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Cluster and Virial Expansions for the Multi-species Tonks Gas

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Abstract

We consider a mixture of non-overlapping rods of different lengths \(\ell _k\) moving in \(\mathbb {R}\) or \(\mathbb {Z}\). Our main result are necessary and sufficient convergence criteria for the expansion of the pressure in terms of the activities \(z_k\) and the densities \(\rho _k\). This provides an explicit example against which to test known cluster expansion criteria, and illustrates that for non-negative interactions, the virial expansion can converge in a domain much larger than the activity expansion. In addition, we give explicit formulas that generalize the well-known relation between non-overlapping rods and labelled rooted trees. We also prove that for certain choices of the activities, the system can undergo a condensation transition akin to that of the zero-range process. The key tool is a fixed point equation for the pressure.

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Notes

  1. The criterion (1) refers to the pressure-activity expansion \(p(\varvec{z})\). The convergence of the density-activity expansion \(\rho _k(\varvec{z})\) in general requires a strict inequality \(\sum _k|z_k|\exp ( a\ell _k) <a\). The same remark applies to the discrete system.

  2. This point of view focuses on qualitative features of the domain of convergence for objects of unbounded size (\(\ell _k\rightarrow \infty \)). A different question is about quantitative estimates on the radius of convergence for hard-sphere systems of bounded size, see e.g. [13].

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Acknowledgments

I am indebted to R. Fernández, S. J. Tate, D. Tsagkarogiannis and D. Ueltschi for many helpful discussions, and to A. van Enter for pointing out connections with the Fisher–Felderhof clusters. This work was completed during a stay at the Institute for Computational and Experimental Research in Mathematics (ICERM), Brown University, for the semester program “Phase Transitions and Emergent Properties.”

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  1. Fakultät für Mathematik, Ruhr-Universität Bochum, 44780, Bochum, Germany

    S. Jansen

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Jansen, S. Cluster and Virial Expansions for the Multi-species Tonks Gas. J Stat Phys 161, 1299–1323 (2015). https://doi.org/10.1007/s10955-015-1367-x

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