Treatment of droplike clusters by means of the classical phase integral in nucleation theory

The translation inconsistency in the theory of nucleation is discussed in historical perspective. A theory is then developed, beginning with the classical phase integral, which not only allows all approximations to be well defined, but also leads to the most natural droplike model for the cluster. The theory makes it possible to apply, in a consistent manner, the thermodynamics of curved surfaces or, alternatively, moleculardynamic numerical computation schemes to the evaluation of the partition function of the cluster. If the cluster is treated as a macroscopic drop (having the free energy of a macroscopic drop), the result for the distribution of clusters differs in only a minor way from that prescribed by the conventional theory of nucleation. It is concluded that for liquid nuclei the conventional theory is consistent, but that a replacement factor may be necessary for solid nuclei. In general, however, the major problems confronting the theory involve the precise evaluation of the work of cluster formation.     

Symmetry properties of integral equations in the theory of classical fluids

The functional formalism described in a preceding paper, leads to a great number of integral equations for molecular distribution functions. Some of them have a certain symmetry property which, in the case of the pair distribution function includes symmetry with respect to the permutation of particles. This symmetry condition is necessary for any self consistent approximation. Among all integral equations known to date only the PY and the HNC equations satisfy this condition. In this paper we derive some new symmetric equations. Integral equations which are obtained with the help of the Percus method, involvingn-particle distribution functions (n > 2), cannot be symmetric with respect to the interchange of particles.     

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.     

On the theory of light scattering in molecular liquids

The theory of light scattering for a system of linear molecules with anisotropic polarizabilities is considered. As a starting point for our theory, we express the result of a scattering experiment in VV and VH symmetry as dynamic correlation functions of tensorial densities ρ _lm(q) with l = 0 and l = 2. l, m denote indices of spherical harmonics. To account for all observed hydrodynamic singularities, a generalization of the theory of Schilling and Scheidsteger [1] for these correlation functions is presented, which is capable to describe the light scattering experiments from the liquid regime to the glassy state. As a microscopic theory it fulfills all sum rules contrary to previous phenomenological theories. We emphasize the importance of the helicity index m for the microscopic theory by showing, that only the existence of m = 1 components lead to the well known Rytov dip in liquids and to the appearance of transversal sound waves in VH symmetry in the deeply supercooled liquid and the glass. Exact expressions for the phenomenological frequency dependent rotation translation coupling coefficients of previous theories are derived. Received 3 July 2000 and Received in final form 7 November 2000     

On the nonuniqueness of solutions for nonlinear integral equations in radiative transfer theory

The nonuniqueness problem is considered for the nonlinear integral equations satisfied by the reflection and transmission matrices of homogeneous plane-parallel atmospheres. The analysis of the problems for semi-infinite and finite atmospheres is based on a recently developed biorthogonolity concept. Explicit expressions for nonphysical solutions are derived. The structure of these solutions reveal that iterative solution procedures may easily yield nonphysical results, if no proper attention is paid to certain linear constraints.     

On the advanced integral equation theory description of dense Yukawa one-component plasma liquids

Different advanced bridge function closures are utilized to investigate the structural and thermodynamic properties of dense Yukawa one-component plasma liquids within the framework of integral equation theory. The isomorph-based empirically modified hypernetted-chain, the variational modified hypernetted-chain, the Rogers–Young, and the Ballone–Pastore–Galli–Gazzillo approaches are compared at the level of thermodynamic properties, radial distribution functions, and bridge functions. The comparison, based on accuracy and computational speed, concludes that the two modified hypernetted-chain approaches are superior and singles out the isomorph-based variant as the most promising alternative to computer simulations of structural properties of dense Yukawa liquids. The possibility of further improvement through artificial crossover to exact asymptotic limits is studied.     

Hamilton-Jacobi functional theory for the integration of classical field equations

By employing the terminology of functional differential calculus, Hamilton-Jacobi theory is extended to apply to classical field equations. It is shown that an asymptotic solution to the Hamilton-Jacobi functional differential equation provides an asymptotic general solution to the associated nonlinear classical field equations.Work supported by a National Science Foundation grant.     

On the memory functions in simple classical liquids

The relation between the two memory function formalisms for correlation functions in classical liquids is discussed. It is found that the kinetic equation formalism of Duderstadt and Akcasu with a simple exponential memory function can account for the double Gaussian form of the memory function in the generalized-hydrodynamics approach. The former therefore gives reasonably good results for the coherent scattering function S(k, ), as is shown for the case of liquid Rb at 315 K in the range 1.25 k 5.5 Å^–.     

Reduction of Chaplygin equations for nonholonomic systems on time scales by means of the cyclic integral and energy integral

In this paper, the reduction of the Chaplygin equations on time scales by means of the cyclic integral and energy integral are proposed and studied. The cyclic integrals and generalized energy integrals for nonholonomic systems on time scales are given. The Chaplygin equations on time scales are reduced by means of the cyclic integrals and the generalized energy integrals. The results show that if the systems have a cyclic integral and an energy integral, then the Chaplygin equations on time scales can be reduced by at least one degree and the form of the Chaplygin equations holds.     

