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Studies in Quantum Statistical Mechanics

Kishore T. Kapale

Published Jan 5, 2010• 0 citations
OpenAlex

Abstract

Part I: The problem of Bose-Einstein condensation of an ideal gas is dealt with to determine the condensate statistics. A brief description of the inter-relations between various ensembles of statistical mechanics is given. Thereafter, various canonical ensemble methods applied to the trapped ideal Bose gas are discussed in great detail. Further, strategies to convert the canonical ensemble results into the microcanonical ensemble description are also discussed. Part II: Various topics bound by a common theme based on the principle of atomic coherence are studied in the second part. First a scheme exploiting interference of incoherent pump processes to achieve strong control over the total spontaneous emission from a three-level atom is discussed. Thereafter various schemes to achieve radiative cooling without the presence of spontaneous emission on the cooling transition are considered. This allows cooling to an internal absolute zero of temperature in an atomic system in very small number of steps. In the last chapter a transient regime for gain without inversion is considered in a three level lambda system that may be easier to implement.

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Part I: The problem of Bose-Einstein condensation of an ideal gas is dealt with to determine the condensate statistics. A brief description of the inter-relations between various ensembles of statistical mechanics is given. Thereafter, various canonical ensemble methods applied to the trapped ideal Bose gas are discussed in great detail. Further, strategies to convert the canonical ensemble results into the microcanonical ensemble description are also discussed. Part II: Various topics bound by a common theme based on the principle of atomic coherence are studied in the second part. First a scheme exploiting interference of incoherent pump processes to achieve strong control over the total spontaneous emission from a three-level atom is discussed. Thereafter various schemes to achieve radiative cooling without the presence of spontaneous emission on the cooling transition are considered. This allows cooling to an internal absolute zero of temperature in an atomic system in very small number of steps. In the last chapter a transient regime for gain without inversion is considered in a three level lambda system that may be easier to implement.
A nonequilibrium approach to the dynamics and statistics of the condensate of an ideal N-atom Bose gas cooling via interaction with a thermal reservoir using the canonical ensemble is developed. We derive simple analytical expressions for the canonical partition function and equilibrium distribution of the number of atoms in the ground state of a trap under different approximations, and compare them with exact numerical results. The N-particle constraint associated with the canonical ensemble is usually a burden. In the words of Kittel, ``in the investigation of the Bose-Einstein\dots{}laws it is very inconvenient to impose the restriction that the number of particles in the subsystem shall be held constant.'' But in the present approach, based on the analogy between a second-order phase transition and laser threshold behavior, the N-particle constraint makes the problem easier. We emphasize that the present work provides another example of a case in which equilibrium (detailed balance) solutions to nonequilibrium equations of motion provide a useful supplementary approach to conventional statistical mechanics. We also discuss some dynamical and mesoscopic aspects of Bose-Einstein condensation. The conclusion is that the present analytical (but approximate) results, based on a nonequilibrium approach, are in excellent agreement with exact (but numerical) results. The present analysis has much in common with the quantum theory of the laser.
Bose-Einstein condensation (BEC) represents a remarkable phase transition, characterized by the formation of a single quantum subsystem. As a result, the statistical properties of the condensate are highly unique. In the case of a Bose gas, while the mean number of condensed atoms is independent of the choice of statistical ensemble, the microcanonical, canonical (CN), or grand CN (GC) variances differ significantly among these ensembles. In this paper, we review the progress made over the past 30 years in studying the statistical fluctuations of BECs. Focusing primarily on the ideal Bose gas, we emphasize the inequivalence of the Gibbs statistical ensembles and examine various approaches to this problem. These approaches include explicit analytic results for primarily one-dimensional systems, methods based on recurrence relations, asymptotic results for large numbers of particles, techniques derived from laser theory, and methods involving the construction of statistical ensembles via stochastic processes, such as the Metropolis algorithm. We also discuss the less thoroughly resolved problem of the statistical behavior of weakly interacting Bose gases. In particular, we elaborate on our stochastic approach, known as the hybrid sampling method. The experimental aspect of this field has gained renewed interest, especially following groundbreaking recent measurements of condensate fluctuations. These advancements were enabled by unprecedented control over the total number of atoms in each experimental realization. Additionally, we discuss the fluctuations in photonic condensates as an illustrative example of GC fluctuations. Finally, we briefly consider the future directions for research in the field of condensate statistics.
We question the validity of the grand canonical ensemble for the description of Bose-Einstein condensation of small ideal Bose gas samples in isolated harmonic traps. While the ground state fraction and the specific heat capacity can be well approximated with the help of the conventional grand canonical arguments, the calculation of the fluctuation of the number of particles contained in the condensate requires a microcanonical approach. Resorting to the theory of restricted partitions of integer numbers, we present analytical and numerical results for such fluctuations in one- and three-dimensional traps, and show that their magnitude is essentially independent of the total particle number.
Abstract Chapter 11 presents a statistical mechanical treatment of the quantum ideal gases, i.e., the ideal Boltzmann, fermion, and boson gases. The discussion begins with the microscopic description and the solution of the eigenvalue problem for a quantum ideal gas. It is argued that calculating the partition function is most readily accomplished in the grand canonical ensemble using a second-quantized formulation. When the particles are distinguishable, the equation of state is identical to that of a classical ideal gas. For fermions and bosons, however, the problem of computing thermodynamic properties is significantly more complex and can only be solved exactly in certain limits. Away from these limits approximations are needed and are discussed in detail. The relevant distributions - the Fermi-Dirac and Bose-Einstein distributions are derived. The local density approximation of density functional theory is derived for the ideal electron gas. For the ideal boson gas, the phenomenon of Bose-Einstein condensation is discussed
We calculate certain features of Bose–Einstein condensation in the ideal gas by using recurrence relations for the partition function. The grand canonical ensemble gives inaccurate results for certain properties of the condensate that are accurately provided by the canonical ensemble. Calculations in the latter can be made tractable for finite systems by means of the recurrence relations. The ideal one-dimensional harmonic Bose gas provides a particularly simple and pedagogically useful model for which detailed results are easily derived. An analysis of the Bose system via permutation cycles yields insight into the physical meaning of the recurrence relations.