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Nigel Goldenfeld

Publications and source records attributed to Nigel Goldenfeld.

108 records · Page 6Linked to original sources
arXiv1999

Crossover Scaling in Dendritic Evolution at Low Undercooling

We examine scaling in two-dimensional simulations of dendritic growth at low undercooling, as well as in three-dimensional pivalic acid dendrites grown on NASA's USMP-4 Isothermal Dendritic Growth Experiment. We report new results on self-similar evolution in both the experiments and simulations. We find that the time dependent scaling of our low undercooling simulations displays a cross-over scaling from a regime different than that characterizing Laplacian growth to steady-state growth.

cond-mat.mtrl-sci↗
arXiv1998

Universal Dynamics of Phase-Field Models for Dendritic Growth

We compare time-dependent solutions of different phase-field models for dendritic solidification in two dimensions, including a thermodynamically consistent model and several ad hoc models. The results are identical when the phase-field equations are operating in their appropriate sharp interface limit. The long time steady state results are all in agreement with solvability theory. No computational advantage accrues from using a thermodynamically consistent phase-field model.

cond-mat.mtrl-sci↗
arXiv1998

Adaptive Mesh Refinement Computation of Solidification Microstructures using Dynamic Data Structures

We study the evolution of solidification microstructures using a phase-field model computed on an adaptive, finite element grid. We discuss the details of our algorithm and show that it greatly reduces the computational cost of solving the phase-field model at low undercooling. In particular we show that the computational complexity of solving any phase-boundary problem scales with the interface arclength when using an adapting mesh. Moreover, the use of dynamic data structures allows us to simulate system sizes corresponding to experimental conditions, which would otherwise require lattices greater that $2^{17}\times 2^{17}$ elements. We examine the convergence properties of our algorithm. We also present two dimensional, time-dependent calculations of dendritic evolution, with and without surface tension anisotropy. We benchmark our results for dendritic growth with microscopic solvability theory, finding them to be in good agreement with theory for high undercoolings. At low undercooling, however, we obtain higher values of velocity than solvability theory at low undercooling, where transients dominate, in accord with a heuristic criterion which we derive.

cond-mat.mtrl-sci↗
arXiv1998

Dynamical Scaling Behavior of the Swift-Hohenberg Equation Following a Quench to the Modulated State

We study the kinetics of phase transitions in a Rayleigh-Benard system after onset of convection using 2D Swift-Hohenberg equation. An initially uniform state evolves to one whose ground state is spatially periodic. We confirmed previous results which showed that dynamical scaling occurs at medium quench ($ε= 0.25$) with scaling exponents 1/5 and 1/4 under zero noise and finite noise respectively. We find logarithmic scaling behavior for a deep quench ($ε= 0.75$) at zero noise. A simple method is devised to measure the proxy of domain wall length. We find that the energy and domain wall length exhibit scaling behavior with the same exponent. For $ε= 0.25$, the scaling exponents are 1/4 and 0.3 at zero and finite noise respectively.

patt-sol↗
arXiv1998

An X-ray Scattering and Simulation Study of the Ordering Kinetics in CuAu

A detailed numerical and experimental study of the ordering of the low temperature tetragonal phase of CuAu is presented. The numerical simulations are based on a coarse-grained free energy derived from electronic structure calculations of CuAu, while the experimental results are obtained from in situ x-ray scattering. Both theoretical and experimental work indicate a subtle kinetic competition between the ordered tetragonal phase and the metastable modulated phase.

cond-mat.stat-mech↗
arXiv1998

Block Spins for Partial Differential Equations

We investigate the use of renormalisation group methods to solve partial differential equations (PDEs) numerically. Our approach focuses on coarse-graining the underlying continuum process as opposed to the conventional numerical analysis method of sampling it. We calculate exactly the coarse-grained or `perfect' Laplacian operator and investigate the numerical effectiveness of the technique on a series of 1+1-dimensional PDEs with varying levels of smoothness in the dynamics: the diffusion equation, the time-dependent Ginzburg-Landau equation, the Swift-Hohenberg equation and the damped Kuramoto-Sivashinsky equation. We find that the renormalisation group is superior to conventional sampling-based discretisations in representing faithfully the dynamics with a large grid spacing, introducing no detectable lattice artifacts as long as there is a natural ultra-violet cut off in the problem. We discuss limitations and open problems of this approach.

cond-mat.stat-mech↗
arXiv1997

Efficient Computation of Dendritic Microstructures using Adaptive Mesh Refinement

We study dendritic microstructure evolution using an adaptive grid, finite element method applied to a phase-field model. The computational complexity of our algorithm, per unit time, scales linearly with system size, rather than the quadratic variation given by standard uniform mesh schemes. Time-dependent calculations in two dimensions are in good agreement with the predictions of solvability theory, and can be extended to three dimensions and small undercoolings

cond-mat.mtrl-sci↗
arXiv1996

Kinetics of Modulated and Ordered Structures in CuAu

A continuum model derived from an atomistic Hamiltonian is used to examine the ordering kinetics in CuAu. A detailed description of the formation of the low and high temperature ordered and modulated superlattice states is given. The metastability of the modulated phase at low temperatures is shown to severely hinder creation of the ordered superlattice. Formation of the modulated superlattice is shown to results in interesting lamellar and labyrinthine structures.

cond-mat.mtrl-sci↗
arXiv1996

Experimental Constraints on the Pairing State of the Cuprate Superconductors: an Emerging Consensus

