Rigorous information-theoretic derivation of quantum-statistical thermodynamics. I

In previous publications we have criticized the usual application of information theory to quantal situations and proposed a new version of information-theoretic quantum statistics. This paper is the first in a two-part series in which our new approach is applied to the fundamental problem of thermodynamic equilibrium. Part I deals in particular with informational definitions of equilibrium and the identification of thermodynamic analogs in our modified quantum statistics formalism.Work supported by a grant from Research Corporation to J. L. P.     

Rigorous derivation of the Gross-Pitaevskii equation

The time-dependent Gross-Pitaevskii equation describes the dynamics of initially trapped Bose-Einstein condensates. We present a rigorous proof of this fact starting from a many-body bosonic Schr?dinger equation with a short-scale repulsive interaction in the dilute limit. Our proof shows the persistence of an explicit short-scale correlation structure in the condensate.     

Rigorous derivation of the long-time asymptotics for reversible binding

Using an iterative solution in Laplace-Fourier space, we supply a rigorous mathematical proof for the long-time asymptotics of reversible binding in one dimension. The asymptotic power law and its concentration dependent prefactor result from diffusional and many-body effects which, unlike for the corresponding irreversible reaction and in classical chemical kinetics, play a dominant role in shaping the approach to equilibrium.     

Rigorous derivation of domain growth kinetics without conservation laws

The time evolution of the Ising model that describes shrinking domains is studied. A singly connected domain of Ising spins, embedded in a sea of the opposite phase, develops atT=0 according to a dynamic rule that does not allow its perimeter to increase. At long enough times the domain disappears; we show that the average lifetime of such a domain is proportional to its area. We also consider theT=0 dynamics of a single infinite quadrant. The area of the quadrant decreases during the time evolution, and we show that the area lost grows linearly with time. We solve a first passage time problem as well. That is, we calculate the average time it takes for the area lost to reach a given value for the first time. Lastly, we map the infinite quadrant model onto a diffusion problem with exclusion in one dimension. This latter problem is mapped onto a critical six-vertex model.     

A kinetic derivation of extended irreversible thermodynamics

The basic postulates of the extended irreversible thermodynamics are derived from the kinetic model for a dilute monoatomic gas. Using the Grad 13-moment method to solve the full nonlinear Boltzmann equation for molecules conceived as soft spheres we obtain the microscopic expressions for the entropy flux, the entropy production, and the generalized Pfaffian for the extended definition of entropy as required by such a theory. Some of the physical implications of these results are discussed.Member, Colegio Nacional.     

Rigorous results on the thermodynamics of the dilute Hopfield model

We study the Hopfield model of an autoassociative memory on a random graph onN vertices where the probability of two vertices being joined by a link isp(N). Assuming thatp(N) goes to zero more slowly thanO(1/N), we prove the following results: (1) If the number of stored patternsm(N) is small enough such thatm(N)/Np(N) 0, asN, then the free energy of this model converges, upon proper rescaling, to that of the standard Curie-Weiss model, for almost all choices of the random graph and the random patterns. (2) If in additionm(N) < ln N/ln 2, we prove that there exists, forT< 1, a Gibbs measure associated to each original pattern, whereas for higher temperatures the Gibbs measure is unique. The basic technical result in the proofs is a uniform bound on the difference between the Hamiltonian on a random graph and its mean value.     

Steady-state thermodynamics for heat conduction: microscopic derivation

Starting from microscopic mechanics, we derive thermodynamic relations for heat conducting nonequilibrium steady states. The extended Clausius relation enables one to experimentally determine nonequilibrium entropy to the second order in the heat current. The associated Shannon-like microscopic expression of the entropy is suggestive. When the heat current is fixed, the extended Gibbs relation provides a unified treatment of thermodynamic forces in the linear nonequilibrium regime.     

Quantum approach to a derivation of the second law of thermodynamics

We reinterpret the microcanonical conditions in the quantum domain as constraints for the interaction of the "gas subsystem" under consideration and its environment ("container"). The time average of a purity measure is found to equal the average over the respective path in Hilbert space. We then show that for typical (degenerate or nondegenerate) thermodynamical systems almost all states within the allowed region of Hilbert space have a local von Neumann entropy S close to the maximum and a purity P close to its minimum, respectively. Typically, thermodynamical systems should obey the second law.     

