Rogers-Ramanujan identities in the hard hexagon model

The hard hexagon model in statistical mechanics is a special case of a solvable class of hard-square-type models, in which certain special diagonal interactions are added. The sublattice densities and order parameters of this class are obtained, and it is shown that many Rogers-Ramanujan-type identities naturally enter the working.Supported in part by the National Science Foundation under Grant No. PHY-79-06376A01.Part of this work was performed while the author was a visiting professor at the Institute for Theoretical Physics, State University of New York, Stony Brook, New York 11794.     

Modular properties of the hard hexagon model

The physical quantities (or powers thereof) in the hard-hexagon model that were computed exactly by Baxter are shown to be modular functions with respect to the number-theoretic group ₁[N]. This allows us to determine the analytic structure of, the partition function per site in the thermodynamic limit, and, the density, as functions of the activityz.     

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.     

Lattice gas generalization of the hard hexagon model. I. Star-triangle relation and local densities

In the solvable hard hexagon model there is at most one particle in every pair of adjacent sites, and the solution automatically leads to various mathematical identities, in particular to the Rogers-Ramanujan relations. These relations have been generalized by Gordon. Here we construct a solvable model with at most two particles per pair of adjacent sites, and find the solution involves the next of Gordon's relations. We conjecture the corresponding solution for a model with at mostn particles per pair of adjacent sites: this involves all Gordon's relations, as well as others that we will discuss in a subsequent paper.     

Lattice gas generalization of the hard hexagon model. II. The local densities as elliptic functions

In a previous paper 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. Here we use various mathematical identities (in particular Gordon's generalization of the Rogers-Ramanujan relations) to express the local densities in terms of elliptic functions. The critical behavior is then readily obtained.Supported in part by the National Science Foundation Grant MCS 8201733.     

A polynomial energy-gap model for molecular linewidths

A simple phenomenological model for predicting high-temperature linewidths is described. The rotationally inelastic relaxation time is calculated by using a polynomial inverse-energy-gap model. From this time, the linewidths may be calculated and ambient pressure linewidths may then be used to determine the polynomial coefficients and predict high-temperature linewidths. A detailed comparison with known nitrogen linewidths is given. The relaxation rates obtainable from the model may also be used to calculate pressure-narrowing effects for vibrational Q branches.     

Orthogonal polynomial expansions for effective interactions in the modified Lipkin model

The effective interaction in a one-dimensional model is calculated in terms of an orthogonal polynomial (OP) expansion derived from spectral distribution considerations. Configuration densities are assumed to be gaussian. The results are compared with those of a Brillouin-Wigner (BW) perturbation expansion for the same model. While for a strong perturbation the BW series diverges, the OP expansion is shown to converge smoothly.     

A hard sphere fluid model for superionic conductors

We develop a theory for the mobile constituent of a superionic conductor using the Ornstein-Zernike integral equation for the pair correlation function of an inhomogeneous fluid. We solve this equation in the Percus-Yevick approximation using a simple decoupling procedure and hard core potentials. Comparison is made with molecular dynamics calculations on α-AgI.     

Application of asymptotic solutions for the computation of the piston model of tsunami

Conditions are derived under which asymptotic solutions of the Cauchy problem for the wave equation can be applied to the computation of a piston model of tsunami in a basin of constant depth. The derivation is carried out for the initial displacement of the bottom in the form of Gaussian exponentials by comparing the results of numerical integration with computations using asymptotic formulas. This work was supported by the Russian Foundation for Basic Research, project 05-01-00968.     

The mean spherical model for charged hard spheres

The reduced mean electrostatic potential v(r) and the radial distribution functions gij(r) for a system of charged hard spheres of equal diameter are calculated from the solution of the mean spherical model equation given by Waisman and Lebowitz. An analytical solution is given for v(r) and the gij(r) are shown to be the sum of the Percus-Yevick uncharged hard-sphere distribution function and an electrostatic term. The correct qualitative behaviour of the mean potential is predicted at high concentrations but the radial distribution functions are only accurate for low valency electrolytes at high concentrations.     

