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Journal of Weinan Normal University

The Discussion of Several Problems in the Bose-Einstein Condense Phenomenon

Yingli Pan

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

The paper takes the ideal Bose-Einstein gas as an example,and the phase transition point equation and the phase transition point curve,the isotherm equation and isotherm,the two-phase equilibrium coexist curve equation and the two-phase equilibrium coexist curve and the Clausius-Clapeyron equation on the Bose-Einstein condense phenomenon based on quantum statistics are given.It is useful to master the Bose-Einstein condense phenomenon in an all-round way.

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The paper takes the ideal Bose-Einstein gas as an example,and the phase transition point equation and the phase transition point curve,the isotherm equation and isotherm,the two-phase equilibrium coexist curve equation and the two-phase equilibrium coexist curve and the Clausius-Clapeyron equation on the Bose-Einstein condense phenomenon based on quantum statistics are given.It is useful to master the Bose-Einstein condense phenomenon in an all-round way.
The peculiar condensation in phase space of ideal Bose-Einstein gases is discussed when the internal energy states of the atoms or molecules is taken into account and when the gas of atoms without internal structure is placed in an external conservative field of force. The inclusion of the internal energy states decreases while the application of the external field increases the condensation temperature from its value in absence of either of these causes. The transition phenomenon, apart from the preceding change in the condensation temperature, proceeds in the same way as in the case of the free gas atoms without internal structure; in particular, the order of the transition remains invariable, as the continuity of the heat capacity at constant volume is maintained at the transition point.
Abstract The theory of quantum-mechanical grand canonical ensembles is used to derive for the case of a perfect Bose-Einstein gas the average number of particles in the different energy levels, the fluctuations in these numbers and the equation of state. The Einstein condensation phenomenon is then discussed, and it is shown that in a p-v diagram (v being the specific volume) the isotherm consists of two analytically different parts in the limit where the number of particles in the system, JV, goes to infinity. It is also shown that for finite N at the critical volume ∂np/∂vn is of the order N1/3(n-2) in accordance with a result obtained by Wergeland & Hove-Storhoug.
In this paper, we present an exactly solvable phase transition model in which the phase transition is purely statistically derived. The phase transition in this model is a generalized Bose–Einstein condensation. The exact expression for the thermodynamic quantity, which can be used to simultaneously describe both the gas phase and the condensed phase, is solved with the help of the homogeneous Riemann–Hilbert problem, so one can judge whether there exists a phase transition and determine the phase transition point mathematically rigorously. A generalized statistics in which the maximum occupation numbers of different quantum states can take on different values is introduced, as a generalization of Bose–Einstein and Fermi–Dirac statistics.
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.