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Communications in Mathematical Physics

, Volume 93, Issue 1, pp 57–121 | Cite as

The inverse problem in classical statistical mechanics

  • J. T. Chayes
  • L. Chayes
  • Elliott H. Lieb
Article

Abstract

We address the problem of whether there exists an external potential corresponding to a given equilibrium single particle density of a classical system. Results are established for both the canonical and grand canonical distributions. It is shown that for essentially all systems without hard core interactions, there is a unique external potential which produces any given density. The external potential is shown to be a continuous function of the density and, in certain cases, it is shown to be differentiable. As a consequence of the differentiability of the inverse map (which is established without reference to the hard core structure in the grand canonical ensemble), we prove the existence of the Ornstein-Zernike direct correlation function. A set of necessary, but not sufficient conditions for the solution of the inverse problem in systems with hard core interactions is derived.

Keywords

Neural Network Correlation Function Inverse Problem Statistical Mechanic Single Particle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • J. T. Chayes
    • 1
  • L. Chayes
    • 1
  • Elliott H. Lieb
    • 2
  1. 1.Department of PhysicsPrinceton UniversityPrincetonUSA
  2. 2.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA

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