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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
Article

Abstract

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.

Keywords

Neural Network Fourier Free Energy Fourier Transform Statistical Physic 
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.

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References

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    Kac, M., Uhlenbeck, G. E., Hemmer, P. C.: J. Math. Phys.4, 216 (1963).Google Scholar
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    Lebowitz, J. L., Penrose, O. (LP): J. Math. Phys.7, 98 (1966).Google Scholar
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    Gates, D. J., Penrose O., (Part I): Commun. Math. Phys.15, 255 (1969).Google Scholar
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    Royden, H. L.: Real analysis, Proposition 17, p. 108, 2. Ed. New York-London: Macmillan 1968.Google Scholar
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    Hardy, G., Littlewood, J. E., Polya, G.: Inequalities. London: Cambridge University Press 1959.Google Scholar

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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