Thermodynamic limit of the canonical partition function with respect to the quark number in QCD

We investigate QCD in the canonical ensemble with respect to the quark number. We reveal that the canonical description in which the quark number is fixed would be reduced to the grand canonical description under the thermodynamic limit. Since the grand canonical ensemble contains fluctuations of the quark number, the idea of the canonical ensemble is of no use for the purpose of defining order parameters for the deconfinement transition. We clarify the origin of such reduction and propose an idea to define the order parameter. The Monte-Carlo simulation by means of the spin system, which is an effective model of QCD at finite temperature, shows prosperous behaviors though the results suffer from the severe statistical error due to the sign problem.