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arXiv:cond-mat/0012498

Quantization of Thermodynamics, Supersecondary Quantization, and a New Variational Principle

Виктор Павлович Маслов

Published Dec 28, 2000• 2 citations
arXivHTMLPDF

Abstract

In solving the problem of finding a temperature distribution which, at zero temperature, corresponds to superfluidity, i.e., to nonzero energy, the author tried to quantize free energy. This was done on the basis of supersecondary quantization whose special case is the usual secondary quantization for bosons and with the help of which new representations of the Schrödinger equation were obtained. The supersecondary quantization allowed the author to construct a variational method whose zero approximation are the Hartree-Fock and Bogolyubov-BCSch variational principles. This method works especially well in the case of not a large number of particles. The new quantization and the variational method are of general character and can be used in the quantum field theory.

DOI
10.48550/arxiv.cond-mat/0012498

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In solving the problem of finding a temperature distribution which, at zero temperature, corresponds to superfluidity, i.e., to nonzero energy, the author tried to quantize free energy. This was done on the basis of supersecondary quantization whose special case is the usual secondary quantization for bosons and with the help of which new representations of the Schrödinger equation were obtained. The supersecondary quantization allowed the author to construct a variational method whose zero approximation are the Hartree-Fock and Bogolyubov-BCSch variational principles. This method works especially well in the case of not a large number of particles. The new quantization and the variational method are of general character and can be used in the quantum field theory.
The density matrix of von Neumann can be used to formulate an exact variational principle for quantum statistics which embodies the principle of maximization of entropy. Using the formalism of second quantization, we can write this variational principle for fermions or bosons and can then derive from it an approximate variational procedure which yields the particle states of a system of interacting bosons or fermions as well as the distribution of particles in these states. These equations yield the generalization of the Hartree-Fock equations for nonzero temperature and the corresponding extension to bosons.
The variational theory of equilibrium boson system state to have been previously developed by the author under the density matrix formalism is applicable for researching equilibrium states and thermodynamic properties of the quantum Bose gas which consists of zero-spin particles. Particle pulse distribution function is obtained and duly employed for calculation of chemical potential, internal energy and gas capacity temperature dependences. It is found that specific phase transition, which is similar to transition of liquid helium to its superfluid state, occurs at the temperature exceeding that of the Bose condensation.
The approach to the theory of many-particle interacting systems from a unified standpoint, based on the variational principle for free energy is reviewed. A systematic discussion is given of the approximate free energies of complex statistical systems. The analysis is centered around the variational principle of Bogoliubov for free energy in the context of its applications to various problems of statistical mechanics. The review presents a terse discussion of selected works carried out over the past few decades on the theory of many-particle interacting systems in terms of the variational inequalities. It is the purpose of this paper to discuss some of the general principles which form the mathematical background to this approach and to establish a connection of the variational technique with other methods, such as the method of the mean (or self-consistent) field in the many-body problem. The method is illustrated by applying it to various systems of many-particle interacting systems, such as Ising, Heisenberg and Hubbard models, superconducting (SC) and superfluid systems, etc. This work proposes a new, general and pedagogical presentation, intended both for those who are interested in basic aspects and for those who are interested in concrete applications.
Based on an approach introduced by Gerjuoy, Rau, and Spruch, we construct variational principles in a systematic way for the nonlinear Schrödinger equation and obtain new variational principles for the case of Ginzburg-Pitaevskii-Gross equation (GPG) which is believed to describe accurately the Bose-Einstein condensation at zero temperature. As an application of these variational methods, a variational iteration method is proposed for calculating eigenvalue (chemical potential) and wave function for the GPG equation 1.
A self-contained pedagogical introduction to the functional Schrödinger picture method of many-body theory is given at a level suitable for graduate students and also for many-body physicists who have not been exposed to the functional Schrödinger picture method previously. Mathematical tools necessary for the functional Schrödinger picture calculation are introduced. The method is first applied to various condensed matter problems including the electron gas, the Hubbard model, the BCS superconductivity, and dilute bose gas within the variational approximation. It is shown that the variational approximation with the Gaussian trial functional invariably leads into the Hartree-Fock results for both zero and finite temperature cases. In order to go beyond the Gaussian results, concepts of variational and optimized perturbation theories are introduced using simple quantum mechanical languages. Then, the variational perturbation theory in the functional Schrödinger picture is applied to the λϕ4λϕ^4λϕ4 model to yield the effective potential to the third order. The optimized perturbation theory is also applied to the λϕ4λϕ^4λϕ4 model to yield a general expression up to the second order.
This paper represents the full version of a paper published earlier in Physica A [246 (1997), 275]. The present paper includes argumentation, proofs and details omitted in the shortened version. The papers are a further development of the approach in quantum statistical mechanics proposed by the author. The hierarchy of equations for reduced density matrices obtained previously is extended to the case corresponding to the Bose-Einstein condensation. The relevant state of the system with a condensate can be superfluid as well as nonsuperfluid. Special attention is given to the thermodynamics of superfluid systems. According to the results of the papers superfluidity is the state of a fluid whose symmetry is spontaneously broken because of a stationary flow. The state corresponds to thermodynamic equilibrium while the magnitude of the flow depends upon the temperature and is determined by thermodynamic considerations. The equations obtained are solved in two simple cases. The physical origin of superfluidity, peculiarities of the phenomenon in closed volumes and the critical velocity are discussed as well.