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On the equilibrium states in quantum statistical mechanics

Rudolf Haag, Nicolaas Marinus Hugenholtz, Marinus Winnink

Communications in Mathematical PhysicsJun 1, 1967762 citations

Boundary conditions and the free energy for statistical mechanical models

Paul Martin

The author shows that the choice of boundary conditions can significantly affect the calculation of the free energy for discrete statistical mechanical models on finite or semi-infinite lattices when the exponentiated coupling is extended into the complex plane. He shows however, that the boundary conditions most consistent with boundary configuration independence in the direction in which a lattice is infinite give a unique analytic function for the free energy. This unique function is the free energy most appropriate for extrapolation to the full thermodynamic limit. The author shows that such boundary conditions are orthogonal to all but one degenerate ferromagnetic ground states at zero temperature.The author shows that the choice of boundary conditions can significantly affect the calculation of the free energy for discrete statistical mechanical models on finite or semi-infinite lattices when the exponentiated coupling is extended into the complex plane. He shows however, that the boundary conditions most consistent with boundary configuration independence in the direction in which a lattice is infinite give a unique analytic function for the free energy. This unique function is the free energy most appropriate for extrapolation to the full thermodynamic limit. The author shows that such boundary conditions are orthogonal to all but one degenerate ferromagnetic ground states at zero temperature.

Journal of Physics A Mathematical and GeneralJun 21, 19879 citations

On the One Variational Principle in Quantum Statistical Mechanics

A. M. Kurbatov, Дмитрий Петрович Санкович

Jan 1, 19800 citations

Anomalous Statistics of Bose-Einstein Condensate in an Interacting Gas: An Effect of the Trap’s Form and Boundary Conditions in the Thermodynamic Limit

Sergey V. Tarasov, Vl. V. Kocharovsky, Vitaly V. Kocharovsky

We analytically calculate the statistics of Bose-Einstein condensate (BEC) fluctuations in an interacting gas trapped in a three-dimensional cubic or rectangular box with the Dirichlet, fused or periodic boundary conditions within the mean-field Bogoliubov and Thomas-Fermi approximations. We study a mesoscopic system of a finite number of trapped particles and its thermodynamic limit. We find that the BEC fluctuations, first, are anomalously large and non-Gaussian and, second, depend on the trap's form and boundary conditions. Remarkably, these effects persist with increasing interparticle interaction and even in the thermodynamic limit-only the mean BEC occupation, not BEC fluctuations, becomes independent on the trap's form and boundary conditions.We analytically calculate the statistics of Bose-Einstein condensate (BEC) fluctuations in an interacting gas trapped in a three-dimensional cubic or rectangular box with the Dirichlet, fused or periodic boundary conditions within the mean-field Bogoliubov and Thomas-Fermi approximations. We study a mesoscopic system of a finite number of trapped particles and its thermodynamic limit. We find that the BEC fluctuations, first, are anomalously large and non-Gaussian and, second, depend on the trap's form and boundary conditions. Remarkably, these effects persist with increasing interparticle interaction and even in the thermodynamic limit-only the mean BEC occupation, not BEC fluctuations, becomes independent on the trap's form and boundary conditions.

EntropyFeb 27, 201811 citations

Thermodynamic Limit and Decoherence: Rigorous Results

Frasca, M

Time evolution operator in quantum mechanics can be changed into a statistical operator by a Wick rotation. This strict relation between statistical mechanics and quantum evolution can reveal deep results when the thermodynamic limit is considered. These results translate in a set of theorems proving that these effects can be effectively at work producing an emerging classical world without recurring to any external entity that in some cases cannot be properly defined. In a many-body system has been recently shown that Gaussian decay of the coherence is the rule with a duration of recurrence more and more small as the number of particles increases. This effect has been observed experimentally. More generally, a theorem about coherence of bulk matter can be proved. All this takes us to the conclusion that a well definite boundary for the quantum to classical world does exist and that can be drawn by the thermodynamic limit, extending in this way the deep link between statistical mechanics and quantum evolution to a high degree.Time evolution operator in quantum mechanics can be changed into a statistical operator by a Wick rotation. This strict relation between statistical mechanics and quantum evolution can reveal deep results when the thermodynamic limit is considered. These results translate in a set of theorems proving that these effects can be effectively at work producing an emerging classical world without recurring to any external entity that in some cases cannot be properly defined. In a many-body system has been recently shown that Gaussian decay of the coherence is the rule with a duration of recurrence more and more small as the number of particles increases. This effect has been observed experimentally. More generally, a theorem about coherence of bulk matter can be proved. All this takes us to the conclusion that a well definite boundary for the quantum to classical world does exist and that can be drawn by the thermodynamic limit, extending in this way the deep link between statistical mechanics and quantum evolution to a high degree.

Nov 2, 20062 citations

On the equations of state in quantum statistical mechanics

Richard Norton

Annals of PhysicsNov 1, 19806 citations

On the independence of the thermodynamic limit on the boundary conditions in quantum statistical mechanics

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