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 R-2 on an ideal gas. The trace of the heat semigroup theta (t) = Sigma (infinity)(v=1) exp(-t mu (v)), where \{mu (r)\}(v=1)(infinity) are the eigenvalues of the negative Laplacian -Delta (2) = - Sigma (2)(k=1) (partial derivative/partial derivativex(k))(2) in the (x(1),x(2))-plane, is studied for a general multi-connected domain Ohm in R-2 surrounding by a simply connected bounded domains Ohm (j) with smooth boundaries a Omega (j) (j = 1,...,q), where a finite number of Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components Gamma (i) (i = 1 + k(j-l),...,k(j)) of partial derivative Omega (j), are considered where partial derivative Omega (j) = Ui=(kj)(l+kj-1) Gamma (i) with k(0) = 0. In this paper, one may extract information on the geometry of Omega by analyzing the asymptotic expansions of theta (t) for short-time t. Some applications of theta (t) for an ideal gas enclosed in Omega are given. Thermodynamic quantities of an ideal gas enclosed in Omega 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. (C) 2001 Elsevier Science Ltd. All rights reserved.