S.V. Tarasov - Post-graduate Student, Junior Researcher, Institute of Applied Physics of the Russian Academy of Sciences. E-mail: email@example.com
The present work develops the mathematical technique which allows one to efficiently investigate the Bose-Einstein condensation of an ideal gas in mesoscopic systems (which have finite size and a strictly defined number of particles) into the critical region, where the system sharply rebuilds and changes its thermodynamic and statistical parameters. The width of this critical region is finite for any real mesoscopic system. However, in the critical region the fluctuations of the order parameter are compared with the value of the order parameter itself, and this fact makes well-known mean-field methods inappropriate, including the grand-canonical-ensemble method.
For a case of an ideal gas the rigorous critical theory has been established only recently. It is shown that the partition function of the mesoscopic system is expressed through a so-called “unconstrained” probability distribution, which is a cumulative probability distribution of the total number of noncondensed particles in the auxiliary problem with unbounded and independent occupations of energy levels. The analysis of that distribution seems to be feasible.
Curiously, the «unconstrained» probability has a following property in the critical region: while the size of the system and the number of particles within are growing (note that the geometry of the trap holds unchanged), it rapidly tends to a self-similar function. The self-similar argument is a difference of the total and critical number of particles divided to a standard deviation of the ”unconstrained” distribution. Taking into account this rapid convergence one could substitute the self-similar distribution instead of the mesoscopic one during calculating the partition function and does not mistake a lot. So, the problem of description of the mesoscopic system reduces to an exact calculation of the self-similar statistics.
The investigation of the critical region is done according to the following way. We start from the one-particle partition function to introduce and analyze so-called trap functions, which are generalized spectral Zeta functions produced by the one-particle energy spectrum of the trap. Then, with a Mellin transform these trap functions links with a characteristic function of the total number of the noncondensed particles. Finally, the wanted “unconstrained” probability distribution is obtained through the characteristic function via Fourier transform.
The exact form of the characteristic function depends qualitatively on the poles’ structure of the trap functions, which is determined by a law of energy level increasing. The position of the rightmost pole in fact defines the classification of all traps into two groups with different self-similar statistics.
In case when one-particle energy levels grow slowly, the “unconstrained” distribution is just a Gaussian one, and the specific shape of the confining potential impacts only a way how the self-similar argument is calculated. These traps form the Gaussian class.
In case when the one-particle energy levels grow rapidly enough, the “unconstrained” distribution is a special mathematical function defined through the one-particle energy spectrum, which is dependent on the specific shape of the confining potential. For these traps forming an anomalous class we develop a regular method which allows one to calculate asymptotics for large self-similar arguments, which simplify a lot the general expression for the “unconstrained” distribution.
The developed technique is suitable for effective investigation of the critical region of the Bose-Einstein condensation. In particular, it is possible to proof and clearly illustrate that the shape of the “unconstrained” distribution for the anomalous class traps is affected to even small perturbations of the spectrum, such as a change in boundary conditions, and this effect does not disappear even in the thermodynamic limit.
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