beltrami hierarchies. An analysis of the persistence of such Beltrami hierarchies is also presented for general solutions of the Navier-Stokes equations.Research partially supported by grant NSF DMS-860-2031. Sloan research fellowship gratefully acknowledgedThis research was partially supported by grants NSF DMS 86-11110 and DARPA — ONR N00014-86-K-0759     

Devil's staircase of incompressible electron states in a nanotube

It is shown that a periodic potential applied to a nanotube can lock electrons into incompressible states. Depending on whether electrons are weakly or tightly bound to the potential, excitation gaps open up either due to the Bragg diffraction enhanced by the Tomonaga-Luttinger correlations, or via pinning of the Wigner crystal. Incompressible states can be detected in a Thouless pump setup, in which a slowly moving periodic potential induces quantized current, with a possibility to pump on average a fraction of an electron per cycle as a result of interactions.     

Equilibrium states of a semi-quantum lattice system

As an intermediate situation between quantum and classical lattice systems we investigate a semi-quantum lattice system (such as an alloy of several kinds of atoms). The relevant observable algebra is a special type of AF algebras. We study equilibrium states of the system and obtain some equivalent equilibrium conditions.     

Equilibrium configurations for a system of N particles in the plane

We bring together work on central configurations of the classical N-body problem and on asymptotic properties of a system of repelling particles. By introducing a parameter β, we consider configurations that occur in these problems to be special cases of a more general system. We describe an asymmetric configuration obtained in the eight-particle system and investigate a bifurcation occurring in the four-particle system.     

A maximum-entropy principle for two-dimensional perfect fluid dynamics

We use Kullback entropy for Young measures to define statistical equilibrium states for a two-dimensional incompressible flow of a perfect fluid. This approach is justified, as it gives a concentration property about the equilibrium state in the phase space. It might give a statistical understanding of the appearance of coherent structures in two-dimensional turbulence.     

Bianchi type-V perfect fluid models

A study of the nontilted diagonal Bianchi type-V perfect fluid models is undertaken in the nonlocally rotationally symmetric case. It is shown that in such a model, the singularity is necessarily velocity dominated and the upper bound for the ratio of the shear to the expansion is 1/3. A class of such exact models is also presented; the structure of the singularity and the effect of the anisotropy on the scale are discussed. Lastly, an attempt is made to solve exactly the equations of motion for the coordinates of the null geodesics.