On an approach to the description of the local thermodynamic state of solid bodies

We introduce extensive thermodynamic state parameters corresponding to the process of deformation of a solid body. We write the basic thermodynamic relations for deformable systems in terms of these parameters. By passing to the specific extensive quantities and their densities we obtain the corresponding local thermodynamic relations.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 57–62.     

A system of parabolic equations in nonequilibrium thermodynamics including thermal and electrical effects

The time-dependent equations for a charged gas or fluid consisting of several components, exposed to an electric field, are considered. These equations form a system of strongly coupled, quasilinear parabolic equations which in some situations can be derived from the Boltzmann equation. The model uses the duality between the thermodynamic fluxes and the thermodynamic forces. Physically motivated mixed Dirichlet-Neumann boundary conditions and initial conditions are prescribed.The existence of weak solutions is proven. The key of the proof is (i) a transformation of the problem by using the entropic variables, or electro-chemical potentials, which symmetrizes the equations, and (ii) a priori estimates obtained by using the entropy function. Finally, the entropy inequality is employed to show the convergence of the solutions to the thermal equilibrium state as the time tends to infinity.     

Sulla regolarità dei processi termodinamici per alcuni materiali con memoria

Summary We study the thermodynamic restrictions for a class of materials which have memory and emphasize that the different regularity of the temperature gradient brings to a different definition of material state and to different properties for thermodynamic potentials.

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Lavoro eseguito nell'ambito del G.N.F.M. del C.N.R.     

Thermodynamics from the differential geometry standpoint

We study the differential-geometric structure of the space of thermodynamic states in equilibrium thermodynamics. We demonstrate that this space is a foliation of codimension two and find variables in which the foliation fibers are flat. We show that we can introduce a symplectic structure on this space: the external derivative of the 1-form of the heat source, which has the form of the skew-symmetric product dT 蝃 dS in the found variables. The entropy S then plays the role of the Lagrange function (or Hamiltonian) in mechanics, completely determining the thermodynamic system. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 1, pp. 141–148, October, 2008.     

Time-Dependent Correlators of Local Spins of the One-Dimensional XY Heisenberg Chain

Time-dependent temperature correlators of the anisotropic Heisenberg XY chain are calculated by the integration techniques with respect to Grassmann variables. For a chain of length M, the correlators are represented as the determinants of 2M × 2M matrices. In the thermodynamic limit, the correlation functions are expressed in terms of the Fredholm determinants of linear integral operators with matrix kernels. Bibliography: 32 titles.     

On the Quasi-Static Evolution of Nonequilibrium Steady States

The quasi-static evolution of steady states far from equilibrium is investigated from the point of view of quantum statistical mechanics. As a concrete example of a thermodynamic system, a two-level quantum dot coupled to several reservoirs of free fermions at different temperatures is considered. A novel adiabatic theorem for unbounded and nonnormal generators of evolution is proven and applied to study the quasi-static evolution of the nonequilibrium steady state (NESS) of the coupled system. Submitted: April 23, 2006. Accepted: October 4, 2006.     

Phase Transition and Eigenvalue Degeneracy of a One Dimensional Anharmonic Oscillator

For a one dimensional lattice model with long range exponential interaction the problem of deriving the thermodynamic properties of the system reduces to the problem of determining the spectrum of a differential equation for a one dimensional anharmonic oscillator with potential αx² + βγx⁴ where γ is the inverse of the range of the interaction and α is a function of temperature which is positive at high temperatures and negative at low temperatures. It is proved that at low temperatures, α < 0, and in the limit γ → 0, (corresponding to an infinitely long range interaction) the spectrum becomes asymptotically degenerate. The relation of degeneracy to the occurrence of a phase transition is discussed and the generalization to higher dimensions is mentioned briefly.     

