Communications in Mathematical Physics

, Volume 17, Issue 3, pp 194–209 | Cite as

The van der Waals limit for classical systems

III. Deviation from the van der Waals-Maxwell theory
  • D. J. Gates
  • O. Penrose


We examine the limiting free energy density\(a(\varrho ,0 + ) \equiv \mathop {\lim }\limits_{\gamma \to 0} a(\varrho ,\gamma )\) of a classical system of particles with the two-body potentialq(r)+γ v Kr), at density ϱ inv dimensions. Starting from a variational formula fora(ϱ, 0 + ), obtained in Part I of these papers, we obtain a new upper bound ona(ϱ, 0 + ) given by
$$a(\varrho ,0 + ) \mathbin{\lower.3ex\hbox{$\buildrel<\over{\smash{\scriptstyle=}\vphantom{_x}}$}} CE\{ ME[a^0 (\varrho ) + {\textstyle{1 \over 4}}\tilde K_{\min } \varrho ^2 ] + ({\textstyle{1 \over 2}}\alpha - {\textstyle{1 \over 4}}\tilde K_{\min } )\varrho ^2 \} $$
. HereM E f, called the mid-point envelope off, is defined for any functionf by
$$MEf(\varrho ) \equiv \mathop {\inf }\limits_h {\textstyle{1 \over 2}}[f(\varrho + h) + f(\varrho - h)];$$
C E f, called the convex envelope off, is defined for anyf as the maximal convex function not exceedingf; also α ≡ εdsK(s) and\(\tilde K_{\min } \) is the minimum of the Fourier transform ofK, whilea0(ϱ) is the free energy density forK = 0.
For the class of functionsK such that\(\tilde K_{\min } \) < 0 and\(\tilde K_{\min } \) <2α, we deduce from this upper bound thata(ϱ, 0 + ) <C E[a0(ϱ) + 1/2αϱ2] for all values of ϱ wherea0(ϱ) + 1/2αϱ2 differs from its convex envelope, or where\(a^0 (\varrho ) + {\textstyle{1 \over 4}}\tilde K_{\min } \varrho ^2 \) differs from its mid-point envelope. Consequently, the generalized van der Waals equation
$$a^0 (\varrho ,0 + ) = CE[a^0 (\varrho ) + {\textstyle{1 \over 2}}\alpha \varrho ^2 ]$$
does not apply in this case. We prove that in a certain sense the local density is non-uniform over distances of order γ−1 in this case, and infer that this density is periodic.

We also give a simpler derivation of other bounds ona(ϱ, 0 + ) obtained by Lebowitz and Penrose.


Neural Network Fourier Free Energy Fourier Transform Statistical Physic 
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Copyright information

© Springer-Verlag 1970

Authors and Affiliations

  • D. J. Gates
    • 1
  • O. Penrose
    • 2
  1. 1.Mathematics DepartmentImperial CollegeLondonEngland
  2. 2.The Open UniversityBletchleyEngland

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