Mayer and Virial Series at Low Temperature

We analyze the Mayer and virial series (pressure as a function of the activity resp. the density) for a classical system of particles in continuous configuration space at low temperature. Particles interact via a finite range potential with an attractive tail. We propose physical interpretations of the Mayer and virial series’ radii of convergence, valid independently of the question of phase transition: the Mayer radius corresponds to a fast increase from very small to finite density, and the virial radius corresponds to a cross-over from monatomic to polyatomic gas. Our results are consistent with the Lee-Yang theorem for lattice gases and with the continuum Widom-Rowlinson model.     

On the analyticity of the pressure in the hierarchical dipole gas

The convergence of the Mayer expansion is proved by estimating directly the convergence radius.     

A Simple Inductive Approach to the Problem of Convergence of Cluster Expansions of Polymer Models

We explain a simple inductive method for the analysis of the convergence of cluster expansions (Taylor expansions, Mayer expansions) for the partition functions of polymer models. We give a very simple proof of the Dobrushin–Kotecký–Preiss criterion and formulate a generalization usable for situations where a successive expansion of the partition function has to be used.     

Decay of correlations for infinite range interactions in unbounded spin systems

In unbounded spin systems at high temperature with two-body potential we prove, using the associated polymer model, that the two-point truncated correlation function decays exponentially (respectively with a power law) if the potential decays exponentially (respectively with a power law). We also give a new proof of the convergence of the Mayer series for the general polymer model.Supported by C.N.R. (G.N.F.M.)     

On Lennard–Jones Type Potentials and Hard-Core Potentials with an Attractive Tail

We revisit an old tree graph formula, namely the Brydges–Federbush tree identity, and use it to get new bounds for the convergence radius of the Mayer series for gases of continuous particles interacting via non-absolutely summable pair potentials with an attractive tail including Lennard–Jones type pair potentials.     

A Correction to a Remark in a Paper by Procacci and Yuhjtman: New Lower Bounds for the Convergence Radius of the Virial Series

In this note we deduce a new lower bound for the convergence radius of the Virial series of a continuous system of classical particles interacting via a stable and tempered pair potential using the estimates on the Mayer coefficients obtained in the recent paper by Procacci and Yuhjtman (Lett Math Phys 107:31–46, 2017). This corrects the wrongly optimistic lower bound for the same radius claimed (but not proved) in the above cited paper (in Remark 2 below Theorem 1). The lower bound for the convergence radius of the Virial series provided here represents a strong improvement on the classical estimate given by Lebowitz and Penrose in 1964.     

Finite vs. Infinite Decompositions in Conformal Embeddings

We revisit two old and apparently little known papers by Basuev (Teoret Mat Fiz 37(1):130–134, 1978, Teoret Mat Fiz 39(1):94–105, 1979) 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 (Yuhjtman in J Stat Phys 160(6): 1684–1695, 2015) 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 10⁵ on the current best lower bound recently obtained in de Lima and Procacci (J Stat Phys 157(3):422–435, 2014).     

Computation of high-order virial coefficients in high-dimensional hard-sphere fluids by Mayer sampling

The Mayer sampling method was used to compute the virial coefficients of high-dimensional hard-sphere fluids. The first 64 virial coefficients for dimensions 12 < D ? 100 were obtained to high precision, and several lower dimensional virial coefficients were computed. The radii of convergence of the virial series in 13, 15, 17 and 19 dimensions agreed well with the analytical results from the Percus–Yevick closure.     

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

We establish new lower bounds for the convergence radius of the Mayer series and the Virial series of a continuous particle system interacting via a stable and tempered pair potential. Our bounds considerably improve those given by Penrose (J Math Phys 4:1312, 1963) and Ruelle (Ann Phys 5:109–120, 1963) for the Mayer series and by Lebowitz and Penrose (J Math Phys 7:841–847, 1964) for the Virial series. To get our results, we exploit the tree-graph identity given by Penrose (Statistical mechanics: foundations and applications. Benjamin, New York, 1967) using a new partition scheme based on minimum spanning trees.     

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

We provide a lower bound for the convergence radius of the Mayer series of the Lennard–Jones 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 22, 175–190 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.     

Derivation of Mayer Series from Canonical Ensemble

Mayer derived the Mayer series from both the canonical ensemble and the grand canonical ensemble by use of the cluster expansion method. In 2002, we conjectured a recursion formula of the canonical partition function of a fluid(X.Z. Wang, Phys. Rev. E66(2002) 056102). In this paper we give a proof for this formula by developing an appropriate expansion of the integrand of the canonical partition function. We further derive the Mayer series solely from the canonical ensemble by use of this recursion formula.     

