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Gibbs equilibrium averages and Bogolyubov measure

Дмитрий Петрович Санкович

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

Application of the functional integration methods in equilibrium statistical mechanics of quantum Bose-systems is considered. We show that Gibbs equilibrium averages of Bose-operators can be represented as path integrals over a special Gauss measure defined in the corresponding space of continuous functions. We consider some problems related to integration with respect to this measure.

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•Oct 13, 2011•22 citations
Application of the functional integration methods in equilibrium statistical mechanics of quantum Bose-systems is considered. We show that Gibbs equilibrium averages of Bose-operators can be represented as path integrals over a special Gauss measure defined in the corresponding space of continuous functions. We consider some problems related to integration with respect to this measure.
Application of the functional integration methods in equilibrium statistical mechanics of quantum Bose-systems is considered. We show that Gibbs equilibrium averages of Bose-operators can be represented as path integrals over a special Gauss measure defined in the corresponding space of continuous functions. This measure arises in the Bogolyubov T-product approach and is non-Wiener. We consider problems related to integration with respect to the Bogolyubov measure in the space of continuous functions and calculate some functional integrals with respect to this measure. Approximate formulas that are exact for functional polynomials of a given degree and also some formulas that are exact for integrable functionals belonging to a broader class are constructed. We establish the nondifferentiability of the Bogolyubov trajectories in the corresponding function space and prove a theorem on the quadratic variation of trajectories and study the properties implied by this theorem for the scale transformations. We construct some examples of semigroups related to the Bogolyubov measure. Independent increments are found for this measure. We consider the relation between the Bogolyubov measure and parabolic partial differential equations. An inequality for some traces is proved.
The first steps in the application of methods for integrating functions defined on abstract sets were taken by Wiener. Most widely, the ideas of functional integration were developed in Feynman's works. The Feynman continual integral is well known to a wide community of physicists. Along with this, there is another approach to the construction of a functional integral in quantum physics. This approach was proposed by Bogolyubov. Bogolyubov's methods are relevant in quantum statistical physics, and have natural ties with probability theory. We review some mathematical results of integration with respect to a special Gaussian measure that arises in the statistical theory for quantum systems. It is shown that the Gibbs equilibrium averages of the chronological products of Bose operators can be represented as functional integrals with respect to this measure (the Bogolyubov measure). Some properties of this measure are studied. We rewrite partition function of many particle Bose systems in terms of Bogolyubov functional integral.
Abstract The functional integral representation of the density matrix at thermal equilibrium in non-relativistic quantum mechanics (QM) with many degrees of freedom, in the grand canonical formulation is introduced. In QM, Hamiltonians H(p,q) can be also expressed in terms of creation and annihilation operators, a method adapted to the study of perturbed harmonic oscillators. In the holomorphic formalism, quantum operators act by multiplication and differentiation on a vector space of analytic functions. Alternatively, they can also be represented by kernels, functions of complex variables that correspond in the classical limit to a complex parametrization of phase space. The formalism is adapted to the description of many-body boson systems. To this formalism corresponds a path integral representation of the density matrix at thermal equilibrium, where paths belong to complex spaces, instead of the more usual position–momentum phase space. A parallel formalism can be set up to describe systems with many fermion degrees of freedom, with Grassmann variables replacing complex variables. Both formalisms can be generalized to quantum gases of Bose and Fermi particles in the grand canonical formulation. Field integral representations of the corresponding quantum partition functions are derived.
Quantum statistical mechanics has developed primarily through two approaches, pioneered by Gibbs and Feynman, respectively. In Gibbs' method one calculates partition functions from phase-space integrations or sums over stationary states. Alternatively, in Feynman's approach, the focus is on the path-integral formulation. The Hubbard-Stratonovich transformation leads to a functional-integral formulation for calculating partition functions. We outline here the functional integral approach to quantum statistical mechanics, including generalizations and improvements to Hubbard's formulation. We show how the dimensionality of the integrals is reduced exactly, how the problem of assuming an unknown canonical transformation is avoided, how the reality of the partition function in the complex representation is guaranteed, and how the extremum conditions are simplified. This formulation can be applied to general systems, including superconductors.
The equilibrium and quasi-equilibrium properties of a system of interacting bosons are studied from a microscopic point of view. For equilibrium, the model of Bogolyubov is generalized to finite temperature by using the grand partition function. The thermodynamic properties and the pair-correlation function are calculated. The statistical mechanics for moving systems is then developed and applied to the problem of a rotating fluid. For quasi-equilibrium, general transport equations are derived from first principles, independent of statistics and model. For the Bogolyubov model, the familiar two-fluid hydrodynamics is then derived, leading to the phenomena of first and second sound.
Introduction. 1: Evolution of States of Quantum Systems of Finitely Many Particles. 1. Principal Concepts of Quantum Mechanics. 2. Evolution of States of Quantum Systems with Arbitrarily Many Particles. 3. Evolution of States in the Heisenberg Representation and in the Interaction Representation. Mathematical Supplement I. References. 2: Evolution of States of Infinite Quantum Systems. 4. Bogolyubov Equations for Statistical Operators. 5. Solution of the Bogolyubov Equations. 6. Gibbs Distributions. Mathematical Supplement II. Mathematical Supplement III. References. 3: Thermodynamic Limit. 7. Thermodynamic Limit for Statistical Operators. 8. Statistical Operators in the Case of Quantum Statistics. 9. Bogolyubov's Principle of Weakening of Correlations. Mathematical Supplement IV. References. 4: Mathematical Problems in the Theory of Superconductivity. 10. Frohlich Model. 11. Bogolyubov's Compensation Principle for 'Dangerous' Diagrams. Compensation Equations. 12. Bardeen--Cooper--Schrieffer (BCS) Hamiltonian. 13. Microscopic Theory of Superfluidity. Mathematical Supplement V. References. 6: Green's Functions. 14. Green's Functions. Equations for Green's Functions. 15. Investigation of the Equations for Green's Functions in the Theory of Superconductivity and Superfluidity. 16. Green's Functions in the Thermodynamic Limit. References. 6: Exactly Solvable Models. 17. Description of the Hamiltonians of Model Systems. 18. Functional Spaces of Translation--Invariant Function. 19. ModelHamiltonians in the Spaces of Translation Invariant Functions. 20. Model BCS Hamiltonian in the Space hT. Equivalence of General and Model Hamiltonians in the Space of Pairs. 21. Equations for Green's Functions and their Solutions. Mathematical Supplement VI. References. 7: Quasiaverages. Theorem on Singularities of Green's Functions of 1/q2-Type. 22. Quasiaverages. 23. Green's Functions and their Spectral Representations. 24. Theorem on Singularities of Green's Functions of 1/q2-Type. References. Subject Index.