Calculation of thermodynamic properties of finite Bose-Einstein systems

We derive an exact recursion formula for the calculation of thermodynamic functions of finite systems obeying Bose-Einstein statistics. The formula is applicable for canonical systems where the particles can be treated as noninteracting in some approximation, e.g. like Bose-Einstein condensates in magnetic traps. The numerical effort of our computation scheme grows only linear with the number of particles. As an example we calculate the relative ground state fluctuations and specific heats for ideal Bose gases with a finite numbers of particles enclosed in containers of different shapes. 03.75.Fi, 05.30.Jp, 32.80.Pj Typeset using REVTEX 1 With the observation of Bose-Einstein condensation (BEC) of magnetically [1–3] and optically [4] trapped atoms new insights into the nature of this state of matter have been given. The experimental situation is in all cases quite different from the ideal gas treated within the grand canonical ensemble, which is the standard textbook example. First, the number of particles within the traps is fixed and finite, which suggests a canonical or microcanonical treatment of the systems. Second, the confining trap potentials greatly influence the condensate properties. Third, although the trapped gases are quite dilute the validity of the treatment as non-interacting particle gases has to be checked from case to case. Even within the approximation of non-interacting particles the calculation of the thermodynamic properties of the Bose-Einstein systems remains a hard to tackle mathematical problem. Recently, some approximate methods to calculate the fluctuation of the ground state occupation number in a trapped Bose-Einstein condensate have been developed [5–8]. Here we present an exact method to calculate all thermodynamic quantities of finite canonical Bose systems, given the one particle density of states. As the starting point we utilize the recursive formula of the canonical partition function for a system of N noninteracting bosons as given in [9]: ZN(β) = 1 N N