Statistical properties of aggregation with injection

We study a generalized aggregation process in which charged particles diffuse and coalesce randomly on a lattice. For one-dimensional and mean-field models, we show that there exists a statistically-invariant steady state when randomly charged particles are continuously injected. The steady-state charge distribution obeys a power law with the exponent depending both on the type of the injection and on the spatial dimension. The response of the system to a perturbation (i.e., relaxation) is characterized by either a power law decay (t ^– ,1) or a compressed exponential decay [exp(–t ),>1].     

Finite-size scaling for Potts models

Recently, Borgs and Kotecký developed a rigorous theory of finite-size effects near first-order phase transitions. Here we apply this theory to the ferromagneticq-state Potts model, which (forq large andd2) undergoes a first-order phase transition as the inverse temperature is varied. We prove a formula for the internal energy in a periodic cube of side lengthL which describes the rounding of the infinite-volume jumpE in terms of a hyperbolic tangent, and show that the position of the maximum of the specific heat is shifted by _m (L)=(Inq/E)L ^–d +O(L ^–2d ) with respect to the infinite-volume transition point _t . We also propose an alternative definition of the finite-volume transition temperature _t (L) which might be useful for numerical calculations because it differs only by exponentially small corrections from _t .     

Novel perturbation expansion for the Langevin equation

We discuss the randomly driven systemdx/dt= -W(x) +f(t), wheref(t) is a Gaussian random function or stirring force withf(t)f(t)=2(t–t), andW(x) is of the formgx ¹⁺². The parameter is a measure of the nonlinearity of the equation. We show how to obtain the correlation functionsx(t)f(t)···x(t( n)) _f as a power series in. We obtain three terms in the expansion and show how to use Padé approximants to analytically continue the answer in the variable. By using scaling relations, we show how to get a uniform approximation to the equal-time correlation functions valid for allg and.     

Thermodynamics of ionic solid solutions: II. Discontinuous series: A new treatment of existing distribution data

A method of finding the activity coefficients of salts, anhydrous or hydrated, in binary solid solutions, described in an earlier paper as it applies to continuous series, has been applied to discontinuous series. The salts must differ with respect to only one ion. The method requires isothermal distribution data for equilibria between liquid (aqueous) and solid solutions in the ternary system consisting of the two salts and water. The following salt pairs were used for illustration: K(I/Br) at 0, 15, 25, 35, and 50°C., (NH₄/K)SCN at 0, 30, 60, and 90°C., (K/Tl)C10₃ at 10°C., and (NH₄/K)SO₃NH₂, (NH₄/K)Br, (Mg/Co)SO₄-7H₂O, and (Mn/Cu) SO₄.n H₂O-all at 25°C. Two kinds of behavior were noted and treated differently: systems in which the two series have the same, and those in which they have different crystal lattices. For two salts, A and B, which have the same lattice, and whose rational activity coefficients, f _A and f _B , can be described by 2-suffix Margules equations (regular solutions), lnf _A =B_sx _B ² and lnf _B =B_sx _A ² to be partially miscible, B_s>2, but this requirement does not apply if the lattices are different. In each series, distribution constants for the equilibria were also determined. Where possible, the calculated activities of the salts or the Gibbs excess energies of the solid solutions were compared with values reported by others who determined them by other methods. All the salt pairs studied show slight or strong positive deviations from ideality.     

On the Scaling Study for the A_f-A_g Type Free Radical Polymerization

根据Ａｆ－Ａｇ自由基加聚反应的数量分布函数，导出凝胶点附近的渐进分布函数和高分子矩的表示式．进一步应用标度变换，得到了描述溶胶－凝胶相变的广义标度律，从而揭示了Ａｆ－Ａｇ自由基加聚的固化反应是一个相变过程．     

The third law of thermodynamics and the degeneracy of the ground state for lattice systems

The third law of thermodynamics, in the sense that the entropy per unit volume goes to zero as the temperature goes to zero, is investigated within the framework of statistical mechanics for quantum and classical lattice models. We present two main results: (i) For all models the question of whether the third law is satisfied can be decided completely in terms of ground-state degeneracies alone, provided these are computed for all possible boundary conditions. In principle, there is no need to investigate possible entropy contributions from low-lying excited states, (ii) The third law is shown to hold for ferromagnetic models by an analysis of the ground states.Dedicated to Pierre Résibois. Work supported in part by NSF grant PHY-7825390 A01.     

On the ν-dimensional one-component classical plasma: The thermodynamic limit problem revisited

We prove theH-stability property and the existence of the thermodynamic limit of the free energy density of the two-dimensional, one-component classical plasma. We give lower and upper bounds on the free energy density in any dimensionv and draw some consequences.     

A Singular Limit Problem for Conservation Laws Related to the Camassa–Holm Shallow Water Equation

We consider a shallow water equation of the Camassa–Holm type, which contains nonlinear dispersive effects as well as fourth order dissipative effects. We prove that as the diffusion and dispersion parameters tend to zero, with a condition on the relative balance between these two parameters, smooth solutions of the shallow water equation converge to discontinuous weak solutions of a scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the L ^p setting.     

Optimal error estimates of a decoupled scheme based on two-grid finite element for mixed Stokes–Darcy model

Although the numerical results suggest the optimal convergence order of the two-grid finite element decoupled scheme for mixed Stokes–Darcy model with Beavers–Joseph–Saffman interface condition in literatures, the numerical analysis only gets the optimal error order for porous media flow and a non-optimal error order that is half order lower than the optimal one in fluid flow. The purpose of this paper is to fill in the gap between the numerical results and the theoretical analysis.     

Performance analysis of linear and nonlinear techniques for automatic scaling of discretized control problems

This paper introduces three (one linear and two nonlinear) automatic scaling techniques for NLPs with states and constraints spread over several orders of magnitude, without requiring complex off-the-shelf external tools. All of these methods have been compared to standard techniques and applied to three problems using SNOPT and IPOPT. The results confirm that the proposed techniques significantly improve the NLP conditioning, yielding more reliable and in some cases, faster NLP solutions.