Diverging seventh virial coefficient of sticky discs

The seventh virial coefficient of a two-dimensional system of particles interacting with a hard-core square-well pair potential is studied. The Ree-Hoover type cluster integrals were examined and it was found that a graph in the form of a hexagonal wheel with all the bonds of the attractive square-well type does not allow Baxter's ‘sticky sphere’ limit to be achieved. The value of that particular cluster integral was calculated. It was shown that when approaching the sticky limit the cluster integral corresponding to the hexagonal wheel diverges linearly with the height of the peak in the Mayer f function at the location of the potential square-well. As a consequence, the seventh virial coefficient of the sticky disc system does not have a finite value.     

方势阱中凝聚体的孤子动力学行为

利用多重尺度法, 解析地研究了方势阱中玻色-爱因斯坦凝聚体的孤子动力学行为. 结果表明, 方势阱对凝聚体中的孤子动力学有重要的影响. 进入方势阱时孤子作加速运动, 逃逸出势阱时孤子作减速运动; 且随着势阱深度的增加, 孤子的速度增加、幅度增加、宽度减小. 这为实验操控孤子的动力学行为提供一定的参考价值. 关键词： 玻色-爱因斯坦凝聚体 孤子 方势阱     

Perturbation theory for the square-well dimer fluid

An analytical equation of state is presented for the square-well dimer fluid of variable well width (1 ≤ λ ≥ 2) based on Barker-Henderson perturbation theory using the recently developed analytical expression for radial distribution function of hard dimers. The integral in the first- and the second-order perturbation terms utilizes the Tang, Y and Lu, B. C.-Y., 1994, J. chem. Phys., 100, 6665 formula for the Hilbert transform. To test the equation of state, NVT and Gibbs ensemble Monte Carlo simulations for square-well dimer fluids are performed for three different well widths (λ = 1.3, 1.5 and 1.8). The prediction of the perturbation theory is also compared with that of thermodynamic perturbation theory in which the equation of state for the square-well dimer is written in terms of that of square-well monomers and the contact value of the radial distribution function.     

Solutions of the Yvon-Born-Green equation for a system of square-well molecules at a temperature below the triple point

The Yvon-Born-Green equation (with superposition approximation) is solved numerically for the pair correlation function for a system of molecules interacting via the square-well potential (with σ₂/σ₁ = 1·85), for an isotherm below the triple point, and over a broad range of densities. The correlation function data and attendant thermodynamics generated for this isotherm are compared with results reported previously by the authors for several supercritical and subcritical isotherms of the square-well fluid. To facilitate the interpretation of these results, particularly in those regions of (T, ρ) space where phase transitions may occur, a geometrical representation of the data is presented (motivated, in part, by recent work by René Thom), and the location of the triple point is discussed in terms of this construction. The differences anticipated between results reported here and those that would be obtained in an exact statistical mechanical analysis, are identified.     

Square-well diatomics Exact low density results

Exact semi-analytical expressions are obtained for the zero density site-site distribution function and the second virial coefficient for homonuclear square-well diatomics. The diatomic molecules considered here are composed of two fused square-well spheres with hard core diameter σ, well diameter λσ, and dimer bond length L, such that 0<Lˇ-σ and 1ˇ-λˇ-1+L/σ. Very accurate, although approximate, analytical expressions are also given for these functions.     

$$\mathcal{P}\mathcal{T}$$ -supersymmetric partner of a short-range square well

In a box of size L, a spatially antisymmetric square-well potential of a purely imaginary strength ig and size l < L is interpreted as an initial element of the SUSY hierarchy of solvable Hamiltonians, the energies of which are all real for g < g _c (l). The first partner potential is constructed in closed form and discussed. Presented at the 3rd International Workshop “Pseudo-Hermitian Hamiltonians in Quantum Physics”, Istanbul, Turkey, June 20–22, 2005.     

Geometrical entropies. The extended entropy

By taking into account a geometrical interpretation of the measurement process [1, 2], we define a set of measures of uncertainty. These measures will be called geometrical entropies. The amount of information is defined by considering the metric structure in the probability space. Shannon-von Neumann entropy is a particular element of this set. We show the incompatibility between this element and the concept of variance as a measure of the statistical fluctuations. When the probability space is endowed with the generalized statistical distance proposed in reference [3], we obtain the extended entropy. This element, which belongs to the set of geometrical entropies, is fully compatible with the concept of variance. Shannon-von Neumann entropy is recovered as an approximation of the extended entropy. The behavior of both entropies is compared in the case of a particle in a square-well potential. Received 4 November 1999     

Quantum corrections to the thermodynamic properties of a two-dimensional fluid with hard-core potential

A simple theory, based on the physical interpretation of the reciprocal of activity, is developed to evaluate the thermodynamic properties of a two-dimensional fluid in the semi-classical limit. The theory is applied to calculate the quantum corrections to the equation of state and excess free energy of two-dimensional fluids, whose molecules interactvia hard-disc and square-well potential. It is found that the quantum effect increases with the increase of density and decrease of temperature.     

