Statistics of an ideal homogeneous Bose gas with a fixed number of particles

The distribution function w₀(n₀) of the number n₀ of particles is found for the condensate of an ideal gas of free bosons with a fixed total number N of particles. It is shown that above the critical temperature (T > T_c) this function has the usual form w₀(n₀) = (1 — e^μ)e^μn₀, where μ is the chemical potential in temperature units. In a narrow vicinity of the critical temperature |T/T_c — 1| ≤ N^-1/3, this distribution changes and at T < T_c acquires the form of a resonance. The width of the resonance depends on the shape of the volume occupied by the gas and it has exponential (but not the Gaussian) wings. As the temperature is lowered, the resonance maximum shifts to larger values of n₀ and its width tends to zero, which corresponds to the suppression of fluctuations. For N → ∞, this change occurs abruptly. The distribution function of the number of particles in excited states for the systems with a fixed and a variable number of particles (when only a mean number of particles is fixed) prove to be identical and have the usual form.