E.M.E. ZAYED
应用数学学报(英文版). 2004, 20(2): 215-230.
The asymptotic expansion of the heat kernel $\Theta
(t)=\sum\limits_{j=1}^\infty \exp (-t\lambda \Sb \\ j \endSb )$
where $\{\lambda \Sb \\ j \endSb \}\Sb
\\ j=1 \endSb ^\infty $ are the eigenvalues of the negative Laplacian $%
-\Delta \Sb \\ n \endSb =-\sum\limits_{k=1}^n(\frac \partial
{\partial x^k})^2$ in $R^n(n=2$ or $3)$ is studied for short-time
$t$ for a general bounded domain $\Omega $ with a smooth boundary
$\partial \Omega .$ In this paper, we consider the case of a
finite number of the Dirichlet conditions $\phi =0$
on $\Gamma \Sb \\ i \endSb \,\,(i=1,...,J)$ and the Neumann conditions $%
\frac{\partial \phi }{\partial \upsilon \Sb \\ i \endSb }=0$ on
$\Gamma \Sb
\\ i \endSb \,\,(i=J+1,\cdots ,k)$ and the Robin conditions $(\frac \partial
{\partial \upsilon \Sb \\ i \endSb }+\gamma \Sb \\ i \endSb
)\phi =0$ on $\Gamma \Sb \\ i \endSb \,\,(i=k+1,\cdots ,m)$
where $\gamma \Sb \\ i \endSb $ are piecewise smooth positive
impedance functions, such that $\partial \Omega $ consists of a
finite number of piecewise smooth components $\Gamma \Sb \\ i
\endSb \,\,(i=1,\cdots ,m)$ where $\partial \Omega
=\bigcup\limits_{i=1}^m\Gamma \Sb \\ i \endSb .$ 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 $\Omega .$ 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.
