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Journal of Mathematical Physics

Thermodynamically Equivalent Hamiltonian Method in Nonequilibrium Statistical Mechanics

Marvin D. Girardeau

Published Mar 1, 1967• 5 citations
Publisher

Abstract

The equilibrium statistical mechanics of the Bardeen-Cooper-Schrieffer model of superconductivity, as well as that of a wide class of similar models, can be evaluated exactly by the ``thermodynamically equivalent Hamiltonian'' method of Bogoliubov, Zubarev, Tserkovnikov, and Wentzel. It has been pointed out by Wentzel that this method can be extended to certain weakly nonequilibrium situations. It is shown here that the method allows an exact evaluation of the nonequilibrium statistical mechanics of the following situations: (a) temporal evolution of the statistical expectation value of an observable O(r) whose initial deviation from equilibrium is spatially localized, but not necessarily small; (b) temporal evolution of the statistical expectation value of an observable O due to a perturbation V which is spatially localized, but not necessarily small.

DOI
10.1063/1.1705204

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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.

•Oct 13, 2011•22 citations
The equilibrium statistical mechanics of the Bardeen-Cooper-Schrieffer model of superconductivity, as well as that of a wide class of similar models, can be evaluated exactly by the ``thermodynamically equivalent Hamiltonian'' method of Bogoliubov, Zubarev, Tserkovnikov, and Wentzel. It has been pointed out by Wentzel that this method can be extended to certain weakly nonequilibrium situations. It is shown here that the method allows an exact evaluation of the nonequilibrium statistical mechanics of the following situations: (a) temporal evolution of the statistical expectation value of an observable O(r) whose initial deviation from equilibrium is spatially localized, but not necessarily small; (b) temporal evolution of the statistical expectation value of an observable O due to a perturbation V which is spatially localized, but not necessarily small.
A general variational approach to the quantum statistics of many-particle systems is developed, based on the Gibbs-Bogoliubov variational principle and the ``thermodynamically equivalent Hamiltonian'' method first applied to the BCS model by Bogoliubov, Zubarev, and Tserkovnikov [Sov. Phys.---Dokl. 2, 535 (1957)] and subsequently generalized by Wentzel [Phys. Rev. 120, 1572 (1960)]. The method retains a judiciously chosen discrete set of momentum transfers in the second-quantized interaction Hamiltonian. It is expected to be applicable to a variety of systems including both crystalline solids and quantum fluids such as liquids and dense plasma phases.
A theory of non-equilibrium thermodynamics is developed. The general method to derive the closed set of kinetic equations for expectation values of physical quantities is developed, by making use of the functional Taylor expansion with respect to expectation values of physical quantities around the reference state. For this purpose, a definition of the thermodynamic state in the system whose state changes with time is presented in order to describe the general state of the many-body system from the unified point of view. The statistical mechanical probability distribution functions for the quantum-mechanical expectation values at the stationary state are developed, without making use of the concept of the constant of the motion. The relation between the expectation value and the most probable value for the probability distribution function is discussed, and it is found that both values are equivalent to each other, if the system is extended infinitely in space and time. Thus the condition for the thermodynamic limit may be defined in a generalized way. Our method is looked as a perturbational one in the equilibrium statistical mechanics, though the choice of the reference state is arbitrary.
The equilibrium properties of a system of N two-level atoms interacting with a quantized field, the so-called Dicke model of superradiance, are evaluated exactly by the 'thermodynamically equivalent Hamiltonian' method of Bogoliubov et at. (1957). The results are identical with those obtained by Hepp and Lieb (1973) and Wang and Hioe (1973). The thermodynamically equivalent Hamiltonian is shown to be equivalent to the mean field Hamiltonian of the Ising model, which indicates why the transition from the 'normal' to the 'superradiant' state of the Dicke model is similar to the transition from the disordered to the ordered state of the Ising model.
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.