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.
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Communicated by A. Jaffe
Work partially supported by NSF grant PHY-8117463
Work partially supported by NSF grant PHY-8116101 A01
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Chayes, J.T., Chayes, L. & Lieb, E.H. The inverse problem in classical statistical mechanics. Commun.Math. Phys. 93, 57–121 (1984). https://doi.org/10.1007/BF01218639
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DOI: https://doi.org/10.1007/BF01218639