Alkali rare-gas line profiles in a square-well potential approximation: Width,shift, and asymmetry

We have studied the profiles of spectral lines of alkalies perturbed by rare gases using the Anderson and Talman theory of line broadening with a square-well potential, chosen for its simplicity in order to understand the influence of potential parameters on the profiles. Although it is obvious that such a simple potential is not able to give an exact quantitative fit to the experimentally measured results, we have been able to give a complete and satisfactory explanation of the variations of width, shift and asymmetry with density by applying previously established results, for the case of well-resolved satellites, to the case of unresolved satellites often encountered experimentally.     

Thermodynamic properties of hard ellipsoid square-well fluid

The perturbation theory with non-spherical reference system is used for molecular fluid with angle-dependent square-well type potential. Simple analytic expressions are given for the thermodynamic properties such as the equation of state, excess free energy per particle, internal energy and internal heat capacity. The effects of anisotropy on the thermodynamic properties are discussed. The anisotropy effects increase with increase of density and decrease of temperature and depends on the anisotropy parameterx ₀.     

Vapour—liquid equilibrium of the square-well fluid of variable range via a hybrid simulation approach

The equilibrium between vapour and liquid in a square-well system has been determined by a hybrid simulation approach combining chemical potentials calculated via the Gibbs ensemble Monte Carlo technique with pressures calculated by the standard NVT Monte Carlo method. The phase equilibrium was determined from the thermodynamic conditions of equality of pressure and chemical potential between the two phases. The results of this hybrid approach were tested by independent NPT and μPT calculations and are shown to be of much higher accuracy than those of conventional GEMC simulations. The coexistence curves, vapour pressures and critical points were determined for SW systems of interaction ranges λ = 1.25, 1.5, 1.75 and 2. The new results show a systematic dependence on the range λ, in agreement with results from perturbation theory where previous work had shown more erratic behaviour.     

Microcanonical-ensemble computer simulation of the high-temperature expansion coefficients of the Helmholtz free energy of a square-well fluid

The microcanonical-ensemble computer simulation method (MCE) is used to evaluate the perturbation terms A_i of the Helmholtz free energy of a square-well (SW) fluid. The MCE method offers a very efficient and accurate procedure for the determination of perturbation terms of discrete-potential systems such as the SW fluid and surpass the standard NVT canonical ensemble Monte Carlo method, allowing the calculation of the first six expansion terms. Results are presented for the case of a SW potential with attractive ranges 1.1 ≤ λ ≤ 1.8. Using semi-empirical representation of the MCE values for A_i, we also discuss the accuracy in the determination of the phase diagram of this system.     

A Global Investigation About Hard Core Attractive Yukawa Approximation and Adhesive Hard Sphere Approximation for Structure of Colloidal Dispersion Systems

The accuracy of hard core attractive Yukawa (HCAY) potential and adhesive hard sphere (AH) potential in representing the structure factor of short range square well potential and Asakura and Oosawa (AO) depletion potential is examined by comparing theoretical predictions with the existing simulation data and the present numerical results from the non-linear optimized random phase approximation closure for Ornstein-Zernike equation. For the case of square-well (SW) potential, it is shown that the structure factor of HCAY potential based on a recently proposed semi-analytical expression for the radial distribution function can describe the structure factor of SW potential with reduced well width λ≤2 only if the reduced contact potential βε_SW≤0.25, while the analytical expression for the structure factor of AH potential under Percus-Yevick (PY) approximation completely fails for the case of λ>1.2. For the case of AO depletion potential, the domain of validity of both HCAY potential and AH potential is complementary. With the above analysis and considering the solid-liquid transition of the AH potential with an adhesive parameter τ below 1.31 cannot be predicted by modified weighted density approximation, the role played by the HCAY potential about the mapping manipulation should not be ignored.     

Elastic scattering and the K matrix: (II). Phase shifts,resonances and few-level approximations

The K-matrix expansions given in part I are applied to the scattering by a square-well potential. Their convergence proves very satisfactory from a physical point of view since few-level approximations allow very good approximations to the phase shifts and cross sections. It also appears that all the complex poles of K_l and the real ones with positive residue should undoubtedly be associated with physical resonances. As for the real ones with negative residue, i.e. the echo poles, they are obviously unrelated to resonances, but they provide a very good parametrization of the background part of the scattering. The time delay is given a major role in the argument. The possibility of having double poles is also discussed and sum rules are given for the energies and residues of the poles.     

