Navrotskaya, Irina
(2016)
Inverse problem in classical statistical mechanics.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
Abstract
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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Details
| Item Type: |
University of Pittsburgh ETD
|
| Status: |
Unpublished |
| Creators/Authors: |
| Creators | Email | Pitt Username | ORCID  |
|---|
| Navrotskaya, Irina | irn6@pitt.edu | IRN6 | |
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| ETD Committee: |
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| 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 |
| URI: |
http://d-scholarship.pitt.edu/id/eprint/28992 |
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