On the thermodynamic foundations of the theory of local-gradient mechanicothermodiffusion systems

In the context of the continuous-thermodynamic approach we generalize the Gibbs equation and obtain the initial relations of local-gradient mechanicothermodiffusion. We state the relation between the thermodynamic flows and forces in the form of functionals. We find influence functions that cause expansion of the phase space that determines the thermodynamic potentials by the gradients of the intensive parameters of the equilibrium state of the system. It is shown that such influence functions are connected with the undamped memory of the body of the action at the initial time. Translated fromMatematychni Metody ta Fizyko-Mekhanichni, Polya, Vol. 41, No. 1, 1998, pp. 62–72.     

On the stability of the equilibrium states for Hamiltonian dynamical systems arising in non–equilibrium thermodynamics

In this paper we consider a thermodynamic system with an internal state variable, and study the stability of its equilibrium states by exploiting the reduced entropy inequality. Remarkably, we derive a Hamiltonian dynamical system ruling the evolution of the system in a suitable thermodynamic phase space. The use of the Hamiltonian formalism allows us to prove the equivalence of the asymptotic stability at constant temperature, at constant entropy and at constant energy, thus extending some classical results by Coleman and Gurtin (J. Chem. Phys., 47, 597–613, 1967).     

Thermodynamic Limits for Hamiltonians Defined as Pseudodifferential Operators

Abstract

If P(h) is a h-pseudodifferential operator in R ⁿ associated to an holomorphic semi-bounded symbol in some neighborhood of the real phase space, with bounded derivatives, we describe the symbol of e ^?tP(h), by inequalities where the constants depend on the bounds for the derivatives of the symbol of P(h), but not on the dimension n. Some applications to thermodynamic limits (free energy) are given.     

Thermodynamic foundations of locally gradient generalized thermomechanics

We consider the thermodynamic foundation of locally gradient generalized thermomechanics. In describing the local situation we take account of the gradient of the temperature field and the chemical potential field, as well as the inertial effects of mechanical, thermal, and self-diffusive forms of motion.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 20–24.     

Thermoelastic processes in deformable solid bodies with point defects

We propose a thermodynamic model and obtain a closed system of differential equations describing the thermoelastic processes jointly with the redistribution and recombination (generation) of point defects in deformable solid bodies.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 28, 1988, pp. 18–21.     

Statistical theory of nonlinear hydrodynamic fluctuations in ionic systems

A system of equations for time correlation functions in ionic systems is obtained and solved on the basis of the Fokker-Planck equation for the distribution function of coarse-grained dynamic variables such as mass density, charge density, momentum density, current density, and total energy. Statistical weight, thermodynamic forces and hydrodynamic velocities are calculated both in Gaussian and higher approximations. Expressions for generalized transfer coefficients depending onk andw, with separate fluctuation contributions, are found.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 108, No. 3, pp. 448–464, September, 1996.     

Equilibrium in quantum systems of particles with magnetic interaction. Fermi and bose statistics

Quantum systems of particles interacting via an effective electromagnetic potential with zero electrostatic component are considered (magnetic interaction). It is assumed that the j th component of the effective potential for n particles equals the partial derivative with respect to the coordinate of the jth particle of “magnetic potential energy” of n particles almost everywhere. The reduced density matrices for small values of the activity are computed in the thermodynamic limit for d-dimensional systems with short-range pair magnetic potentials and for one-dimensional systems with long-range pair magnetic interaction, which is an analog of the interaction of three-dimensional Chern-Simons electrodynamics (“magnetic potential energy” coincides with the one-dimensional Coulomb (electrostatic) potential energy). Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Published in Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 5, pp. 691–698, May 1997.     

