Ursell Functions Research Articles

Monotonicity of Ursell Functions in the Ising Model

In this paper, we consider Ising models with ferromagnetic pair interactions. We prove that the Ursell functions $$u_{2k}$$ satisfy: $$(-1)^{k-1}u_{2k}$$ is increasing in each interaction. As an application, we prove a 1983 conjecture by Nishimori and Griffiths about the partition function of the Ising model with complex external field h: its closest zero to the origin (in the variable h) moves towards the origin as an arbitrary interaction increases.

Well-Posedness of the Iterative Boltzmann Inversion

The iterative Boltzmann inversion is an iterative scheme to determine an effective pair potential for an ensemble of identical particles in thermal equilibrium from the corresponding radial distribution function. Although the method is reported to work reasonably well in practice, it still lacks a rigorous convergence analysis. In this paper we provide some first steps towards such an analysis, and we show under quite general assumptions that the algorithm is well-defined in a neighborhood of the true pair potential, assuming that such a potential exists. On our way we establish important properties of the cavity distribution function and provide a proof of a statement formulated by Groeneveld concerning the rate of decay at infinity of the Ursell function associated with a Lennard-Jones type potential.

Fréchet differentiability of molecular distribution functions II: the Ursell function

For a grand canonical ensemble of classical point-like particles at equilibrium in continuous space we investigate the functional relationship between a stable and regular pair potential describing the interaction of the particles and the thermodynamical limit of the Ursell or pair correlation function. For certain admissible perturbations of the pair potential and sufficiently small activity we rigorously establish Frechet differentiability of the Ursell function in the $L^1$ norm. Furthermore, concerning the thermodynamical limit of the pair distribution function we explicitly compute its Fr\'echet derivative as a sum of a multiplication operator and an integral operator.

Coherent wave propagation in viscoelastic media with mode conversions and pair-correlated scatterers

The influence of correlation between scatterers on coherent waves propagation is studied in the case of a viscoelastic medium hosting a random configuration of either spherical or cylindrical scatterers. A distinction is made between the hole correction and the additional disturbances to the pair correlation function beyond the excluded volume via a radial and concentration dependent Ursell function. The effect of the Ursell function on the effective wavenumber is shown to be of order 3 in concentration and order 2 in scattering, and the corresponding formulas generalize those of Caleap et al. (2012) for an ideal fluid host medium. The whole order 3 in concentration is calculated; its other part is of order 3 in scattering. Both parts of the order 3 in concentration are the sum of two terms, one related to mode conversions, the other not. The numerical study is performed mostly for aluminum spheres in epoxy, which is a rather illustrative situation of the different phenomena that participate to the coherent propagation. The Ursell function effect is enhanced at low frequency, while counteracted partly at higher frequency, by the other term of order 3 in concentration. The most visible effects of both terms are on the attenuation. The Ursell term related to mode conversions is larger than the one with no mode conversions included in the low frequency regime.

Expansion of the correlation functions of the grand canonical ensemble in powers of the activity

A study is made of the grand canonical ensemble of single-component systems of particles in a region Λ. A new representation of the Ursell functions is given. In it an Ursell function is represented as a sum of products of Mayer and Boltzmann functions over the subset of connected graphs labeled by trees. Such a representation greatly reduces the complexity of the structure of these functions. A new definition of all-round tending of the region Λ to infinity is given. The relationship between this definition and the well-known definition of tending of the set Λ to infinity in the sense of Fisher is demonstrated in examples. It is shown that in the case of all-round tending of the set Λ to infinity a term-by-term passage to the limit can be made in the series in Ruelle's representation of the correlation functions as a finite sum of finite products of convergent series. The domain of convergence of the obtained expansions is discussed. As examples, the expansions of the single-particle and binary correlation functions are obtained.

Equilibrium Properties of Ionic Mixtures: Derivation of Fluctuation Formulae from the Grand‐Canonical Ensemble

AbstractThe grand‐canonical ensemble is used to derive fluctuation formulae for the particle densities, the pressure and the energy density of a multi‐component ionic mixture. As an intermediate step sum rules are established for the two‐particle Ursell functions.