On statistics and thermodynamics II

The paper is an attempt on the foundation of statistical thermodynamics on the basis of the thermodynamic theory of measurement.The author wishes to thank U.G.C. for financial assistance.     

Non-linear supersymmetry for non-Hermitian,non-diagonalizable Hamiltonians: II. Rigorous results

We continue our investigation of the non-linear SUSY for complex potentials started in part I [A.A. Andrianov, F. Cannata, A.V. Sokolov, math-ph/0610024] and prove the theorems characterizing its structure in the case of non-diagonalizable Hamiltonians. This part provides the mathematical basis of previous studies. The classes of potentials invariant under SUSY transformations for non-diagonalizable Hamiltonians are specified and the asymptotics of formal eigenfunctions and associated functions are derived. Several results on the normalizability of associated functions at infinities are rigorously proved. Finally the Index Theorem on relation between Jordan structures of intertwined Hamiltonians depending of the behavior of elements of canonical basis of supercharge kernel at infinity is proven.     

Rigorous derivation of the distribution of the eigenstates of the quantum Heisenberg-Ising chain with XY-like anisotropy

The distribution of the eigenstates of the quantum Heisenberg-Ising chain with XY-like anisotropy is obtained rigoruously. It is shown that the assumptions made by Takahashi and Suzuki can be derived from the boundary conditions for the wave functions.     

On the dynamics of a quantum-statistical condensate

The dynamical equation for a condensate at finite temperatures is derived. The derivation of the equation is based on the model due to Wu. The equation derived is the nonlinear Schrödinger equation of diffusion type.     

On the derivation of quantum kinetic equations. II. Nonlocal Uehling-Uhlenbeck equation

The binary and triple collision terms of the quantum kinetic equation derived previously are analyzed in the weak coupling approximation. In this approximation the equation appears to be a nonlocal Markovian extension of the kinetic equation due to Uehling and Uhlenbeck. After linearization, its relationship with non-Markovian formulations found in the literature is studied.This investigation is part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (FOM), which is financially supported by the Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek (ZWO).     

Thermodynamics in quantum-statistical theory of a crystal

Thermodynamics of a crystal is developed on the basis of a system of equations with a self-consistent field for the particle number density in the quantum case [1]. The internal energy and pressure are found. It is demonstrated that these quantities satisfy the second law of thermodynamics and the entropy is found. Moreover, the method for calculating thermodynamic quantities by expansion in powers of is also considered.Translated from Izvestiya Uchebnykh Zavedenii, Fizika, No. 6, pp. 25–29, June, 1980.     

On statistical thermodynamics of multicomponent systems II

The equivalent neighbour (EN) model of interacting continuous spherically symmetric spins with the length equal or smaller than unity is solved exactly for all values of the spin dimensionality, D. A surprising result is that only for D 628 the system undergoes phase transitions at temperatures lower than those of the EN model of interacting spins with the hypercubic symmetry. In the D → ∞ limit the equation of state in zero field is derived analytically and gives the Landau type behaviour in the whole range of temperatures.     

On the second law of thermodynamics II. Inequalities for cyclic processes

Consequences of and relations among the classical statements of the second law of thermodynamics are derived for non-equilibrium thermodynamic systems not necessarily satisfying the first law. It is shown that each classical version of the second law implies one or two inequalities for cyclic processes which yield the Clausius inequality for cyclic processes if the first law holds. The inequalities for cyclic processes are derived by means of a general theorem stated and proved in the first part.The author wishes to express his deep thanks to Dr. Jan Kratochvíl for reading a previous draught of the paper and for discussing the topics dealt with in it.     

A solvable model of microscopic frequency modulation: II. Rigorous treatment of the damping operator

The solvable model of quantum-statistical mechanical frequency modulation, which was proposed previously by the authors, is solved exactly by rigorous treatment of the damping operator for the cases of boson and two-state irrelevant subsystems to obtain the intensity distribution and the response function for a certain operator of the relevant subsystem. The results are compared with the previous ones which were obtained within the conventional treatment of the damping operator.