We consider a quantum gas in its Feynman-Kac representation, which is a gas of
interacting Brownian loops, and study the asymptotic expansion of the logarithm
of the grand partition function of this loop gas in a bounded domain as this
domain is dilated to infinity. Under suitable restrictions on the pair interaction
we obtain the volume and boundary terms as well as the third term that is
proportional to the integral mean curvature which in two-dimensional case
is the Euler-Poincare characteristic. The proof is based on the cluster expansion method.
This result can be viewed as a natural generalization of the famous Kac problem of
computing the asymptotics of the partition function Tr exp(a Delta) with the Dirichlet
Laplacian Delta as a tends to 0.