Analytic solution of the Percus-Yevick equation for sticky hard sphere potential

It is shown that within the Percus-Yevick approximation the radial distribution function for sticky (i.e. with a surface adhesion) hard spheres satisfies a linear differential equation with retarded right-hand side. Using the theory of distributions and the Green's function technique the analytic solution of this equation is found and explicit formulas are given enabling one to evaluate the radial distribution function both for sticky and non-attractive hard spheres for any distance and any density.     

Approximate solution of the Percus-Yevick equation for a binary mixture of hard spheres with non-additive diameters

We present an approximate solution of the Percus-Yevick integral equation for a binary mixture of hard spheres with non-additive diameters. Defining R_ij the distance of closest approach between particles of species i and j by R ₁₂ = ½(R ₁₁ + R ₂₂) + α, we obtain a closed set of equations for the direct correlation functions c_ij (r) when 0 < α ? min [½(R ₂₂ - R ₁₁), ½R ₁₁]. Our expressions for c_ii (r), and for c ₁₂(r) in the range 0 < r ? ½[R ₂₂ - R ₁₁] - α, agree with those previously obtained by Lebowitz and Zomick.     

Exact solution of the Percus-Yevick equation for a hard-core fluid in odd dimensions

It is shown that the Percus-Yevick integral equation for the pair distribution function of a fluid interacting with a hard-core potential can be solved not only in one and three dimensions, where the solution is well known, but more generally in all odd dimensions. The nonlinear integral equation is reduced to an algebraic equation of order d?3 for odd dimensions d greater than three. As an example the direct correlation function in five dimensions is derived explicitly.     

Percus-Yevick theory for the system of hard spheres with a square-well attraction

Analytic solution of the Percus-Yevick equation for the system of hard spheres with a square-well attraction is proposed provided the range of attraction,γ, is much smaller than the hard sphere diameter. It is shown that forγ close to zero the system exhibits the first-order phase transition similar to that found for sticky hard spheres; for attraction ranges greater than a certainγ _m the triple point and the line of solidification appear as well.     

Percus-Yevick equation for an inhomogeneous fluid: Critical region

The critical region of a locally nonuniform fluid with gaussian density inhomogeneity is investigated. Nonclassical behaviour is found; critical exponents agree very well with those obtained from the perturbation theory of liquids. The action of an external field shifts the critical region and weakens the critical behaviour. Discussed effects could be verified experimentally.     

A note on the Boltzmann equation for hard spheres

A simple form of the Boltzmann kinetic equation for hard spheres is proposed.     

Isotropic fluid phases of dipolar hard spheres

Monte Carlo simulations are used to calculate the equation of state and free energy of dipolar hard sphere fluids at low temperatures and densities. Evidence for the existence of isotropic-fluid-isotropic-fluid phase transitions is presented and discussed. Condensation in the dipolar hard sphere fluid is unusual in that it is not accompanied by large energy or entropy changes. An explanation of this behavior is put forward.     

Integral equation approximations for fluids of hard spheres with linear quadrupoles

In this paper we solve numerically several integral equation theories for the dense quadrupolar hard-sphere fluid. Closure approximations obtained by expanding the hypernetted-chain equation are shown to give pair-correlation functions and internal energies in good agreement with Monte Carlo calculations. The mean spherical approximation, however, is found to be extremely poor.     

High-density properties of hard spheres within a modified Percus-Yevick theory: The role of thermodynamic consistency

Using an integral-equation approach based upon an approximation for the tail function, the equilibrium properties of a system of hard spheres are studied with special concern for the behavior in the region of close packing. The closure adopted is such that full, internal consistency is ensured in the thermodynamics of the model with respect to both the two zero-separation theorems as well as to the more standard virial and fluctuation routes to the equation of state. The scheme also makes use of the continuity properties of the tail function and of the cavity distribution function at contact. These properties are explictly tested in the low-density limit up to the fourth derivative. The theory generates an equilibrium branch bounded on the high-density side by a point corresponding to a packing fraction0.78, a value which closely matches Rogers' least upper bound for the densest packing of spheres. The pair structure of the fluid in the state of random close packing is also compared to the type of local order predicted by the theory at similar densities.     

Integral equation approximations for fluids of hard spheres with dipoles and quadrupoles

The LHNC, QHNC and mean spherical approximations are solved for fluids of hard spheres with dipoles and linear quadrupoles. The theories are evaluated by comparison with new Monte Carlo results. The dielectric constant is found to decrease rapidly with increasing quadrupole moment.     

Analytic solution of the multiloop Baxter equation

The asymptotic spectrum of anomalous dimensions of gauge-invariant operators in maximally supersymmetric Yang–Mills theory is believed to be described by a long-range integrable spin chain model. We focus in this study on its sl(2)$sl(2)$ subsector spanned by the twist-two single-trace Wilson operators, which are shared by all gauge theories, supersymmetric or not. We develop a formalism for the solution of the perturbative multiloop Baxter equation encoding their asymptotic anomalous dimensions, using Wilson polynomials as basis functions and Mellin transform technique. These considerations yield compact results which allow analytical calculations of multiloop anomalous dimensions bypassing the use of the principle of maximal transcendentality. As an application of our method we analytically confirm the known four-loop result. We also determine the dressing part of the five-loop anomalous dimensions.     

Analytic solution for Telegraph equation by differential transform method

In this article differential transform method (DTM) is considered to solve Telegraph equation. This method is a powerful tool for solving large amount of problems (Zhou (1986) [1], Chen and Ho (1999) [2], Jang et al. (2001) [3], Kangalgil and Ayaz (2009) [4], Ravi Kanth and Aruna (2009) [5], Arikoglu and Ozkol (2007) [6]). Using differential transform method, it is possible to find the exact solution or a closed approximate solution of an equation. To illustrate the ability and reliability of the method some examples are provided. The results reveal that the method is very effective and simple.     

The triplet distribution function in a fluid of hard spheres

The distribution functions of a fluid can be expressed as a series in powers, of the density. The coefficient of the fourth power of the density in the expansion of the triplet distribution function is obtained as a function of the separations of the molecules for a fluid of hard spheres. The coefficient is negative. It is used to estimate the amount by which the true triplet distribution function is less than the value attributed to it by the superposition approximation.     

Dissipative dynamics for hard spheres

The dynamics for a system of hard spheres with dissipative collisions is described at the levels of statistical mechanics, kinetic theory, and simulation. The Liouville operator(s) and associated binary scattering operators are defined as the generators for time evolution in phase space. The BBGKY hierarchy for reduced distribution functions is given, and an approximate kinetic equation is obtained that extends the revised Enskog theory to dissipative dynamics. A Monte Carlo simulation method to solve this equation is described, extending the Bird method to the dense, dissipative hard-sphere system. A practical kinetic model for theoretical analysis of this equation also is proposed. As an illustration of these results, the kinetic theory and the Monte Carlo simulations are applied to the homogeneous cooling state of rapid granular flow.     

Solution of the Percus-Yevick equation for the Widom-Rowlinson model

The Percus-Yevick equation for the Widom-Rowlinson model is solved exactly in one and three dimensions. In one dimension the direct correlation function is obtained explicitely. In three dimensions only the thermodynamic properties have been obtained so far implicitely in terms of elliptic integrals, and there is a maximum density beyond which the P.Y. equation has no solution and that before that density is ‘critical density’ at which the homogenous state becomes unstable.