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arXiv:1602.06536

On the Thermodynamic Limit of Bogoluibov's Theory of Bose Gas

Levent Akant, Ebru Dogan, Ertugrul, Emine, Turgut, O. Teoman

Published Feb 21, 2016• 0 citations
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Abstract

Assuming that Bogoliubov's theory of weakly interacting dilute Bose gas defines a self-consistent model Hamiltonian, we investigate its thermodynamic limit as we take the volume to infinity. The infinite volume is taken via a sequence of scaled convex regions with piecewise smooth boundary and the volumes staying proportional to the cube of the diameter of the region. To get a strict bound on the behavior of the thermodynamic limit, we use the recent formulation of Bogoliubov's theory of condensation in terms of heat kernels for a given domain as well as an estimate of the difference of traces between the heat kernel with Neumann boundary conditions on this domain and the infinite space result. We cannot control the limiting process by the area term; however, we can come arbitrarily close to it.

DOI
10.48550/arxiv.1602.06536

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Assuming that Bogoliubov's theory of weakly interacting dilute Bose gas defines a self-consistent model Hamiltonian, we investigate its thermodynamic limit as we take the volume to infinity. The infinite volume is taken via a sequence of scaled convex regions with piecewise smooth boundary and the volumes staying proportional to the cube of the diameter of the region. To get a strict bound on the behavior of the thermodynamic limit, we use the recent formulation of Bogoliubov's theory of condensation in terms of heat kernels for a given domain as well as an estimate of the difference of traces between the heat kernel with Neumann boundary conditions on this domain and the infinite space result. We cannot control the limiting process by the area term; however, we can come arbitrarily close to it.
Abstract Assuming that Bogoliubov’s theory of weakly interacting dilute Bose gas defines a self-consistent model Hamiltonian, we investigate its thermodynamic limit as we take the volume to infinity. The infinite volume is taken via a sequence of scaled convex regions with piecewise smooth boundary and the volumes staying proportional to the cube of the diameter of the region. To get a strict bound on the behavior of the thermodynamic limit, we use the recent formulation of Bogoliubov’s theory of condensation in terms of heat kernels for a given domain as well as an estimate of the difference of traces between the heat kernel with Neumann boundary conditions on this domain and the infinite space result. We cannot control the limiting process by the area term; however, we can come arbitrarily close to it.
The exactness of the Bogoliubov approximation is discussed from the point of view of the condensation properties of the Bose gas. For the imperfect Bose gas, the authors find that the approximation yields the correct condensate density when the thermodynamic limit is taken by isotropic dilation; this is not the case in general for other ways of going to the infinite volume limit. They also prove the existence of Bose-Einstein condensation for a class of Bose gases with weak interactions and an energy gap.
The thermodynamic properties and finite-size scaling of weakly interacting Bose gases with a finite number of particles confined in a cubic box of volume L3{L}^{3}L3 with periodic boundary conditions are investigated theoretically based on an exact recursion relation for the canonical-ensemble partition function. In a similar manner to the case of an ideal Bose gas, the recursive scheme of the interacting finite system at any temperatures is established in the framework of the Bogoliubov theory. The temperature dependence of the condensate fraction and specific heat with different particles and interactions is obtained numerically. By defining two different transition temperatures of the finite systems, the effect of interactions on the transition temperatures yields the different and nonmonotonous behaviors. The finite-size scaling condensate fraction at the transition temperature for the systems is also discussed. It is shown that the calculated finite-size scaling is universally independent on the various system sizes and transition temperatures.
In 1947, Bogoliubov suggested a heuristic theory to compute the excitation spectrum of weakly interacting Bose gases. Such a theory predicts a linear excitation spectrum and provides expressions for the thermodynamic functions which are believed to be correct in the dilute limit. Thus far, there are only a few cases where the predictions of Bogoliubov can be obtained by means of rigorous mathematical analysis. A major challenge is to control the corrections beyond Bogoliubov theory, namely, to test the validity of Bogoliubov’s predictions in regimes where the approximations made by Bogoliubov are not valid. In these notes, we discuss how this challenge can be addressed in the case of a system of N interacting bosons trapped in a box with volume one in the Gross-Pitaevskii limit, where the scattering length of the potential is of the order 1/N and N tends to infinity. This is a recent result obtained in Boccato et al. [Commun. Math. Phys. (to be published); preprint arXiv:1812.03086 and Acta Math. 222, 219–335 (2019); e-print arXiv:1801.01389], which extends a previous result obtained in Boccato et al. [Commun. Math. Phys. 359, 975 (2018)], removing the assumption of a small interaction potential.
We re-examine the way in which Bogoliubov's theory of a dilute Bose gas at T=0T=0T=0 has been extended to describe the statistical mechanics of interacting bosons at finite temperature. We show explicitly that the field-theoretic calculation of the grand partition function in this formulation amounts to a canonical trace over the eigenfunctions of the Bogoliubov Hamiltonian at fixed total number of bosons NNN, and that the additional trace over NNN that is required in the grand-canonical formalism is never carried out. This implies that what usually passes as the grand-canonical treatment of the Bogoliubov Hamiltonian is not quite grand-canonical, and is in fact a canonical one. We also show that the discontinuity in the condensate density predicted by previous formulations of this theory as the temperature TTT goes past the critical transition temperature TcT_cTc​ is a direct consequence of an inappropriate generalization of the Bogoliubov prescription to finite temperatures, and that this discontinuity disappears when this prescription is either used as a zero temperature approximation or avoided altogether. Armed with the above findings, we reformulate the statistical mechanics of interacting bosons in the canonical ensemble and derive the thermodynamics of the system. We then show how the canonical treatment can be used to setup a truly grand-canonical description of the statistical mechanics of a weakly interacting Bose gas where the average number of bosons in the system varies with temperature, unlike existing formulations where the total number of bosons NNN is taken to be a constant that does not depend on TTT. Consequences on the physics of interacting bosons are briefly discussed.
Aiming at simplicity of explicit equations and, at the same time, controllable accuracy of the theory, we present our results for all the thermodynamic quantities and correlation functions for a weakly interacting Bose gas at short-to-intermediate distances obtained within an improved version of Beliaev's diagrammatic technique. With a controllably small (but essentially finite) Bogoliubov's symmetry-breaking term, Beliaev's diagrammatic technique becomes regular in the infrared limit. Up to higher-order terms (for which we present parametric order-of-magnitude estimates), the partition function and entropy of the system formally correspond to those of a non-interacting bosonic (pseudo-)Hamiltonian with a temperature-dependent Bogoliubov-type dispersion relation. Away from the fluctuation region, this approach provides the most accurate—in fact, the best possible within the Bogoliubov-type pseudo-Hamiltonian framework—description of the system with controlled accuracy. It produces accurate answers for the off-diagonal correlation functions up to distances where the behavior of correlators is controlled by generic hydrodynamic relations and, thus, can be accurately extrapolated to arbitrarily large scales. In the fluctuation region, the non-perturbative contributions are given by universal (for all weakly interacting U(1) systems) constants and scaling functions, which can be obtained separately—by simulating classical U(1) models—and then used to extend the description of the weakly interacting Bose gas to the fluctuation region. The technique works in all spatial dimensions, and we explicitly checked the validity of this technique against first-principle Monte Carlo simulations for various thermodynamic properties and the single-particle density matrix.