Bose–Einstein distribution as a problem of analytic number theory: The case of less than two degrees of freedom

Microcanonical Entropy, Partitions of a Natural Number into Squares and the Bose–Einstein Gas in a Box

“…Nevertheless, mathematicians do have made significant and insightful progress in partitions theory over several decades, which are not always encompassed in the traditional overviews of statistical mechanics. There are exceptions [ 34 , 35 , 36 ], and in fact a growing interest has accompanied the need to describe mesoscopic quantum systems within microcanonical ensemble theory [ 4 , 5 , 6 , 7 , 8 ]. Following those steps, we have tried to exploit a small but significant subset of results in partition theory [ 22 , 37 ] to incorporate them in statistical mechanics.…”

“…which is the counterpart to the inaccurate Equation (34). Notice that the difference Equation (80) has little meaning very close to the ground state.…”

“…This is already widely covered even in textbooks on the subject, thus we only touch upon it. There are, however, plenty of results in the mathematical community and in number theory dealing with asymptotic (mainly large m) estimations of the number of partitions or compositions of an integer into a sequence of other assigned integers [25][26][27][28][29][30][31][32][33], many of which have direct physical applications [4][5][6][7][8][34][35][36]. 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].…”