V.E Mean Field Theory of Condensation

In principle, all properties of the interacting system, including phase separation, are contained within the thermodynamic potentials that can be obtained by evaluating Z(T,N) or Q(T, μ). Phase transitions, however, are characterized by discontinuities in various state functions and must correspond to the appearance of singularities in the par­ tition functions. At first glance, it is somewhat surprising that any singular behavior should emerge from computing such well behaved integrals (for short-ranged interactions) as   ∫ ∏N N 2 dp~id ~qi ∑ p ∑ Z(T,N, V ) = i=1 exp −β i − β V(q~i − ~qj) . (V.57) N !h3N 2m i=1 i<j