用于电磁驱动真空-等离子体系统数值模拟的弛豫磁流体力学模型

提出了一个弛豫磁流体力学模型,特别适合电磁驱动真空-等离子体系统的数值模拟。该模型和Seyler采用的弛豫模型有相似之处,即采用全电磁模型,不同的是采用忽略电子惯性项的广义欧姆定律直接作为本构来封闭麦克斯韦方程,减少了独立变量,是适合此类问题的最简模型。分析了磁流体力学模型电磁部分的色散关系,从而论证了其在真空区退化为电磁传播,在等离子体物质区退化为磁扩散近似,并且相速和群速是有上界的。改进了Seyler采用的时间离散方式,从而将时间精度从1阶提高到3阶,时间步长不受刚性源项约束,只受系统最大的特征速度确定的柯朗-弗里德里奇-列维(CFL)条件约束,便于显式计算和大规模并行化。     

用于电磁驱动真空-等离子体系统数值模拟的弛豫磁流体力学模型

提出了一个弛豫磁流体力学模型,特别适合电磁驱动真空-等离子体系统的数值模拟。该模型和Seyler采用的弛豫模型有相似之处,即采用全电磁模型,不同的是采用忽略电子惯性项的广义欧姆定律直接作为本构来封闭麦克斯韦方程,减少了独立变量,是适合此类问题的最简模型。分析了磁流体力学模型电磁部分的色散关系,从而论证了其在真空区退化为电磁传播,在等离子体物质区退化为磁扩散近似,并且相速和群速是有上界的。改进了Seyler采用的时间离散方式,从而将时间精度从1阶提高到3阶,时间步长不受刚性源项约束,只受系统最大的特征速度确定的柯朗-弗里德里奇-列维(CFL)条件约束,便于显式计算和大规模并行化。     

Convergent polynomial expansion,scaling of differential cross-section and computation of scaling functions for elastic diffraction scattering processes

Existing data on the differential cross-section ratio at high energies for pp, p, π^±p andK ^±p scatering have been fitted by the proposed convergent polynomial expansion to determine the unknown coefficients in the scaling function. It is found that the data are very well represented within and somewhat outside the peak regions by only four or five terms in the proposed series in terms of Laguerre polynomials.     

基于多项式模型的混沌系统平方和算法脉冲控制

针对一大类混沌系统,提出一种新颖的基于多项式模型的脉冲控制方法.首先,建立系统的多项式模型.其状态方程由系统状态的多项式矩阵与其单项式组成列向量构成.与其他建模方法相比,该方法不必使用任何预制的假设.其次,提出基于平方和优化算法的脉冲控制方法,使得混沌系统的状态能够实现渐近稳定.基于该算法的脉冲控制与基于线性矩阵不等式凸优化算法的结果相比,能得到更大的脉冲间距,从而可以使用较少的控制能量实现同样的控制效果.最后,仿真实验结果验证了本方法的有效性.     

Solution of the BGK model kinetic equation for very hard particle interaction

We study the Bhatnagar-Gross-Krook model kinetic equation with a velocity-dependent collision frequency. We derive the conditions that must be verified in order to keep the main physical properties of the Boltzmann equation, i.e.,H-theorem and conservation laws. The particular case of the so-called VHP interaction is considered, and the resulting kinetic equation is solved for a homogeneous and isotropic gas. Overpopulation phenomena are observed and analyzed for some kinds of initial conditions. The results are compared, where possible, with the exact solution of the Boltzmann equation.     

彩色打印机特性化多元回归模型的研究

多元回归法是彩色打印机特性化普遍采用的方法,实验对影响多元回归精度的阶数、色空间选择、以及训练样本选取进行了研究.研究结果表明:(1)二、三、四阶的特性化精度分别在10.9±0.3、6.6±0.2、6.0±0.2个CIELAB色差单位;(2)回归模型采用RGB到CIELAB或CIEXYZ的预测精度基本相同,大概相差1个...     

An approximate equation for the radial distribution function in the hard sphere model

We investigate the spatially homogeneous solution of a new approximate equation for the radial distribution function of classical hard-core systems. Its bifurcation may be connected with a phase transition.     

Equation of state for the hard sphere model in the third virial coefficient approximation

Exactly solvable systems of infinite-dimensional integral equations are investigated. These systems approximate the hard sphere model and the Ising model for antiferromagnetism. An equation of state for the hard sphere model in the approximation of the third virial coefficient is obtained.