The Mayer Series of the Lennard–Jones Gas: Improved Bounds for the Convergence Radius

Abstract

We provide a lower bound for the convergence radius of the Mayer series of the LennardJones gas which strongly improves on the classical bound obtained by Penrose and Ruelle 1963. To obtain this result we use an alternative estimate recently proposed by Morais et al. (J. Stat. Phys. 2014) for a restricted class of stable and tempered pair potentials (namely those which can be written as the sum of a non-negative potential plus an absolutely integrable and stable potential) combined with a method developed by Locatelli and Schoen (J. Glob. Optim. 2002) for establishing a lower bound for the minimal interatomic distance between particles interacting via a Morse potential in a cluster of minimum-energy configurations.

On Stable Pair Potentials with an Attractive Tail, Remarks on Two Papers by A. G. Basuev

“…In particular, we show how these results yield improvements on all known results for a wide class of pair potentials, focusing in particular on the important and physically relevant subclass of the Lennard Jones type potentials. We also provide new results for the specific case of the classical Lennard-Jones potential V (r) = 1/r 12 − 2/r 6 which strongly improve on the results given by two of us in a recent paper [19]. To get this last result on the specific case of the Lennard-Jones potential, we used the very recent results obtained by one of us [38] on the stability constant and minimal inter-particle distance in lowest energy configurations of the Lennard-Jones gas.…”

“…In section 4 we (re)state the Basuev criterion (Theorem 4), whose proof is given in Appendix B, and (re)derive from the Basuev tree graph identity the bounds on the Ursell coefficient for particle systems interacting via a Basuev pair potential. Finally, in Section 5 we present the estimates of the Mayer coefficients and convergence radius for Lennard-Jones type potentials and classical Lennard-Jones potential, showing how this estimates improve on the recent bounds obtained in [24] and [19].…”

“…In this final section we perform a comparison between bounds (4.3) and (4.5) deduced from the Basuev Tree-graph identity and the recent bounds for Lennard-Jones type potentials given in [24] and [19]. Let us first introduce the standard definition of a Lennard-Jones type potential (a.k.a.…”

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We revisit two old and apparently little known papers by Basuev [3] [4] and show that the results contained there yield strong improvements on current lower bounds of the convergence radius of the Mayer series for continuous particle systems interacting via a very large class of stable and tempered potentials which includes the Lennard-Jones type potentials. In particular we analyze the case of the classical Lennard-Jones gas under the light of the Basuev scheme and, using also some new results [38] on this model recently obtained by one of us, we provide a new lower bound for the Mayer series convergence radius of the classical Lennard-Jones gas which improves by a factor of the order 10 5 on the current best lower bound recently obtained in [19].

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Convergence of Mayer and Virial expansions and the Penrose tree-graph identity

“…As observed in Section 2, the improvement of Theorem 1 respect to the old Penrose-Ruelle bound is manifestly evident, as far as general stable and tempered pair potentials are concerned. There have been however recent works [14,10,11] which also obtain improvements on the convergence radius of the Mayer series for systems of particles interacting via a restricted class of stable and tempered pair potentials. In particular, in a recent paper [11], which is a development of an early work by Basuev [2], the authors obtain new bounds for the Mayer series convergence radius which are better than any precedent bound given in the literature (see in [11] the new bounds of Theorem 5, formula (4.5)), as far as the particles in the system interact via a so-called Basuev potential (see Definition 3 in [11]).…”