Statistics of an ideal homogeneous Bose gaswith a fixed number of particles

Abstract

The distribution function w0(n0) of the number n0of particles is found for the condensate of an ideal gas of freebosons with a fixed total number N of particles. It is shownthat above the critical temperature (T > Tc) this function hasthe usual form w0(n0)=(1—eμ)eμn0, where μ is the chemicalpotential in temperature units. In a narrow vicinity of thecritical temperature |T/Tc — 1| ≤ N-1/3, this distributionchanges and at T < Tc acquires the form of a resonance.The width of the resonance depends on the shape of thevolume occupied by the gas and it has exponential (but notthe Gaussian) wings. As the temperature is lowered, theresonance maximum shifts to larger values of n0 and its widthtends 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 statesfor the systems with a fixed and a variable number of particles (when only a mean number of particles is fixed) prove tobe identical and have the usual form.