Quantum Corrections to the Dynamics of the Expanding Universe

For many-particle quantum systems, calculating thermodynamic quantities in the canonical ensemble is a very hard task, while this is tractable in the grand canonical ensemble.The second ensemble is then used.The results are supposed to be the same, at least in the thermodynamic limit.Is this actually the case?In this work, we consider a system of N noninteracting bosons distributed among few energy levels.We can calculate the canonical partition function in this case and deduce the canonical mean energy.We compare it to the mean energy deduced from the grand canonical ensemble for the same number of particles.We consider the case of a large number of particles.