The Kirkwood-Salsburg equations: Solutions and spectral properties

It is shown that the Kirkwood-Salsburg equations for a classical lattice gas are equivalent to the Dobrushin-Lanford-Ruelle equilibrium equations. The term “Kirkwood-Salsburg equations” is used here in a restricted sense, and thus the known result for a larger system of equations is improved (see Table 1). Some information on the spectrum of the Kirkwood-Salsburg operator is found in connection with zeros of partition functions. An example is given to show that the Kirkwood-Salsburg equations can have other solutions than states in the space of uniformly bounded correlation functions.     

The kirkwood-salsburg equations for random continuum percolation

We develop two different hierarchies of Kirkwood-Salsburg equations for the connectedness functions of random continuum percolation. These equations are derived by writing the Kirkwood-Salsburg equations for the distribution functions of thes-state continuum Potts model (CPM), taking thes1 limit, and forming appropriate linear combinations. The first hierarchy is satisfied by a subset of the connectedness functions used in previous studies. It gives rigorous, low-order bounds for the mean number of clusters n _c and the two-point connectedness function. The second hierarchy is a closed set of equations satisfied by the generalized blocking functions, each of which is defined by the probability that a given set of connections between particles is absent. These auxiliary functions are shown to be a natural basis for calculating the properties of continuum percolation models. They are the objects naturally occurring in integral equations for percolation theory. Also, the standard connectedness functions can be written as linear combinations of them. Using our second Kirkwood-Salsburg hierarchy, we show the existence of an infinite sequence of rigorous, upper and lower bounds for all the quantities describing random percolation, including the mean cluster size and mean number of clusters. These equations also provide a rigorous lower bound for the radius of convergence of the virial series for the mean number of clusters. Most of the results obtained here can be readily extended to percolation models on lattices, and to models with positive (repulsive) pair potentials.     

Derivation and low density solution of a nonlinear Kirkwood-Salsburg equation

Nonlinear Kirkwood-Salsburg equations which are parametrized by the density ? are derived from the linear ones by elimination of the activity z. Upper bounds on ? are derived below which the solution of these equations is unique. Narrow upper and lower bounds on z(?) are obtained as well as upper bounds on the pair distribution function. Nonlinear Mayer-Montroll equations are also briefly discussed.     

Spectral properties of the Kirkwood-Salsburg operator

A mathematically precise definition of the “infinite-volume” Kirkwood-Salsburg operator as a bounded linear operator in a Banach space is given. It is shown that this operator has a bounded inverse for a bounded, stable and regular pair potential. These facts are exploited to establish the connection between the Kirkwood-Salsburg and the Mayer-Montroll equations and to give a classification of the spectra and resolvents of the Kirkwood-Salsburg operator and of its inverse. The theorems proved in this article constitute a framework for the derivation of any more precise results for special potentials.     

Cluster expansion for abstract polymer models

A new direct proof of convergence of cluster expansions for polymer (contour) models is given in an abstract setting. It does not rely on Kirkwood-Salsburg type equations or combinatorics of trees. A distinctive feature is that, at all steps, the considered clusters contain every polymer at most once.     

Spectral properties of Kirkwood-Salsburg and Kirkwood-Ruelle operators

A method for solving Kirkwood-type equations in Banach spacesE () andE ^S () is applied to derive spectral properties of Kirkwood-Salsburg and Kirkwood-Ruelle operators in these spaces. For stable interactions these operators are shown to have, besides the point spectrum, a residual one. We establish also that the residual spectrum may disappear if a superstable (singular) interaction between particles is switched on. In this case the bounded Kirkwood-Salsburg operator is proved to have a zero Fredholm radius.     

Equivalence of integral equations in the molecular theory of fluids

It is proven that, under physically reasonable conditions, the correlation functions satisfying the BBGKY equations for an infinite system are also solutions of the Mayer-Montroll and Kirkwood-Salsburg equations. The relation between these correlation functions and the probability distributions for finding a fixed number of particles in a given finite region of an infinite system is investigated. The Gibbsian nature of these probability distributions is shown to depend on the range of the intermolecular forces.     

The Kirkwood-Salsburg integral equation and the virial coefficients of Lennard-Jones molecules

The fifth virial coefficients E_c and E_p are derived for a classical fluid composed of molecules interacting according to the Lennard-Jones potential. The calculations are based on the Kirkwood-Salsburg integral equation and the superposition approximation. It is found that, unlike the systems considered in a previous communication, satisfactory agreement between the present results and the known values is possible only at high temperatures.     

The Kirkwood-Salsburg equations for lattice gases with hard cores

The results due to Gallavotti and Miracle-Sole on the uniqueness of the solution of the Kirkwood-Salsburg equations for the correlation functions of a lattice gas are extended to cover lattice gases with hard cores.     