We present a critical discussion of recent experimental probes of the pairing state of the high temperature superconductors, focusing primarily, but not exclusively, on \Yba, where the best data currently exist. Penetration depth measurements near \Tc\ give no indication of an extra transition, indicating that the pairing state is a one-dimensional representation of the crystal symmetry. Penetration depth measurements at low temperatures provide strong evidence for a change in sign of the gap function over the Fermi surface. Quantum mechanical phase interference experiments generally confirm this and in addition show that the nodal positions are consistent with a \dx2y2\ pairing state. This pairing state is consistent with photoemission measurements of the gap function, Raman scattering, the effect on \Tc\ of impurities, and many other data (reviewed by two of us previously) which indicate the presence of low lying excitations in the superconducting state. We also discuss evidence that apparently does not fit in with a \dx2y2\ pairing state, and we describe possible alternative scenarios.

cond-mat↗
arXiv1995

Does Fully-Developed Turbulence Exist? Reynolds Number Independence versus Asymptotic Covariance

By analogy with recent arguments concerning the mean velocity profile of wall-bounded turbulent shear flows, we suggest that there may exist corrections to the 2/3 law of Kolmogorov, which are proportional to $(\ln\,\Re)^{-1}$ at large Re. Such corrections to K41 are the only ones permitted if one insists that the functional form of statistical averages at large Re be invariant under a natural redefinition of Re. The family of curves of the observed longitudinal structure function $D_{LL}(r, \Re)$ for different values of Re is bounded by an envelope. In one generic scenario, close to the envelope, $D_{LL}(r, \Re)$ is of the form assumed by Kolmogorov, with corrections of $O((\lnRe)^{-2})$. In an alternative generic scenario, both the Kolmogorov constant $C_K$ and corrections to Kolmogorov's linear relation for the third order structure function $D_{LLL} (r)$ are proportional to $(\ln\,\Re)^{-1}$. Recent experimental data of Praskovsky and Oncley appear to show a definite dependence of $C_K$ on Re, which if confirmed, would be consistent with the arguments given here.

cond-mat↗
arXiv1995

The Renormalization Group and Singular Perturbations: Multiple-Scales, Boundary Layers and Reductive Perturbation Theory

Perturbative renormalization group theory is developed as a unified tool for global asymptotic analysis. With numerous examples, we illustrate its application to ordinary differential equation problems involving multiple scales, boundary layers with technically difficult asymptotic matching, and WKB analysis. In contrast to conventional methods, the renormalization group approach requires neither {\it ad hoc\/} assumptions about the structure of perturbation series nor the use of asymptotic matching. Our renormalization group approach provides approximate solutions which are practically superior to those obtained conventionally, although the latter can be reproduced, if desired, by appropriate expansion of the renormalization group approximant. We show that the renormalization group equation may be interpreted as an amplitude equation, and from this point of view develop reductive perturbation theory for partial differential equations describing spatially-extended systems near bifurcation points, deriving both amplitude equations and the center manifold.

hep-th↗
arXiv1994

Numerical Renormalization Group Calculations For Similarity Solutions and Travelling Waves

We present a numerical implementation of the renormalization group (RG) for partial differential equations, constructing similarity solutions and travelling waves. We show that for a large class of well-localized initial conditions, successive iterations of an appropriately defined discrete RG transformation in space and time will drive the system towards a fixed point. This corresponds to a scale-invariant solution, such as a similarity or travelling-wave solution, which governs the long-time asymptotic behavior. We demonstrate that the numerical RG method is computationally very efficient.

chao-dyn↗
arXiv1994

Analogies between Scaling in Turbulence, Field Theory and Critical Phenomena

We discuss two distinct analogies between turbulence and field theory. In one analogue, the field theory has an infrared attractive renormalization-group fixed point and corresponds to critical phenomena. In the other analogue, the field theory has an ultraviolet attractive fixed point, as in quantum chromodynamics.

cond-mat↗
arXiv1994

Renormalization Group Theory for Global Asymptotic Analysis

We show with several examples that renormalization group (RG) theory can be used to understand singular and reductive perturbation methods in a unified fashion. Amplitude equations describing slow motion dynamics in nonequilibrium phenomena are RG equations. The renormalized perturbation approach may be simpler to use than other approaches, because it does not require the use of asymptotic matching, and yields practically superior approximations.

cond-mat↗
arXiv1994

Renormalization Group Theory And Variational Calculations For Propagating Fronts

We study the propagation of uniformly translating fronts into a linearly unstable state, both analytically and numerically. We introduce a perturbative renormalization group (RG) approach to compute the change in the propagation speed when the fronts are perturbed by structural modification of their governing equations. This approach is successful when the fronts are structurally stable, and allows us to select uniquely the (numerical) experimentally observable propagation speed. For convenience and completeness, the structural stability argument is also briefly described. We point out that the solvability condition widely used in studying dynamics of nonequilibrium systems is equivalent to the assumption of physical renormalizability. We also implement a variational principle, due to Hadeler and Rothe, which provides a very good upper bound and, in some cases, even exact results on the propagation speeds, and which identifies the transition from ` linear'- to ` nonlinear-marginal-stability' as parameters in the governing equation are varied.

cond-mat↗
arXiv1993

Selection, Stability and Renormalization

We illustrate how to extend the concept of structural stability through applying it to the front propagation speed selection problem. This consideration leads us to a renormalization group study of the problem. The study illustrates two very general conclusions: (1) singular perturbations in applied mathematics are best understood as renormalized perturbation methods, and (2) amplitude equations are renormalization group equations.

cond-mat↗
arXiv1993

Structural Stability and Renormalization Group for Propagating Fronts

A solution to a given equation is structurally stable if it suffers only an infinitesimal change when the equation (not the solution) is perturbed infinitesimally. We have found that structural stability can be used as a velocity selection principle for propagating fronts. We give examples, using numerical and renormalization group methods.

cond-mat↗