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Foundations of Physics

The principle of thermodynamic equivalence in statistical mechanics: The method of approximating Hamiltonian

Alexander P. Bakulev, N. N. Bogoljubov, A. M. Kurbatov

Published Sep 1, 1986• 1 citation
Publisher
DOI
10.1007/bf00765335

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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.