An inverse problem of the heat equation for a general multi-connected drum with applications in Physics

We study the influence of a multi-connected bounded container in R2 on an ideal gas. The trace of the heat semigroup θ(t)=∑v=1∞exp(−tμv), where {μv}v=1∞ are the eigenvalues of the negative Laplacian in the (x1,x2)-plane, is studied for a general multi-connected domain Ω in R2 surrounding by a simply connected bounded domains Ωj with smooth boundaries ∂Ωj (j=1,…,q), where a finite number of Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components Γi (i=1+kj−1,…,kj) of ∂Ωj, are considered where with k0=0. In this paper, one may extract information on the geometry of Ω by analyzing the asymptotic expansions of θ(t) for short-time t . Some applications of θ(t) for an ideal gas enclosed in Ω are given. Thermodynamic quantities of an ideal gas enclosed in Ω are determined. We use an asymptotic expansion for high temperatures to obtain the partition function of an ideal gas showing the leading corrections to the internal energy due to a finite container. We show that the ideal gas cannot feel the shape of its container, although it can feel some geometrical properties of it.