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Inverse problem in classical statistical mechanics

Navrotskaya, Irina (2016) Inverse problem in classical statistical mechanics. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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This thesis concentrates on the inverse problem in classical statistical mechanics and its applications. Let us consider a system of identical particles with the total energy W + U, where W is a fixed scalar function, and V is an additional internal or external potential in the form of a sum of m-particle interactions u. The inverse conjecture states that any positive, integrable function ρ(m) is the equilibrium m-particle density corresponding to some unique potential u. It has been proved for all m ≥ 1 in the grand canonical ensemble by Chayes and Chayes in 1984. Chapter 2 of this thesis contains the proof of the inverse conjecture for m ≥ 1 in the canonical formulation. For m = 1, the inverse problem lies at the foundation of density functional theory for inhomogeneous fluids. More generally, existence and differentiability of the inverse map for m ≥ 1 provides the basis for the variational principle on which generalizations to density functional theory can be formulated. Differentiability of the inverse map in the grand canonical ensemble for m ≥ 1 is proved here in Section 3.2. In particular, this result leads to the existence of a hierarchy of generalized Ornstein-Zernike equations connecting the 2m-,...,m-particle densities and generalized direct correlation functions. This hierarchy is constructed in Section 3.3.


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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
Navrotskaya, Irinairn6@pitt.eduIRN6
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairManfredi, Juan
Committee CoChairSwigon,
Committee MemberHajlasz,
Committee MemberCoalson,
Date: 3 October 2016
Date Type: Publication
Defense Date: 16 June 2016
Approval Date: 3 October 2016
Submission Date: 28 July 2016
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Number of Pages: 73
Institution: University of Pittsburgh
Schools and Programs: Dietrich School of Arts and Sciences > Mathematics
Degree: PhD - Doctor of Philosophy
Thesis Type: Doctoral Dissertation
Refereed: Yes
Uncontrolled Keywords: statistical mechanics, measure theory, functional analysis
Date Deposited: 03 Oct 2016 13:52
Last Modified: 15 Nov 2016 14:35


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