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     

Microstructure characterization and bulk properties of disordered two-phase media

A general formalism is developed to statistically characterize the microstructure of porous and other composite media composed of inclusions (particles) distributed throughout a matrix phase (which, in the case of porous media, is the void phase). This is accomplished by introducing a new and generaln-point distribution functionH _n and by deriving two series representations of it in terms of the probability density functions that characterize the configuration of particles; quantities that, in principle, are known for the ensemble under consideration. In the special case of an equilibrium ensemble, these two equivalent but topologically different series for theH _n are generalizations of the Kirkwood-Salsburg and Mayer hierarchies of liquid-state theory for a special mixture of particles described in the text. This methodology provides a means of calculating any class of correlation functions that have arisen in rigorous bounds on transport properties (e.g., conductivity and fluid permeability) and mechanical properties (e.g., elastic moduli) for nontrivial models of two-phase disordered media. Asymptotic and bounding properties of the general functionH _n are described. To illustrate the use of the formalism, some new results are presented for theH _n and it is shown how such information is employed to compute bounds on bulk properties for models of fully penetrable (i.e., randomly centered) spheres, totally impenetrable spheres, and spheres distributed with arbitrary degree of impenetrability. Among other results, bounds are computed on the fluid permeability, for assemblages of impenetrable as well as penetrable spheres, with heretofore unattained accuracy.     

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).     

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.     

Rigorous estimates of general Mayer graphs and applications: I: Uniform upper bounds

We obtain new upper bounds on Mayer graphs with n root-points. These bounds are an improvement over those obtained by Groeneveld. They work also for some classes of graphs occuring in the theory of ionized and polar systems, where the Groeneveld bounds fail entirely.     

On the ν-dimensional one-component classical plasma: The thermodynamic limit problem revisited

We prove theH-stability property and the existence of the thermodynamic limit of the free energy density of the two-dimensional, one-component classical plasma. We give lower and upper bounds on the free energy density in any dimensionv and draw some consequences.     

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.     

Binary ionic mixtures and a solvable model

The theory of solutions of McMillan and Mayer is applied to the jellium model of a binary ionic mixture: two species of charged particles, with charges e and Ze, immersed in a neutralizing background. The density ρ₂ of the particles of charge Ze is considered as small, and is used as an expansion parameter. The free energy, the pair distribution functions, the internal energy, and the pressure of the mixture are expressed as power series in ρ₂; the coefficients are integrals of correlation functions defined in the system at ρ₂=0 (the reference system). Explicit expressions are obtained in the two-dimensional case, at a special temperature, since in that case the reference system (the two-dimensional, one-component plasma) is a solvable model.     

The Gas Phase of Continuous Systems of Hard Spheres Interacting via $n$-Body Potential

We show that a system of classical continuous hard spheres interacting through a general n body potential satisfying suitable integrability conditions, admits a high temperature-low activity gas phase. We find explicitly a condition on the activity λ and the inverse temperature β which ensures that the Mayer series for the pressure is absolutely convergent uniformly in the volume. Received: 9 September 1999 / Accepted: 4 January 2000     

Convergence of Mayer expansions

The tree graph bound of Battle and Federbush is extended and used to provide a simple criterion for the convergence of (iterated) Mayer expansions. As an application estimates on the radius of convergence of the Mayer expansion for the two-dimensional Yukawa gas (nonstable interaction) are obtained.     

The “screening phase transitions” in the two-dimensional Coulomb gas

The Mayer series of a Coulomb gas with fixed ultraviolet cutoff is studied in two dimensions. In particular, we show the existence of infinitely many thresholdsT _n =(e ²/8k)(1-1/2n)^–1 k=Boltzmann's constant,e=electric charge,n=1, 2,..., which are conjectured to reflect a sequence of transitions from pure multipole phase (the Kosterlitz-Thouless region) to a plasma phase (the Debye screening region) via an infinite number of intermediate phases. Mathematically we prove that the Mayer series' coefficients of order up to 2n are finite if the temperatureT is <T _n. ForT<T all the coefficients are finite and the gas can be formally interpreted as a multipole gas with multipoles with finite activity.The first author was supported in part by NSF grant No. MCS-8108814 (A03).     

On the absence of intermediate phases in the two-dimensional Coulomb gas

For a simple, continuum two-dimensional Coulomb gas (with soft cutoff), Gallavotti and Nicoló [J. Stat. Phys. 38:133–156 (1985)] have proved the existence of finite coefficients in the Mayer activity expansion up to order 2n below a series of temperature thresholdsT _n =T [1+(2n–1)^–1] (n=1, 2,...). With this in mind they conjectured that an infinite sequence of intermediate, multipole phases appears between the exponentially screened plasma phase aboveT ₁ and the full, unscreened Kosterilitz-Thouless phase belowT T _KT. We demonstrate that Debye-Hückel-Bjerrum theory, as recently investigated ford=2 dimensions, provides a natural and quite probably correct explanation of the pattern of finite Mayer coefficients while indicating the totalabsence of any intermediate phases at nonzero density ; only the KT phase extends to >0.     

