### 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.

Original language | English (US) |
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Pages (from-to) | 57-121 |

Number of pages | 65 |

Journal | Communications in Mathematical Physics |

Volume | 93 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1 1984 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Physics and Astronomy(all)
- Mathematical Physics

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## Cite this

*Communications in Mathematical Physics*,

*93*(1), 57-121. https://doi.org/10.1007/BF01218639