Distribution function of the number of particles in the condensate of an ideal Bose gas confined by a trap

The distribution function W₀(n₀) of the number n₀ of particles in the condensate of an ideal Bose gas confined by a trap is found. It is shown that at the temperature below the critical temperature T_c this function has a Gaussian shape and depends on the trap potential via two parameters only. The center of this function shifts to larger values of n₀ with decreasing temperature and its width tends to zero, which corresponds to the suppression of fluctuations. In the narrow vicinity of the critical temperature \(\left| {T - {T_c}} \right| \leqslant {T_c}/\sqrt N \), where N is the number of particles in the trap, the distribution function changes and at the temperature above the critical one it takes the usual form W₀(n₀) = 1 ? exp(μ)]exp(μn₀), where μ is the chemical potential in temperature units. In the limit N→∞, this change occurs at a jump.