Iterated Mayer expansion for classical gases at low temperatures

We derive iterated Mayer expansions for classical gases and establish recursive bounds which control their convergence. These bounds are useful for gases with two body forces which are strong and possibly attractive at distances that are short compared to their range. Our procedure is based on splitting the potential into pieces of decreasing strength and increasing range. This may be called a renormalization group treatment of a classical gas. We apply our results to Yukawa lattice gas models and obtain convergence of series expansions for the pressure for a range of parameters (temperature, fugacities, range of the interaction) that was inaccessible before. Application to 3-dimensionalU(1) lattice gauge theory (Coulomb gas,Z-ferromagnet) will be made elsewhere.Work supported in part by Deutsche Forschungsgemeinschaft     

Cluster Expansion in the Canonical Ensemble

We consider a system of particles confined in a box ${\Lambda \subset \mathbb{R}^d}$ interacting via a tempered and stable pair potential. We prove the validity of the cluster expansion for the canonical partition function in the high temperature - low density regime. The convergence is uniform in the volume and in the thermodynamic limit it reproduces Mayer??s virial expansion providing an alternative and more direct derivation which avoids the deep combinatorial issues present in the original proof.     

Coulomb interaction symmetries and the Mayer series in the two-dimensional dipole gas

We study the Mayer series of the two-dimensional dipole gas in the high-temperature, low-density regime. Without performing any multiscale analysis, we obtain bounds showing that the Mayer coefficients are finite in the thermodynamic limit. These bounds are obtained by exploiting a particular partial symmetry of the interaction (which we nameO-symmetry), already used in some problems related to the two-dimensional Coulomb gas. By direct bounds on some Mayer graphs we also conjecture that any technique based uniquely on theO-symmetry will not be sufficient to prove analyticity of the series.     

Mayer coefficients in two-dimensional coulomb systems

We show that, for neutral systems of particles of arbitrary charges in two dimensions, with hard cores, coefficients of the Mayer series for the pressure exist in the thermodynamic limit below certain thresholds in the temperature. Our methods apply also to correlation functions and yield bounds on the asymptotic behavior of their Mayer coefficients.     

From Natural History to the Nuclear Shell Model: Chemical Thinking in the Work of Mayer,Haxel, Jensen,and Suess

In 1949 the nuclear shell model was discovered simultaneously in the United States and Germany. Both discoveries were the result of a nuclear scientist looking at geochemical and nuclear data with the eyes of a chemist. Maria Goeppert Mayer in the United States and Hans Suess in Germany both brought a chemists perspective to the problem; the theoretical solution was subsequently supplied independently by Mayer and Hans Jensen.Karen E. Johnson is Priest Associate Professor at St.Lawrence University, where she teaches physics and history of science. She is currently writing a dual biography of Maria Goeppert Mayer and Joseph E.Mayer.     

Mayer expansions and the Hamilton-Jacobi equation

We review the derivation of Wilson's differential equation in (infinitely) many variables, which describes the infinitesimal change in an effective potential of a statistical mechanical model or quantum field theory when an infinitesimal integration out is performed. We show that this equation can be solved for short times by a very elementary method when the initial data are bounded and analytic. The resulting series solutions are generalizations of the Mayer expansion in statistical mechanics. The differential equation approach gives a remarkable identity for connected parts and precise estimates which include criteria for convergence of iterated Mayer expansions. Applications include the Yukawa gas in two dimensions past the=4 threshold and another derivation of some earlier results of Göpfert and Mack.     

Mayer function perturbation theory of effective interaction of charged colloids

In this paper, the effective interaction between charged colloids has been studied based on the standard Mayer function perturbation theory. With the formalism developed in this paper, the effective interaction as a function of Mayer functions and the correlation functions of the homogeneous microions is obtained. The asymptotic behaviour of the effective interaction at large distance is analysed in detail. It is found that at large distance the effective interaction is Yukawa like, provided the bare charge is replaced by the renormalised one. Exact expressions for the renormalised charge and the decay length as functions of the short-range part of the Mayer function and that of the correlation function of the homogeneous microions are obtained. With perturbation methods, it is easy to see how the effective interaction at large distance is affected by microion correlations and nonlinearity.     

Continuous Particles in the Canonical Ensemble as an Abstract Polymer Gas

We revisit the expansion recently proposed by Pulvirenti and Tsagkarogiannis for a system of N continuous particles in the Canonical Ensemble. Under the sole assumption that the particles interact via a tempered and stable pair potential and are subjected to the usual free boundary conditions, we show the analyticity of the Helmholtz free energy at low densities and, using the Penrose tree graph identity, we establish a lower bound for the convergence radius which happens to be identical to the lower bound of the convergence radius of the virial series in the Grand Canonical ensemble established by Lebowitz and Penrose in 1964. We also show that the free energy can be written as a series in powers of the density whose k-th order coefficient coincides, modulo terms o(N)/N, with the k-th order virial coefficient divided by k+1, according to its expression in terms of the m-th order (with m≤k+1) simply connected cluster integrals first given by Mayer in 1942. We finally give an upper bound for the k-th order virial coefficient which slightly improves, at high temperatures, the bound obtained by Lebowitz and Penrose.     

On the gas-liquid phase transition

A new approach to the problem of the gas-liquid phase transition, based on the Mayer cluster expansion of the partition function, is proposed. It is shown that the necessary and sufficient condition for phase transition to occur is that there exist a temperatureT= T_c > 0 such that forT T _c, all theb _l (except perhaps a finite number of them) are positive, where theb _l, are the cluster integrals (as defined by Mayer) in the thermodynamic limit. Explicit expressions for the isotherms for gas-saturated vapor and liquid phases are given.