Communications in Mathematical Physics

, Volume 40, Issue 3, pp 197–213 | Cite as

An upper bound on the free energy for classical systems with Coulomb interactions in a varying external field

  • E. R. Smith
  • O. Penrose


The thermodynamic limit is taken using a sequence of regions all the same shape as a given region ω of volume |ω|, with a specified distribution of normal field component on ∂ω. We show that with magnetostatic interactions the limiting free energy density is bounded above by jhen where\(\bar f\)(ϱ,B) is the free energy density for a system of density ϱ in a uniform external fieldB and the “inf” is taken over all divergence-free fieldsB with given normal component on ∂ω and all densities ϱ(x) compatible with particle number constraints of the form\(\int\limits_{\Gamma _i } {\varrho (x)d^3 x = \left| {\Gamma _i } \right|\varrho _i } \) where Γi is a sub-region of ω. A physical argument suggests that this upper bound is the true thermodynamic limit, and that it takes account demagnetization effects. Electrostatic interactions can be treated similarly.


Neural Network Free Energy Energy Density Electrostatic Interaction Nonlinear Dynamics 
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Copyright information

© Springer-Verlag 1975

Authors and Affiliations

  • E. R. Smith
    • 1
    • 2
  • O. Penrose
    • 1
    • 2
  1. 1.Mathematics DepartmentUniversity of NewcastleAustralia
  2. 2.Mathematics DepartmentThe Open UniversityMilton KeynesEngland

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