Bose-Einstein Condensation and Partition Theory
Bose-Einstein Condensation and Partition Theory
The physics of ultracold atomic gases meanwhile has become one of the most prolific areas of contemporary research. We have contributed to this field, among others, by investigating the unfamiliar fact that the fluctuation of the number of particles in a trapped, isolated Bose-Einstein condensate differs within the standard statistical ensembles. While the grand canonical ensemble, requiring the presence of both a particle reservoir and a heat bath, gives rise to what we have termed a fluctuation catastrophe, and the canonical ensemble, requiring the presence of a heat bath, likewise does not apply to isolated systems, it is actually the difficult microcanonical ensemble which is needed to evaluate the condensate statistics.
After developing the theoretical framework required to deal properly with this physics-motivated problem, we have also applied our insights to the mathematical theory of partitions. Namely, determining the number of ways a given amount of energy can be shared among the constituents of an isolated system - that is, counting the microstates of that system - is closely related to the determination of the number of possibilities \( p(n) \) to represent an integer number \( n \) as a sum of smaller integers. A famous formula for the asymptotic behavior of \( p(n) \) was announced by the mathematicians Hardy and Ramanujan in 1917. Going beyond that, we have employed our microcanonical tools for studying the probability distribution for finding \( M \) summands in a randomly chosen partition of \( n \), as graphed for \( n=1000 \) and \( n=5000 \) in the above figure published in Europhys. Lett. 59, 486 (2002).
In a detailed rigorous study reported in J. Phys. A: Math. Gen. 36, 1827 (2003), we have calculated higher moments of this distribution. Somewhat amazingly, the leading term of the rms fluctuation of the number of addends in a partition of asymptotically large \( n \) equals \( \sqrt{n} \), while the skewness and the kurtosis approach the constant values 1.1395... (a number involving the Apéry constant) and 2.4, respectively, for truly large \( n \).
Thus, microcanonical statistical physics, as developed for predicting the fluctuation of the number of particles in an ideal isolated Bose-Einstein condensate, flourishes to yield fairly nontrivial results in analytic number theory!