Interacting Fermi Liquid in Two Dimensions at Finite Temperature. Part II: Renormalization

This is a companion paper to [DR1]. Using the method of continuous renormalization group around the Fermi surface and the results of [DR1], we achieve the proof that a two-dimensional jellium system of interacting Fermions at low temperature T is a Fermi liquid above the BCS temperature. Following [S], this means proving analyticity in the coupling constant λ for , where K is some numerical constant, and some uniform bounds on the derivatives of the self-energy. Received: 27 July 1999 / Accepted: 31 May 2000     

General properties of polymer systems

We prove the existence of the thermodynamic limit for the pressure and show that the limit is a convex, continuous function of the chemical potential.The existence and analyticity properties of the thermodynamic limit for the correlation functions is then derived; we discuss in particular the Mayer Series and the virial expansion.In the special case of Monomer-Dimer systems it is established that no phase transition is possible; moreover it is shown that the Mayer Series for the density is a series of Stieltjes, which yields upper and lower bounds in terms of Padé approximants.Finally it is shown that the results obtained for polymer systems can be used to study classical lattice systems.Work presented in partial fullfilment of the Ph. D. Thesis.     

Fermionic behavior of ideal anyons

We prove upper and lower bounds on the ground-state energy of the ideal two-dimensional anyon gas. Our bounds are extensive in the particle number, as for fermions, and linear in the statistics parameter \(\alpha \). The lower bounds extend to Lieb–Thirring inequalities for all anyons except bosons.     

Lattice gas generalization of the hard hexagon model. III.q-Trinomial coefficients

In the first two papers in this series we considered an extension of the hard hexagon model to a solvable two-dimensional lattice gas with at most two particles per pair of adjacent sites, and we described the local densities in terms of elliptic theta functions. Here we present the mathematical theory behind our derivation of the local densities. Our work centers onq-analogs of trinomial coefficients.     

$\mathsf{La_{0.5}Sr_{0.5}CoO_{3}}$: A ferromagnet with strong irreversibility

Large spin systems as given by magnetic macromolecules or two-dimensional spin arrays rule out an exact diagonalization of the Hamiltonian. Nevertheless, it is possible to derive upper and lower bounds of the minimal energies, i.e. the smallest energies for a given total spin S. The energy bounds are derived under additional assumptions on the topology of the coupling between the spins. The upper bound follows from “n-cyclicity", which roughly means that the graph of interactions can be wrapped round a ring with n vertices. The lower bound improves earlier results and follows from “n-homogeneity", i.e. from the assumption that the set of spins can be decomposed into n subsets where the interactions inside and between spins of different subsets fulfill certain homogeneity conditions. Many Heisenberg spin systems comply with both concepts such that both bounds are available. By investigating small systems which can be numerically diagonalized we find that the upper bounds are considerably closer to the true minimal energies than the lower ones. Received 22 October 2002 / Received in final form 4 April 2003 Published online 20 June 2003 RID="a" ID="a"e-mail: jschnack@uos.de     

Interacting Fermi Liquid in Two Dimensions¶at Finite Temperature.¶Part I: Convergent Attributions

Using the method of a continuous renormalization group around the Fermi surface, we prove that a two-dimensional interacting system of Fermions at low temperature T is a Fermi liquid in the domain , where K is some numerical constant. According to [S1], this means that it is analytic in the coupling constant λ, and that the first and second derivatives of the self energy obey uniform bounds in that range. This is also a step in the program of rigorous (non-perturbative) study of the BCS phase transition for many Fermion systems; it proves in particular that in dimension two the transition temperature (if any) must be non-perturbative in the coupling constant. The proof is organized into two parts: the present paper deals with the convergent contributions, and a companion paper (Part II) deals with the renormalization of dangerous two point subgraphs and achieves the proof. Received: 27 July 1999 / Accepted: 31 May 2000     

Gross-Pitaevskii Theory of the Rotating Bose Gas

 We study the Gross-Pitaevskii functional for a rotating two-dimensional Bose gas in a trap. We prove that there is a breaking of the rotational symmetry in the ground state; more precisely, for any value of the angular velocity and for large enough values of the interaction strength, the ground state of the functional is not an eigenfunction of the angular momentum. This has interesting consequences on the Bose gas with spin; in particular, the ground state energy depends non-trivially on the number of spin components, and the different components do not have the same wave function. For the special case of a harmonic trap potential, we give explicit upper and lower bounds on the critical coupling constant for symmetry breaking. Received: 1 December 2001 / Accepted: 19 April 2002 Published online: 6 August 2002     

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.