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Representative Ensembles in Statistical Mechanics

Vyacheslav I. Yukalov

Published Aug 3, 2013• 22 citations
OpenAlex

Abstract

The notion of representative statistical ensembles, correctly representing statistical systems, is strictly formulated. This notion allows for a proper description of statistical systems, avoiding inconsistencies in theory. As an illustration, a Bose-condensed system is considered. It is shown that a self-consistent treatment of the latter, using a representative ensemble, always yields a conserving and gapless theory.

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The notion of representative statistical ensembles, correctly representing statistical systems, is strictly formulated. This notion allows for a proper description of statistical systems, avoiding inconsistencies in theory. As an illustration, a Bose-condensed system is considered. It is shown that a self-consistent treatment of the latter, using a representative ensemble, always yields a conserving and gapless theory.
Bose atoms in optical lattices are considered at low temperatures and weak interactions, when Bose-Einstein condensate is formed.A self-consistent approach, based on the use of a representative statistical ensemble, is employed, ensuring a gapless spectrum of collective excitations and the validity of conservation laws.In order to show that the approach is applicable to both weak and tight binding, the problem is treated in the Bloch as well as in the Wannier representations.Both these ways result in similar expressions that are compared for the self-consistent Hartree-Fock-Bogolubov approximation.A convenient general formula for the superfluid fraction of atoms in an optical lattice is derived.
The Statistical Interpretation of quantum theory is formulated for the purpose of providing a sound interpretation using a minimum of assumptions. Several arguments are advanced in favor of considering the quantum state description to apply only to an ensemble of similarily prepared systems, rather than supposing, as is often done, that it exhaustively represents an individual physical system. Most of the problems associated with the quantum theory of measurement are artifacts of the attempt to maintain the latter interpretation. The introduction of hidden variables to determine the outcome of individual events is fully compatible with the statistical predictions of quantum theory. However, a theorem due to Bell seems to require that any such hidden-variable theory which reproduces all of quantum mechanics exactly (i.e., not merely in some limiting case) must possess a rather pathological character with respect to correlated, but spacially separated, systems.
In classical mechanics the state of a dynamical system at any particular time can be described by the values of a set of coordinates and their conjugate momenta, thus, if the system has n degrees of freedom, by 2n numbers. In quantum mechanics, on the other hand, we have to describe a state of the system by a wave function involving a set of coordinates, thus by a function of n variables. The quantum description is, therefore, much more complicated than the classical one. Let us consider, however, an ensemble of systems in Gibbs' sense, i.e. not a large number of actual systems which could, perhaps, interact with one another, but a large number of hypothetical systems which are introduced to describe one actual system of which our knowledge is only of a statistical nature. The basis of the quantum treatment of such an ensemble has been given by Neumann. The description obtained by Neumann of an ensemble on the quantum theory is no more complicated than the corresponding classical description. Thus the quantum theory, which appears to such a disadvantage on the score of complication when applied to individual systems, recovers its own when applied to an ensemble. It is the object of the present note to examine this question more closely and to show how complete the analogy is between the quantum and classical treatments of an ensemble.
The necessity of accurately taking into account the existence of nonequivalent operator representations associated with canonical transformations is discussed. It is demonstrated that Bose systems in the presence of the Bose-Einstein condensate and without it correspond to different Fock spaces, orthogonal to each other. A composite representation for the field operators is constructed allowing for a self-consistent description of Bose-condensed systems. Equations of motion are derived from the given Hamiltonian, which guarantees the validity of conservation laws and thermodynamic self-consistency. At the same time, the particle spectrum obtained either from diagonalizing this Hamiltonian or from linearizing the field-operator equations of motion has no gap. The condition of the condensate existence assures the absence of the gap in the spectrum, irrespectively to the approximation involved. The suggested self-consistent theory is both conserving and gapless.