Virial Coefficients Expressed By Heat Kernel Coefficients

Virial expansion is an important method to study the effects of interactions on the thermodynamic properties of multi-particle systems.In this thesis,we first discuss the relationship between the virial coefficient of the ideal quantum gas and the heat kernel coefficient of the corresponding Laplace operator,and give the expression of the virial coefficient represented by the heat kernel coefficient.According to the results,we find that when the heat kernel expansion contains half-integer terms,the term that has the greatest influence on the virial coefficient comes from the heat kernel coefficient B_1/2,and its contribution is proportional to?¹.When the heat kernel expansion does not contain half-integer terms,its maximum influence on the virial coefficient is?⁴.Then,using the general results of the virial coefficients,we discuss the effects of boundaries and curvature on the properties of quantum gases.As examples,we discuss the virial coefficients of d-dimensional confined space with boundary,the smooth curved space and the d-dimensional constant curvature space,we also find a condition that two sys-tems with diffferent heat kernels have the same virial expansion.Our results show that the relative modification of the boundary to the second virial coefficient is independent of the dimension,and it always enhances the interaction of quantum exchange under the Dirichlet boundary conditions.However,in the d-dimensional spheres,the effect of curvature enhances the quantum exchange interaction in two dimensions,but weakens it in higher dimensions(d>3).Then in the discussion of the dimensional expansion of the curved space,we find that the main corrections to all of the virial coefficients are proportional to the factor B₁²-2V B₂,so we specifically discuss the approximate equiv-alence condition of quantum gases in different systems.We give the specific conditions for flat space with an external potential between free flat space and sphere,respectively.Finally,we further use a different method to obtain the explicit expression of the virial expansion of the ideal Bose gas,and discuss the applicability range of the virial expan-sion.The virial expansion is often regarded as a near-classical approximation,but our results show that its scope of application is wider than the common understanding.In fact,to require that the virial expansion must satisfy the condition n?^d?1 is unnec-essary.Our calculation shows that even if n?^dis around 1,the accuracy of the virial expansion is high and the error is small.