Acoustic scattering of a high-order Bessel beam by an elastic sphere

The exact analytical solution for the scattering of a generalized (or “hollow”) acoustic Bessel beam in water by an elastic sphere centered on the beam is presented. The far-field acoustic scattering field is expressed as a partial wave series involving the scattering angle relative to the beam axis and the half-conical angle of the wave vector components of the generalized Bessel beam. The sphere is assumed to have isotropic elastic material properties so that the nth partial wave amplitude for plane wave scattering is proportional to a known partial-wave coefficient. The transverse acoustic scattering field is investigated versus the dimensionless parameter ka(k is the wave vector, a radius of the sphere) as well as the polar angle θ for a specific dimensionless frequency and half-cone angle β. For higher-order generalized beams, the acoustic scattering vanishes in the backward (θ = π) and forward (θ = 0) directions along the beam axis. Moreover it is possible to suppress the excitation of certain resonances of an elastic sphere by appropriate selection of the generalized Bessel beam parameters.     

A generalized Schwinger boson mapping with a physical subspace

We investigate the existence of a physical subspace for generalized Schwinger boson mapping of SO(2n+1)⊃SO(2n) in view of previous observations by Marshalek and the recent construction of such a mapping and subspace for SO(8) by Kaup. It is shown that Kaup's construction can be attributed to the existence of a unique SO(8) automorphism. We proceed to construct a generalized Schwinger-type mapping for SO(2n+1)⊃SO(2n) which, in contrast to a similar attempt by Yamamura and Nishiyama, indeed has a corresponding physical subspace. This new mapping includes in the special case of SO(8) the mapping by Kaup which is equivalent to the one given by Yamamura and Nishiyama for n=4. Nevertheless, we indicate the limitations of the generalized Schwinger mapping regarding its applicability to situations where one seeks to establish a direct link between phenomenological boson models and an underlying fermion microscopy.     

Generalized statistics variational perturbation approximation using q-deformed calculus

A principled framework to generalize variational perturbation approximations (VPAs) formulated within the ambit of the nonadditive statistics of Tsallis statistics, is introduced. This is accomplished by operating on the terms constituting the perturbation expansion of the generalized free energy (GFE) with a variational procedure formulated using q-deformed calculus. A candidate q-deformed generalized VPA (GVPA) is derived with the aid of the Hellmann-Feynman theorem. The generalized Bogoliubov inequality for the approximate GFE are derived for the case of canonical probability densities that maximize the Tsallis entropy. Numerical examples demonstrating the application of the q-deformed GVPA are presented. The qualitative distinctions between the q-deformed GVPA model vis-á-vis prior GVPA models are highlighted.     

The effect of finite response–time in coupled dynamical systems

The paper investigates synchronization in unidirectionally coupled dynamical systems wherein the influence of drive on response is cumulative: coupling signals are integrated over a time interval τ. A major consequence of integrative coupling is that the onset of the generalized and phase synchronization occurs at higher coupling compared to the instantaneous (τ?=?0) case. The critical coupling strength at which synchronization sets in is found to increase with τ. The systems explored are the chaotic Rössler and limit cycle (the Landau–Stuart model) oscillators. For coupled Rössler oscillators the region of generalized synchrony in the phase space is intercepted by an asynchronous region which corresponds to anomalous generalized synchronization.     

Equilibrium states for classical systems

It is shown that the Dobrushin-Lanford-Ruelle equations for the probability measure μ, and the Kirkwood-Salsburg type equations for the lattice or continuum correlation functions ?, and for the spin correlation functions σ, are essentially equivalent for all ?, σ, and μ satisfying certain boundedness conditions. It is also noted that the lattice equations are identical to the equations for the stationary states of a certain Markoff process. This extends previous results of Ruelle, Brascamp and Holley who proved some of these equivalences for states.     

Relative equilibria for the generalized rigid body

This paper gives necessary and sufficient conditions for the (n-dimensional) generalized free rigid body to be in a state of relative equilibrium. The conditions generalize those for the case of the three-dimensional free rigid body, namely that the body is in relative equilibrium if and only if its angular velocity and angular momentum align, that is, if the body rotates about one of its principal axes. For the n-dimensional rigid body in the Manakov formulation, these conditions have a similar interpretation. We use this result to state and prove a generalized Saari’s Conjecture (usually stated for the N-body problem) for the special case of the generalized rigid body.     

On a generalization of Wick-powers for generalized free-fields in terms of Jacobi-fields

Within the frame of Jacobi-fields a generalization of Wick-powers of generalized free fields is proposed. The key notion is that of a (generalized) contraction map. Those contraction maps F which yield a relativistic quantum field A^F are characterized. Using some simplifying assumptions the general form of a contraction map F which yields a relativistic quantum field A^F is determined. Furthermore, those contraction maps F are characterized for which A^F is in the Borchers class of the generalized free field A we start with. The Wick-product of generalized free fields appears as a particular example of this construction.     

Semiclassical Limit for Generalized KdV Equations Before the Gradient Catastrophe

We study the semiclassical limit of the (generalized) KdV equation, for initial data with Sobolev regularity, before the time of the gradient catastrophe of the limit conservation law. In particular, we show that in the semiclassical limit the solution of the KdV equation: i) converges in H ^s to the solution of the Hopf equation, provided the initial data belongs to H ^s , ii) admits an asymptotic expansion in powers of the semiclassical parameter, if the initial data belongs to the Schwartz class. The result is also generalized to KdV equations with higher order linearities.