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


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.


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Lieb, E.H.: Density functionals for Coulomb systems. In: Physics as Natural Philosophy: Essays in Honor of Laszlo Tisza on His 75th Birthday, Shimony, A., Feshback, H. (eds.) Cambridge: M.I.T. Press 1982, p. 111Google Scholar
  2. 2.
    Levy, M.: Electron densities in search of Hamiltonians. Phys. Rev.26 A, 1200 (1982)Google Scholar
  3. 3.
    Englisch, H., Englisch, R.: Hohenberg-Kohn theorem and non-V-representable densities. Physica A (in press)Google Scholar
  4. 4.
    Hugenholtz, N.M.: On the inverse problem in statistical mechanics. Commun. Math. Phys.85, 27 (1982)Google Scholar
  5. 5.
    Evans, R.: The nature of the liquid-vapour interface and other topics in the statistical mechanics of non-uniform fluids. Adv. Phys.28, 143 (1979)Google Scholar
  6. 6.
    Percus, J.K.: Equilibrium state of a classical fluid of hard rods in an external field. J. Stat. Phys.15, 505 (1976)Google Scholar
  7. 7.
    Percus, J.K.: One-dimensional classical fluid with nearest-neighbor interaction in arbitrary external field. J. Stat. Phys.28, 67 (1982)Google Scholar
  8. 8.
    Robledo, A., Varea, C.: On the relationship between the density functional formalism and the potential distribution theory for nonuniform fluids. J. Stat. Phys.26, 513 (1981)Google Scholar
  9. 9.
    Mazur, S.: Über die kleinste konvexe Menge, die eine gegebene kompakte Menge enthält. Studia Math.2, 7 (1930)Google Scholar
  10. 10.
    Ruelle, D.: Statistical mechanics. Reading, MA.: W. A. Benjamin, Inc. 1969Google Scholar

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

Personalised recommendations