Bound and scattering wave functions for a velocity-dependent Kisslinger potential for l > 0

Using formal scattering theory, the scattering wave functions are extrapolated to negative energies corresponding to bound-state poles. It is shown that the ratio of the normalized scattering and the corresponding bound-state wave functions, at a bound-state pole, is uniquely determined by the bound-state binding energy. This simple relation is proved analytically for an arbitrary angular momentum quantum number l > 0, in the presence of a velocity-dependent Kisslinger potential. The extrapolation relation is tested analytically by solving the Schr?dinger equation in the p-wave case exactly for the scattering and the corresponding bound-state wave functions when the Kisslinger potential has the form of a square well. A numerical resolution of the Schr?dinger equation in the p-wave case and of a square-well Kisslinger potential is carried out to investigate the range of validity of the extrapolated connection. It is found that the derived relation is satisfied best at low energies and short distances. Received: 17 October 2001 / Accepted: 4 January 2002     

Network Forming Fluids: Yukawa Square-Well <Emphasis Type="Italic">m</Emphasis>-Point Model

Thermal and connectivity properties of the Yukawa square-well m-point (YSWmP) model of the network forming fluid are studied using solution of the multidensity Ornstein-Zernike and connectedness Ornstein-Zernike equations supplemented by the associative mean spherical approximation (AMSA). The model is represented by the multicomponent mixture of Yukawa hard spheres with m_s^am_{s}^{a} square-well sites, located on the surface of each hard sphere. To validate the accuracy of the theory, computer simulation is used to calculate the structure, thermodynamic and connectivity properties of the one-component YSW4P version of the model which is compared against corresponding theoretical data. In addition, connectivity properties of the model were studied using Flory-Stockmayer (FS) theory. Predictions of the AMSA for the thermal properties of the model (radial distribution functions (RDF), internal energy, pressure, fractions of the particles in different bonding states) are in good agreement with computer simulation predictions. Similarly, good agreement was found for the connectedness RDF (CRDF), except for the statepoints located close to the percolation threshold, where the theory fails to reproduce the long-range behavior of the CRDF. Results of both theories (AMSA and FS) for the mean cluster size are reasonably accurate only at low degrees of association. Predictions of the FS theory for the percolation lines are in a good agreement with computer simulation predictions. AMSA predictions of percolation are much less accurate, where corresponding percolation lines are located at a temperatures approximately 25% lower then those calculated using computer simulation.     

Kinetic Arrest Originating in Competition Between Attractive Interaction and Packing Force

We discuss the situation where attractive and repulsive portions of the interparticle potential both contribute significantly to glass formation. We introduce the square-well potential as prototypical model for this situation, and reject the Baxter model as a useful model for comparison to experiment on glasses, based on our treatment within mode coupling theory. We present explicit results for various well widths, and show that, for narrow wells, there is a useful analytical formula that would be suitable for experimentalists working in the field of colloidal science. We raise the question as to whether, in a more exact treatment, the sticky-sphere limit might have an infinite glass transition temperature or a high but finite one.     

Generalized Heisenberg algebra and algebraic method: The example of an infinite square-well potential

We discuss the role of generalized Heisenberg algebras (GHA) in obtaining an algebraic method to describe physical systems. The method consists in finding the GHA associated to a physical system and the relations between its generators and the physical observables. We choose as an example the infinite square-well potential for which we discuss the representations of the corresponding GHA. We suggest a way of constructing a physical realization of the generators of some GHA and apply it to the square-well potential. An expression for the position operator x in terms of the generators of the algebra is given and we compute its matrix elements.     

The effect of steepness of soft-core square-well potential model on some fluid properties

The effect of repulsive steepness of the soft-core square well (SCSW) potential model on the second virial coefficient, critical behaviour (two- phase region and the position of critical point), and coordination number are investigated. The soft-core thermodynamic perturbation theory (TPT) presented by Weeks-Chandler-Anderson (WCA) recently developed by Ben-Amotz and Stell (BAS) has been used for the reference system, and the Barker-Henderson TPT for the perturbed system. The Barker-Henderson macroscopic compressibility approximation has been used for all order perturbation terms in which the second-order one is improved by assuming that the molecules in every two neighbouring shells are correlated upon the original assumption. By using the hard-sphere isothermal compressibility consistency for the radial distribution function (RDF), an analytical closed expression has been derived for the Helmholtz free energy function contained effective hard-sphere diameter. The accuracy of the model has been examined for the hard-core system, and an appropriate range found for the attractive width of the potential well (R), then the effect of steepness parameter on the critical quantities, coordination number, and the inversion temperature of the second virial coefficient, has been investigated qualitatively. The predicted results are in good agreement with the computer simulation data for the critical constants, and coordination number at the limit of the hard-core square-well potential model at least qualitatively, and for the attractive range 1.55 ≤ R ≤ 1.7, quantitatively. It was found that the steepness of the potential model has a marginal effect on the critical behaviour, and also every thermodynamic quantity at low and medium temperatures for which the molecular penetration is negligible, but since the penetration at high temperatures is significant, the role of the steepness of potential on the inversion temperature of the second virial coefficient and coordination number is highlighted.