Limit Gaussian Behavior of the Solutions of the Multidimensional Burger'; Equation with Weak-Dependent Initial Conditions

Asymptotic normality of the solutions of the initial-value problems for multidimensional Burgers' equation with weakly dependent possible non-Gaussian initial condition is proved.     

The dynamics of random interfaces in phase transitions

A review is given of recent developments involving the dynamics of random interfaces formed in the evolution of metastable and unstable systems. Topics which are discussed include interface growth and nonequilibrium dynamical scaling. The possibility of there being dynamical universality classes in first-order phase transitions is also discussed. There are a large number of systems of experimental interest which include binary alloys, binary fluids, and polymer mixtures. Other systems studied by computer simulation include the kinetic Ising, Potts, andZ _N models.Work supported by NSF grant No. DMR-8013700.     

Dynamic aspects of phase decomposition in reactive blends of polycarbonate and syndiotactic polymethyl methacrylate

Dynamics of phase separation in bisphenol-A polycarbonate (PC)/syndiotactic polymethyl methacrylate (sPMMA) blends has been investigated by means of time-resolved light scattering. Solvent-cast films of the PC/sPMMA blends were transparent, suggestive of miscible character. Several temperature jumps were carried out at a 50/50 PC/sPMMA composition from a homogeneous state (room temperature) into a two-phase regime. The process of phase separation first occurred for some considerable period, then it was followed by phase dissolution driven by chemical reaction. The thermodegradative reaction of sPMMA triggered the dissolution process by probably forming PC/sPMMA graft or random copolymers at the interface, which eventually resulted in a single phase. However, annealing at elevated temperatures for an extended period could lead to cross-linking, and thus a two-phase structure could be fixed permanently. The early stage of spinodal decomposition was interpreted in terms of the linearized Cahn-Hilliard theory. In the late stages of spinodal decomposition, the relationship between scattering peak wavenumber and time was found to obey a power law, but the exponents showed a strong dependence on temperature jumps. The temporal universal scaling failed due to the influence of the chemical reaction. © 1995 John Wiley & Sons, Inc.     

Global symmetry breaking in the nonconserved order parameter system during phase ordering

We study global symmetry breaking in the 2D system of scalar nonconserved order parameter following a quench to zero temperature. We show that the instant of time when the symmetry is broken and the final morphology is chosen corresponds to the saturation of the order parameter inside the domains. There are three possible final morphologies: the positive and negative order parameter final morphology, and the state of the coexisting positive and negative order parameter subsystems with a flat interface between them. We find also that each type of the final morphology constitutes about 1/3 of all cases, what agrees with the results obtained recently by Spirin et al. [Phys. Rev. E 65, 016119 (2001)]. Our results are pertinent for the two dimensional systems, but we suspect that there is also a way to apply similar arguments for the three dimensional ones.     

Exact Anomalous Dimensions of Composite Operators in the Obukhov–Kraichnan Model

We consider two stochastic equations that describe the turbulent transfer of a passive scalar field (x)(t,x) and generalize the known Obukhov–Kraichnan model to the case of a possible compressibility and large-scale anisotropy. The pair correlation function of the field (x) is characterized by an infinite collection of anomalous indices, which have previously been found exactly using the zero-mode method. In the quantum field formulation, these indices are identified with the critical dimensions of an infinite family of tensor composite operators that are quadratic in the field (x), which allows obtaining exact values for the latter (the values not restricted to the -expansion) and then using them to find the corresponding renormalization constants. The identification of the correlation function indices with the composite-operator dimensions itself is supported by a direct calculation of the critical dimensions in the one-loop approximation.     

Conformally Invariant Quantization at Order Three

Let (M, g) be a pseudo-Riemannian manifold and the space of densities of degree on M. Denote the space of differential operators from to of order k and S _k with = – the corresponding space of symbols. We construct (the unique) conformally invariant quantization map . This result generalizes that of Duval and Ovsienko.     

Scaling Cosmology

We show that with the help of a suitable coupling between dark energy and cold dark matter it is possible to reproduce any scaling solution _{X M} a , where _X and _M are the densities of dark energy and dark matter, respectively. We demonstrate how the case = 1 alleviates the coincidence problem. Future observations of supernovae at high redshift as well as quasar pairs which are planned to discriminate between different cosmological models will also provide direct constraints on the coupling between dark matter and dark energy.     

Sparse matrix multiplications for linear scaling electronic structure calculations in an atom-centered basis set using multiatom blocks

A sparse matrix multiplication scheme with multiatom blocks is reported, a tool that can be very useful for developing linear-scaling methods with atom-centered basis functions. Compared to conventional element-by-element sparse matrix multiplication schemes, efficiency is gained by the use of the highly optimized basic linear algebra subroutines (BLAS). However, some sparsity is lost in the multiatom blocking scheme because these matrix blocks will in general contain negligible elements. As a result, an optimal block size that minimizes the CPU time by balancing these two effects is recovered. In calculations on linear alkanes, polyglycines, estane polymers, and water clusters the optimal block size is found to be between 40 and 100 basis functions, where about 55-75% of the machine peak performance was achieved on an IBM RS6000 workstation. In these calculations, the blocked sparse matrix multiplications can be 10 times faster than a standard element-by-element sparse matrix package.     

Transient rheology of discotic mesophases

This paper presents an analysis of the role of orientation on the rheology of discotic mesophases subjected to slow shear start-up flows, using a projection of the Landau-de Gennes equations of nematodynamics. Analysis of the shear stress surface as a function of tilt and twist orientation with respect to the shear plane shows that the stress surface is dense in well-oriented and periodically located sets of maxima and minima. Thus overshoots and undershoot stress responses to shear-start up are predicted to be the rule rather than the exception. In-plane (within the shear plane) shear start-up stress responses can exhibit multiple, single, or no overshoots, depending on the number of maxima traversed on the way to steady state. Responses originating from orientations close to the vorticity axis lead to stress undershoots. Complex stress responses, such as a weak overshoot-strong undershoot sequence, are found for intermediate tilt-twist initial states. In-plane modes lead to amplitude and strain scaling. Out-of-plane modes do not display amplitude or strain scaling. These results provide will be useful to interpret and use transient shear rheological data of carbonaceous mesophases and highly filled suspensions of disc-like particles.     

Monodromy Approach to the Scaling Limits in Isomonodromy Systems

The isomonodromy deformation method is applied to the scaling limits in the linear N×N matrix equations with rational coefficients to obtain the deformation equations for the algebraic curves that describe the local behavior of the reduced versions for the relevant isomonodromy deformation equations. The approach is illustrated by the study of the algebraic curve associated with the n-large asymptotics in the sequence of the biorthogonal polynomials with cubic potentials.