Abstract
From basic principles, we review some fundamentals of entropy calculations, some of which are implicit in the literature. We mainly deal with microcanonical ensembles to effectively compare the counting of states in continuous and discrete settings. When dealing with non-interacting elements, this effectively reduces the calculation of the microcanonical entropy to counting the number of certain partitions, or compositions of a number. This is true in the literal sense, when quantization is assumed, even in the classical limit. Thus, we build on a moderately dated, ingenuous mathematical work of Haselgrove and Temperley on counting the partitions of an arbitrarily large positive integer into a fixed (but still large) number of summands, and show that it allows us to exactly calculate the low energy/temperature entropy of a one-dimensional Bose–Einstein gas in a box. Next, aided by the asymptotic analysis of the number of compositions of an integer as the sum of three squares, we estimate the entropy of the three-dimensional problem. For each selection of the total energy, there is a very sharp optimal number of particles to realize that energy. Therefore, the entropy is ‘large’ and almost independent of the particles, when the particles exceed that number. This number scales as the energy to the power of -rds in one dimension, and -ths in three dimensions. In the one-dimensional case, the threshold approaches zero temperature in the thermodynamic limit, but it is finite for mesoscopic systems. Below that value, we studied the intermediate stage, before the number of particles becomes a strong limiting factor for entropy optimization. We apply the results of moments of partitions of Coons and Kirsten to calculate the relative fluctuations of the ground state and excited states occupation numbers. At much lower temperatures than threshold, they vanish in all dimensions. We briefly review some of the same results in the grand canonical ensemble to show to what extents they differ.
Highlights
One tenet of equilibrium statistical mechanics is that one can derive physical predictions from calculating the number of microstates of a system compatible with some macroscipic measurements that can be performed on it [1,2,3]
Building on previous work [22], we investigate the relative fluctuations of the ground state and excited states occupancy numbers, in the microcanonical ensemble Bose–Einstein condensation theory, both in one and three dimensions
Microcanonical ensemble calculations in statistical mechanics are comparatively resilient to full exact solutions, even in the simple case of non-interacting particles
Summary
One tenet of equilibrium statistical mechanics is that one can derive physical predictions from calculating the number of microstates of a system compatible with some macroscipic measurements that can be performed on it [1,2,3]. In the case of partitions of m into integers, it was only in relatively recent times that decades-old mathematical result have been exploited to calculate thermodynamic relations, namely for Bose–Einstein particles trapped in a one-dimensional harmonic potential [4,5,24]. We have discovered an interesting result in number theory, dating back to more than 60 years ago [37], which to our knowledge has not been discussed in the Physics community It deals with the number of partitions of m into the sum of N squares of positive integers, when m is arbitrarily large and N grows with m in some ways.
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