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

作     者：E.M.E.ZAYED 

作者机构：Mathematics DepartmentFaculty of ScienceZagazig UniversityZagazigEgypt 

出 版 物：《Acta Mathematicae Applicatae Sinica》 (应用数学学报（英文版）)

年 卷 期：2004年第20卷第2期

页      面：215-230页

核心收录：

学科分类：07[理学] 070102[理学-计算数学] 0701[理学-数学] 

主　　题：Inverse problem heat kernel Eigenvalues short-time asymptotics special ideal gas one-particle partition function 

摘      要：The asymptotic expansion of the heat kernel Θ(t)=sum from ∞ to j=1 exp(-tλ_j) where {λ_j}_(j=1)~∞ are the eigen-values of the negative Laplacian -Δ_n=-sum from n to k=1((?))~2 in R^n(n=2 or 3) is studied for short-time t for a generalbounded domain Ω with a smooth boundary (?)Ω.In this paper,we consider the case of a finite number of theDirichlet conditions φ=0 on Γ_i (i=1,...,J) and the Neumann conditions (?)=0 on Γ_i (i=J+1,...,k) andthe Robin conditions ((?)+γ_i)φ=0 on Γ_i (i=k+1,...,m) where γ_i are piecewise smooth positive impedancefunctions,such that (?)Ω consists of a finite number of piecewise smooth components Γ_i(i=1,...,m) where(?)Ω=(?)Γ_*** construct the required asymptotics in the form of a power series over *** senior coefficients inthis series are specified as functionals of the geometric shape of the domain Ω.This result is applied to calculatethe one-particle partition function of a“special ideal gas,i.e.,the set of non-interacting particles set up in abox with Dirichlet,Neumann and Robin boundary conditions for the appropriate wave *** the thermodynamic quantities for the ideal gas such as the internal energy,pressure and specific heat revealsthat these quantities alone are incapable of distinguishing between two different shapes of the *** seems to be intuitively clear because it is based on a limited information given by a one-particlepartition function;nevertheless,its formal theoretical motivation is of some interest.