# The van der Waals limit for classical systems

III. Deviation from the van der Waals-Maxwell theory

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## 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 potential. Here

*q*(**r**)+γ^{ v }*K*(γ**r**), at density ϱ in*v*dimensions. Starting from a variational formula for*a*(ϱ, 0 + ), obtained in Part I of these papers, we obtain a new upper bound on*a*(ϱ, 0 + ) given by$$a(\varrho ,0 + ) \mathbin{\lower.3ex\hbox{$\buildrel<\over{\smash{\scriptstyle=}\vphantom{_x}}$}} CE\{ ME[a^0 (\varrho ) + {\textstyle{1 \over 4}}\tilde K_{\min } \varrho ^2 ] + ({\textstyle{1 \over 2}}\alpha - {\textstyle{1 \over 4}}\tilde K_{\min } )\varrho ^2 \} $$

*M E f*, called the mid-point envelope of*f*, is defined for any function*f*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 of*f*, is defined for any*f*as the maximal convex function not exceeding*f*; also α ≡ ε*d***s***K*(**s**) and\(\tilde K_{\min } \) is the minimum of the Fourier transform of*K*, while*a*^{0}(ϱ) is the free energy density for*K*= 0.For the class of functions does not apply in this case. We prove that in a certain sense the local density is non-uniform over distances of order γ

*K*such that\(\tilde K_{\min } \) < 0 and\(\tilde K_{\min } \) <2α, we deduce from this upper bound that*a*(ϱ, 0 + ) <*C E*[*a*^{0}(ϱ) + 1/2αϱ^{2}] for all values of ϱ where*a*^{0}(ϱ) + 1/2αϱ^{2}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$$a^0 (\varrho ,0 + ) = CE[a^0 (\varrho ) + {\textstyle{1 \over 2}}\alpha \varrho ^2 ]$$

^{−1}in this case, and infer that this density is periodic.We also give a simpler derivation of other bounds on*a*(ϱ, 0 + ) obtained by Lebowitz and Penrose.

## Keywords

Neural Network Fourier Free Energy Fourier Transform Statistical Physic
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## References

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© Springer-Verlag 1970