Equilibrium properties of a multi-component ionic mixture: I. Sum rules for correlation functions

Equilibrium statistical methods are used to derive sum rules for two- and three-particle correlation functions of a multi-component ionic mixture. Some of these rules are general consequences of the electrostatic character of the interaction, whereas others depend on specific thermodynamic properties of the system. The first group of rules follows from the BBGKY hierarchy and a clustering hypothesis for Ursell functions. The sum rules of the second group are obtained by describing the system with the help of a restricted grand-canonical ensemble in which the particle numbers of the various components in the mixture fluctuate under the condition that the total charge in the system remains constant.

Signs of the Ising model Ursell functions

It is proven that the Ursell functionsU 2k of the Ising model have the conjectured signs: (−1) k+1 U 2k ≧0. The proof is based on Aizenman's random current representation and combinatorics.

Hamiltonian of the phase separation border and phase transitions of the first kind. I

This paper considers ensembles of contours of a multiphase system with nonfinite interaction. The results of the paper provide, on the basis of equilibrium statistical mechanics, a proper mathematical justification for the condition of equality of the phase pressures in equilibrium in a multiphase system and Gibbs's phase rule and make it possible to investigate the analytic properties of the thermodynamic and correlation functions, and also the phase diagram of the system. The technique proposed in the paper makes it possible to investigate the Gibbs states of discrete lattice systems at low and intermediate temperatures. The Pirogov-Sinai theory of phase transitions of the first kind is generalized to the case when the ground states of the Hamiltonian of the model are interacting random fields. Border Hamiltonians and corresponding Ursell functions are introduced.

Fluctuation susceptibility relations for classical spin systems

We prove identities between integrated Ursell functions and derivatives of the pressure in the thermodynamic limit, for multicomponent classical spin systems which obey the Lee-Yang theorem and some form of Gaussian domination, when the susceptibility is finite (T>Tc). Following Refs. 3 and 4, we view the moment generating function of the magnetization as the inverse of an infinitely divisible characteristic function. Fluctuation susceptibility relations of all orders then follow by bounding the corresponding cumulants, taken in zero external field. High-order cumulants are bounded in terms of the susceptibility using Gaussian and Simon's inequalities for short-range interactions.

Infinitely divisible distribution functions of class L and the Lee-Yang theorem

It is shown that the free-energy density of a large class of ferromagnets satisfying the Lee-Yang property is to be connected with the limit characteristic function of a suitably renormalized sum of independent and non-identically distributed random variables. Using the canonical representation formulae of such characteristic functions, various chains of inequalities are derived for the Ursell functions.

New inequalities for Ising ferromagnets

It is shown that for Ising ferromagnets which obey the Lee-Yang theorem the Ursell functions or cumulants of the magnetization variable at nonzero external field satisfy series of inequalities. Several relations connecting Ursell functions with nonzero and zero field are derived.

Cluster expansion of the distribution functions for a ground state Fermi system

The spin-averaged Slater sum of the fermion system is expanded in terms of the square of the ground state wavefunction of a boson system and the ‘‘antisymmetry’’ Ursell function. This expansion is used to obtain the cluster series for the radial distribution function of the fermion system in terms of (−𝒞(n)/S), where 𝒞(n) is sum of chains of (−f/S) and (−fhB2/S) bonds. The series is further expressed in a more compact form using a function L(n) defined by Eq. (55), and the ‘‘modified’’ FHNC approximation for the radial distribution function is presented.

Stochastic aspects in the theory of spectral-line broadening. II. Noncommutative cluster expansions

Generalizing ideas of von Waldenfels we develop a systematic procedure to define truncatedn-point operators which are reminiscent of Ursell functions of statistical mechanics. The truncation procedure is adapted to factorization relations obeyed by the operators in question. The results are applied to spectral-line broadening in plasmas. We derive cluster expansions for the line-shape function in terms of these truncated operators, where the ions are treated quasistatistically. The first order approximation for the line-shape function is discussed. The results are carried over to several moving perturber species, in particular to nonquasistatic ions.

