Short-time Asymptotics of the Heat Kernel on Bounded Domain with Piecewise Smooth Boundary Conditions and Its Applications to an Ideal Gas

Abstract

The asymptotic expansion of the heat kernel $$ \Theta {\left( t \right)} = {\sum\limits_{j = 1}^\infty {\exp {\left( { - t\lambda _{j} } \right)}} } $$ where $$ {\left\{ {\lambda _{j} } \right\}}^{\infty }_{{j = 1}} $$ are the eigenvalues of the negative Laplacian $$ - \Delta _{n} = - {\sum\limits_{k = 1}^n {{\left( {\frac{\partial } {{\partial x^{k} }}} \right)}} }^{2} $$ in R n (n = 2 or 3) is studied for short-time t for a general bounded domain Ω with a smooth boundary $$ \partial \Omega $$ . In this paper, we consider the case of a finite number of the Dirichlet conditions φ = 0 on Γ i (i = 1, ..., J) and the Neumann conditions $$ \frac{{\partial \phi }} {{\partial \upsilon _{i} }} = 0 $$ on Γ i (i = J +1, · · · , k) and the Robin conditions $$ {\left( {\frac{\partial } {{\partial \upsilon _{i} }} + \gamma _{i} } \right)}\phi = 0 $$ on Γ i (i = k+1, · · ·,m) where γ i are piecewise smooth positive impedance functions, such that $$ \partial \Omega $$ consists of a finite number of piecewise smooth components Γ i (i = 1, · · ·,m) where $$ \partial \Omega = {\bigcup\limits_{i = 1}^m {\Gamma _{i} } } $$ . We construct the required asymptotics in the form of a power series over t. The senior coefficients in this series are specified as functionals of the geometric shape of the domain Ω. This result is applied to calculate the one-particle partition function of a “special ideal gas”, i. e., the set of non-interacting particles set up in a box with Dirichlet, Neumann and Robin boundary conditions for the appropriate wave function. Calculation of the thermodynamic quantities for the ideal gas such as the internal energy, pressure and specific heat reveals that these quantities alone are incapable of distinguishing between two different shapes of the domain. This conclusion seems to be intuitively clear because it is based on a limited information given by a one-particle partition function; nevertheless, its formal theoretical motivation is of some interest.