On the derivation of Smoluchowski equations with corrections in the classical theory of Brownian motion

Differential equations governing the time evolution of distribution functions for Brownian motion in the full phase space were first derived independently by Klein and Kramers. From these so-called Fokker-Planck equations one may derive the reduced differential equations in coordinate space known as Smoluchowski equations. Many such derivations have previously been reported, but these either involved unnecessary assumptions or approximations, or were performed incompletely. We employ an iterative reduction scheme, free of assumptions, and calculate formally exact corrections to the Smoluchowski equations for many-particle systems with and without hydrodynamic interaction, and for a single particle in an external field. In the absence of hydrodynamic interaction, the lowest order corrections have been expressed explicitly in terms of the coordinate space distribution function. An additional application of the method is made to the reduction of the stress tensor used in evaluating the intrinsic viscosity of particles in solution. Most of the present work is based on classical Brownian motion theory, but brief consideration is given in an appendix to some recent developments regarding non-Markovian equations for Brownian motion.Supported by the National Science Foundation.     

Derivation of the classical theory of correlations in fluids by means of functional differentiation

The k-particle, infinite-volume distribution functions $\text{n}\text{?}{}_{\mathrm{k}}\text{(}\text{r}{}_1\text{, …,}\text{r}{}_{\mathrm{k?1}}\text{, γ)}$ and modified Ursell correlation functions $\text{U}\text{?}{}_{\mathrm{k}}\text{(}\text{r}{}_1\text{, …,}\text{r}{}_{\mathrm{k?1}}\text{, γ)}$ of a classical system of particles with the two-body potential $\text{q(}\text{r}\text{) + γ}{}^{\mathrm{\nu }}\text{K(γ}\text{r}\text{)}$ are considered. The limiting values of the functions $\text{n}\text{?}{}_{\mathrm{k}}\text{(}\text{r}{}_1\text{, …,}\text{r}{}_{\mathrm{k?1}}\text{, γ),}\text{n}\text{?}{}_{\mathrm{k}}\text{(}\text{S}{}_1\text{/γ, …,}\text{S}{}_{\mathrm{k?1}}\text{/γ, γ)}$ and $\text{γ(}{}^{\mathrm{1?k}}\text{)ν}\text{U}\text{?}{}_{\mathrm{k}}\text{(}\text{S}{}_1\text{/γ, …,}\text{S}{}_{\mathrm{k?1}}\text{/γ, γ)}$ in the limit γ → 0 are calculated, under fairly weak conditions on q and K, by a method involving functional differentiation. These limiting functions are used to describe the molecular structure of the various states of the system both in the range of the potential q(r) and in the rage of the potential γ^νK(γr). The direct correlation function c? (r, γ) is also considered and it is shown that for $\text{S}\text{≠ 0}$, $\text{lim}{}_{\mathrm{\gamma \to 0}}\text{γ}{}^{\mathrm{?\nu }}\text{c}\text{?}\text{(}\text{S}\text{γ}\text{, γ) = ?βK (}\text{S}\text{)}$, for all one-phase states, where β is the reciprocal temperature. Special cases of our results confirm those of other authors, including the well-known results of Ornstein and Zernike.     

On the numerical solution of Sturm-Liouville equations in the theory of molecular diffusion

The Sturm-Liouville equation from Budo's Theory of diffusion in the presence of potential wells is solved numerically for the normalised complex polarisability across the complete range of well depth (Vo/kT) from zero to effectively infinite. For Vo/kT = 0 and Vo/kT = ∞ Debye process is recovered, in excellent agreement with available analytical limits. At intermediate Vo/kT the original Debye process for Vo/kT (free diffusion) is supplemented by a further loss process on the high frequency side. The numerical method used allow us to investigate the origin of this process in terms of the eigenvalues and eigenfunctions of the original Sturm-Liouville equation.     

Existence theory of the linear equations appearing in the Chapman-Enskog solutions to two kinetic equations for liquids

The linear operators appearing in the Chapman-Enskog solutions to Kirkwood's Fokker-Planck kinetic equation and to Rice and Allnatt's kinetic equation are studied in this article. Existence proofs are given for the linearized Chapman-Enskog equations involving either the Fokker-Planck or the Rice-Allnatt operators. It is shown that the Fokker-Planck and Rice-Allnatt operators, defined in the domain appropriate to kinetic theory, are essentially self-adjoint. It is also shown that the spectrum of either of these operators coincides with the spectrum of the self-adjoint extension of the corresponding operator.Sloan Foundation Fellow 1968–70. Guggenheim Fellow 1969–70.     

广义经典力学中Hamilton-Tabarrok-Leech正则方程的对称性理论

提出广义Hamilton-Tabarrok-Leech正则方程的对称性理论.列写系统的运动方程.研究系统的Noether对称性、形式不变性和Lie对称性，并求出相应的守恒量.举例说明结果的应用. 关键词： 广义经典力学 H-T-L 正则方程 对称性 守恒量     

Method of integral equations in the theory of microstructured optical fibers

A method for designing microstructured optical fibers that is based on exact integral equations for the transverse components of the magnetic field of the mode is proposed. A solution to the vector waveguide problem for fibers with a finite number of circular capillaries in the round cavity of the cladding can be refined successively. Quartz fibers with hexagonal capillary rings are also studied.     

On the solutions of correlation equations for classical continuous systems

A method for solving the finite-volume Kirkwood-type correlation equations for tempered boundary conditions is developed. The central idea is an analytic continuation in the activity of the resolvent formulas for the solutions. The uniqueness theorem is proved for activities in a larger domain of the complex plane than the “standard” circle of analyticity¹). A connection with the eigenvector problem for the corresponding Kirkwood-type operators is discussed. We compare also the correlation equation method with the “equilibrium equation” one handling directly with the Gibbs probability measure.