Graduate Thesis Or Dissertation
 

Probability densities and correlation functions in statistical mechanics

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  • The work in this thesis falls into two parts. The first part presents a rigorous theory of probability densities and correlation functions within the framework of the exponential spaces of D. S. Carter. The second part extends this discussion to include infinite systems. The first part begins by considering an algebra of complex-valued functions defined on the exponential of a set. Multiplication in this algebra corresponds to the star product used by Schwartz and Ruelle. A functional calculus is defined on this algebra, which provides for the discussion of Ursell functions. By introducing a special subspace of integrable functions, a systematic theory is obtained, in which probability densities and correlation functions belonging to this-subspace are related by a pair of mutually inverse integral operators. The second part deals with various ways of representing the statistical state of an infinite system. The existence of states of an infinite system is established as limits of a special class of states of finite systems. The equilibrium states considered by Ruelle are shown to be included in this class.
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