Abstract
The fourth virial coefficient is calculated exactly for a fluid of hard spheres in even dimensions. For this purpose the complete star cluster integral is expressed as the sum of two three-folded integrals only involving spherical angular coordinates. These integrals are solved analytically for any even dimension d, and working with existing expressions for the other terms of the fourth cluster integral, we obtain an expression for the fourth virial coefficient \(B_{4}(d)\) for even d. It reduces to the sum of a finite number of simple terms that increases with d.
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Acknowledgements
I am grateful to an anonymous reviewer for the suggestion of the Eq. (60) that describes the asymptotic behavior of \(B_{4}\) and to the other anonymous reviewer for his comment about the Conjecture done above.
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Appendices
Appendix A: Explicit Form of
To evaluate this Mayer diagram we transform Eq. (8) as
with \(H=\intop _{0}^{\arccos x}(\sin \gamma )^{2\nu }d\gamma \), and also
where we used Eq. (C1). We integrate by parts to obtain
where the first term on the right gives \(\frac{2^{-2\nu }}{2\nu }\left\{ \frac{(2\nu -1)!!}{2^{\nu }\nu !}\frac{\pi }{3}-\frac{\sqrt{3}}{6\nu } \left[ \left( \frac{3}{4}\right) ^{\nu }+\sum _{k=1}^{\nu -1} \frac{\prod _{i=1}^{k}(2\nu -2i+1)}{2^{k}\prod _{i=1}^{k}(\nu -i)} \left( \frac{3}{4}\right) ^{\nu -k}\right] \right\} ^{2}\). To solve the remaining integral on the right of Eq. (A1) we change variable by replacing \(x=\cos y\) to obtain
Here, there are terms that can be written using \(D\left( 2l+1,2k+1\right) \) analyzed in Eq. (39), and a term \(\intop _{\frac{\pi }{3}}^{\frac{\pi }{2}} y\left( \sin y\cos y\right) ^{2\nu }dy\) \(=2^{-2\nu -2}\intop _{\frac{2\pi }{3}}^{\pi }y\left( \sin y\right) ^{2\nu }dy\). This last integral is solved using Eq. (C2) and gives
Thus, one finds
The final result is given in Eq. (11).
Appendix B: Origin of \(\cos \varphi \) Term
Here we derive the identity Eq. (21) showing the origin of the \(\cos \varphi \) term. Let us start from Eq. (17)
we apply the derivative \(\frac{\partial }{\partial \sigma }\) to both sides, to obtain
Here the dashed bond corresponds to \(\delta '(\sigma -r)\), being \(f''=\frac{\partial ^{2}f}{\partial ^{2}r}=-\delta '(\sigma -r)\). Some terms on the left are readily integrated on, and . For the diagram with a dashed bond we have
We replace Eq. (B2) in Eq. (B1) to obtain
Appendix C: Integration of the Different Terms in L
Here we solve the split terms of L (see Eq. (35)), the four integrals: \(\left( \sqrt{3}K+\frac{4\pi }{3}\right) \int _{0}^{\frac{\pi }{3}} \sin \left( \varphi \right) ^{2m+2}d\varphi \), \(-\left( \frac{\sqrt{3}}{2}K+\frac{2\pi }{3}\right) \int _{0}^{\frac{\pi }{3}}\cos \varphi \left( \sin \varphi \right) ^{2m+2}d\varphi \), \(-2\int _{0}^{\frac{\pi }{3}}\varphi \left( \sin \varphi \right) ^{2m+2}d\varphi \), and \(\int _{0}^{\frac{\pi }{3}}\varphi \cos \varphi \left( \sin \varphi \right) ^{2m+2}d\varphi \). To solve the first one we useFootnote 5
to obtain
The second one is straightforward, it gives
To solve the third one, \(-2\int _{0}^{\frac{\pi }{3}}\varphi \left( \sin \varphi \right) ^{2m+2}d\varphi \), we applyFootnote 6
iteratively, to obtain
and finally
To solve the fourth one we useFootnote 7
In particular, for \(q=1\) it reduces to
which gives the result
To solve the last integral we used Eq. (38) (with \(q=0\)) to obtain
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Urrutia, I. The Fourth Virial Coefficient for Hard Spheres in Even Dimension. J Stat Phys 187, 29 (2022). https://doi.org/10.1007/s10955-022-02913-7
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DOI: https://doi.org/10.1007/s10955-022-02913-7