Bimodality and long-range order in ideal Bose systems

The cluster expansion for the classical and the quantum canonical partition function are related to the Bell polynomials. This observation is exploited in derivation of a set of recursion relations that render tractable numerical evaluation of quantities such as mean cluster size distributions and pressure isotherms. The exact volume dependences of properties of an ideal Bose gas are calculated under periodic boundary conditions. Numerical calculations with volume-independent cluster integrals show bimodal distributions in the mean cluster weight for two- and three-dimensional ideal Bose gases at sufficiently low temperature and high density. The variation of the size at which the liquid (condensate) peak appears indicates that the liquid clusters are macroscopic in macroscopic systems. The similarity between the Bose-Einstein condensation and the sol→gel transition in nonlinear chemically polymerizing systems is discussed. When the exact volume dependence of the cluster integrals is taken into account, the mean cluster weight distribution becomes “chair shaped” rather than bimodal and displays no diagonal long-range order in the canonical ensemble. The “Kac density” for an ideal Bose gas implies that in the canonical ensemble the Ursell function satisfies a cluster property in the limit in which the coordinates of the particles are widely separated.

Perturbation theory for quantum fluids at high temperature

A perturbation series is obtained which expresses the pressure, Ursell functions and distribution functions of a quantum fluid in terms of those of the unperturbed fluid having the same temperature and fugacity. The pair interaction potential of the unperturbed fluid is a non-analytic function of the interparticle separation. The advantage of this theory over the usual constant-density theory is that the general term of each series can be given. It is the integral of a product involving generalised Ursell functions of the unperturbed fluid and modified Ursell functions of the quantum fluid and can be conveniently represented by a graph. When this theory is applied to check the results obtained by the constant-density theory, it is found that the first quantum correction to the radial distribution function of the unperturbed fluid obtained by the latter theory needs modification.

Representation for Ursell functions and cluster estimates

Representation for Ursell functions and cluster estimates

Dielectric screening and susceptibility fluctuations in optical scattering

An internally consistent theory of optical scattering in molecular fluids is developed in terms of a ‘screened dipole radiator’ and a screened dipole photon propagator, related by a Bohr-Peierls-Placzek-type relation. The screened dipole radiator describes the propagation of scattered radiation in the dielectric medium, the transition at the surface of the scattering sample, and the ultimate propagation in vacuum outside the material system. The scattering is obtained as a series in terms of Ursell functions, explicit to all orders. In an approximation describing light scattering by matter in bulk, the screened dipole radiator is obtained in closed form involving an intrinsic directional average. The directional average has important consequences for depolarization, for critical scattering, and for questions of convergence. A microscopic representation of local susceptibility fluctuations emerges, and it is shown that polarization fluctuations can naturally and rigorously be decomposed into one contribution from local susceptibility fluctuations and one from field fluctuations. An approximation in closed form to the scattering cross section can be interpreted in macroscopic terms on the basis of this decomposition. In this approximation, critical scattering deviates from Ornstein-Zernike behaviour, but depolarized scattering is incompletely included.

Estimates of Ursell functions, group functions, and their derivatives

Estimates of Ursell functions, group functions, and their derivatives

Renormalization group structure for translationally invariant ferromagnets

We introduce a correlation function description of the renormalization group approach to critical phenomena. Our work is based on treating the renormalization group operator as a linear mapping on the set of Ursell functions, rather than as a nonlinear mapping on the space of Hamiltonians. We mainly consider the ’’mean-spin’’ renormalization group, but the closely related ’’decimation’’ transformation is also considered. Using this approach, we demonstrate for a suitable class of system that the spectrum of the renormalization group operator is bounded and countably infinitely degenerate. We give counterexamples to the notion that there must be convergence to a renormalization group fixed point. Our formulation of the renormalization group is sufficiently general so that convergence to a fixed point does not necessarily imply hyperscaling, i.e., the vanishing of the anomalous dimension of the vacuum, ω*. In the case of convergence to a fixed point we find δ= (d+σ−ω*)/(d−σ−ω*), with ω*?0 necessarily. Above the critical temperature we are able to prove that convergence is obtained to the ’’infinite temperature fixed point,’’ which result generalizes the central limit theorem to ferromagnetically correlated variables.