The number of Bose-condensed particles in a weakly nonideal Bose gas
In the statistical physics of weakly nonideal Bose gas [see \textit{N. N. Bogolyubov}, Izv. Akad. Nauk SSSR, Ser. Fiz. 11, 77-90 (1947) and \textit{E. M. Lifshits} and \textit{L. P. Pitaevskij}, ``Statistical physics. Vol. 2. Theory of condensed state'' (Moscow, 1976)], the number of particles with zero momentum, which is called the number of Bose-condensed particles, plays an important role. It can be expressed as follows via the wave function \(\Psi_N(x_1,\dots,x_N)\) of the system of \(N\) particles (here \(x_1,\dots,x_N\) are the particle coordinates, which range in a measure space \(M)\): \[ n_f=N\int dx dy d\xi_1\dots d\xi_{N-1}\Psi_N(x,\xi_1,\dots,\xi_{N-1})\Psi^*_N(y,\xi_1,\dots,\xi_{N-1}){f^*(x)f(y)\over\int d\xi| f(\xi)|^2}\tag{1} \] for \(f(x)=1\). In the present paper, we construct an asymptotic formula for the number of Bose-condensed particles for the case of an \(N\)-particle system with particle interaction potential of order \(1/N\) in an external field of order 1.
scientific article; zbMATH DE number 1238056