Droplet Excitations for the Spin-1/2 XXZ Chain with Kink Boundary Conditions

We give a precise definition for excitations consisting of a droplet of size n in the XXZ chain with various choices of boundary conditions, including kink boundary conditions and prove that, for each n, the droplet energies converge to a boundary condition independent value in the thermodynamic limit. We rigorously compute an explicit formula for this limiting value using the Bethe Ansatz. Submitted: January 26, 2006; Accepted: April 10, 2006     

Multiperiodic multifractal martingale measures

A nonnegative 1-periodic multifractal measure on is obtained as infinite random product of harmonics of a 1-periodic function W(t). Such infinite products are statistically self-affine and generalize certain Riesz products with random phases. They are martingale structures, therefore converge. The criterion on W for nondegeneracy is provided. It differs completely from those for other known random measures constructed as martingale limits of multiplicative processes. In particular, it is very sensitive to small changes in W(t). When these infinite products are interpreted in the framework of thermodynamic formalism for random transformations, logW is a potential function when W>0. For regular enough potentials, in case of degeneracy, the natural normalization makes the sequence of measures converge. Moreover, this normalization is neutral for nondegenerate martingales. The multifractal analysis of the limit martingale measure is performed for a class of potential functions having a dense countable set of jump points.     

Vacuum of the electroweak gauge theory in a strong magnetic field in A heat bath

Expressions are obtained for the thermodynamic potential of a gas of electrically charged vector bosons in a magnetic field and the effective potential of the Weinberg-salam theory in a strong magnetic field in a heat bath. In the single-loop approximation, an estimate is obtained for the value of the critical temperatureT _cr corresponding to the phase transition in the Weinberg-Salam theory in a magnetic field. It is shown that it is possible to determine correctly the total probability for production of aW ⁺W^– boson pair from the vacuum by an ultrastrong magnetic field in a heat bath, and an expression is obtained for this probability. It is argued that in the Weinberg-Salam theory the restoration of the spontaneously broken symmetry atT=0,B>B _cr ⁽¹⁾ is simultaneously accompanied by dynamical symmetry breaking of the theory.State University, Moscow. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 102, No. 3, pp. 446–456, March, 1995.     

Quantum Monte Carlo study of magnetic and superconducting properties of graphene

We apply quantum Monte Carlo technique to address the issue of high controllability of ferromagnetism in graphene and the issue of electron correlation driven superconductivity in graphene, by simulating the t − U − V Hubbard model on a honeycomb lattice. In the region of low electron band filling below the Van Hove singularity, the system shows a short‐range ferromagnetic correlation, which is strengthened slightly by the on‐site Coulomb interaction and markedly by the next nearest‐neighbor hoping integral. The strong dependence of ferromagnetism on the electron band filling can be manipulated by applying electric gate voltage. For V=0 and close to the half filling, we find that pairing with d + id symmetry dominates pairing with extended‐s symmetry. However, as the system size increases the long‐range part of the d + id pairing correlation decreases and tends to vanish in the thermodynamic limit. An inclusion of nearest‐neighbor interaction V, either repulsive or attractive, has a small effect on the extended‐s pairing correlation, but strongly suppresses the d + id pairing correlation. Copyright © 2013 John Wiley & Sons, Ltd.     

Thermodynamic Formalism and Singular Invariant Measures for Critical Circle Maps

As is well known, the renormalization group transformation in the space of analytic circle homeomorphisms with one cubic critical point and rotation number equal to the golden section has a single fixed point T ₀. We construct the thermodynamic formalism for the critical map T ₀ and use it to calculate the Hölder indices for the singular invariant measure of T ₀.     

Superconductivity based on a model with three overlapping energy bands and a nonphonon mechanism

A study is made of the thermodynamic properties of a three-band superconductor with an electron (or hole) mechanism of superconductivity. Interband carrier interaction is considered as the main reason for the occurrence of superconducting pairs. Expressions are obtained for the order parameters _n (T) (n=1, 2, 3), the specific heatC _S, and the thermodynamic critical fieldH _c in the region of low temperatures and also the temperatureT _c of the superconducting transition, the jump in the specific heat (C _S–C_N) at the pointT=T _c; and 2 _n (0)/T _c . It is shown that a characteristic feature in the behavior ofT _c, (C _S–C_N), and _n (0) as functions of the carrier concentrationn is the presence of a smeared step. It is also shown that (C _S–C_N)/C _N1.43 and that this ratio depends strongly on the relationship between the densities of the electron states of the considered bands.Institute of Applied Physics, Moldavian Republic Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 3, pp. 483–499, June, 1992.