Benjamin Vollmayr-Lee

Fast and accurate coarsening simulation with an unconditionally stable time step.

We present Cahn-Hilliard and Allen-Cahn numerical integration algorithms that are unconditionally stable and so provide significantly faster accuracy-controlled simulation. Our stability analysis is based on Eyres theorem and unconditional von Neumann stability analysis, both of which we present. Numerical tests confirm the accuracy of the von Neumann approach, which is straightforward and should be widely applicable in phase-field modeling. For the Cahn-Hilliard case, we show that accuracy can be controlled with an unbounded time step Delta t that grows with time t as Delta t approximately t(alpha). We develop a classification scheme for the step exponent alpha and demonstrate that a class of simple linear algorithms gives alpha=1/3. For this class the speedup relative to a fixed time step grows with N, the linear size of the system, as N/ln N. With conservative choices for the parameters controlling accuracy and finite-size effects we find that an 8192(2) lattice can be integrated 300 times faster than with the Euler method.

Applications of field-theoretic renormalization group methods to reaction-diffusion problems

We review the application of field-theoretic renormalization group (RG) methods to the study of fluctuations in reaction–diffusion problems. We first investigate the physical origin of universality in these systems, before comparing RG methods to other available analytic techniques, including exact solutions and Smoluchowski-type approximations. Starting from the microscopic reaction–diffusion master equation, we then pedagogically detail the mapping to a field theory for the single-species reaction kA → �A (� < k ). We employ this particularly simple but non-trivial system to introduce the field-theoretic RG tools, including the diagrammatic perturbation expansion, renormalization and Callan–Symanzik RG flow equation. We demonstrate how these techniques permit the calculation of universal quantities such as density decay exponents and amplitudes via perturbative � = dc − d expansions with respect to the upper critical dimension dc. With these basics established, we then provide an overview of more sophisticated applications to multiple species reactions, disorder effects, L´ evy flights, persistence problems and the influence of spatial boundaries. We also analyse field-theoretic approaches to non-equilibrium phase transitions separating active from absorbing states. We focus particularly on the generic directed percolation universality class, as well as on the most prominent exception to this class: even-offspring branching and annihilating random walks. Finally, we summarize the state of the field and present our perspective on outstanding problems for the future.

Anisotropic coarsening: Grain shapes and nonuniversal persistence

The geometrical Wulff construction [1] gives an explicit relation between the anisotropic surface tension and the resulting equilibrium crystal shape. This marks an early and dramatic success in quantitatively connecting morphology to the interfacial properties of a material. However, distinct Wulff microcrystallites must be in “splendid isolation” — with negligible exchange between them in comparison to the internal dynamics required to equilibrate [2]. In contrast, dilute phase separating alloys and coarsening polycrystallites exhibit growing microcrystalline droplets or grains with non-negligible interactions. While it has been shown that anisotropy influences the morphology for these and other coarsening systems [3,4], such effects have not been quantitatively understood for even the simplest models of curvature-driven growth. The understanding of the late-stage coarsening of interacting isotropic phases (see, e.g., [5]) was significantly advanced by the models of Lifshitz and Slyozov [6] and Wagner [7] for locally conserved diffusive and globally conserved curvature-driven coarsening, respectively. These mean-field theories correctly capture a remarkable amount of coarsening phenomenology, and are exact in the dilute limit. With this inspiration, we generalize Wagner’s model — an interacting ensemble of coarsening droplets, evolving to continually lower their surface energy without local conservation laws, but with conserved total volume —to include arbitrary anisotropy in the surface tension and the interface mobility. We solve the model perturbatively in anisotropy strength, and relate the interfacial properties to the resulting nontrivial grain shapes. These “growth shapes” are contrasted with those of equilibrium (Wulff-constructed) grains to highlight the connection between dynamics and microcrystallite morphology. We then compare our results on the ensemble of grains to Wagner’s isotropic solution to demonstrate anisotropy effects on coarsening correlations, including the effect on

Comment on "Scaling laws for a system with long-range interactions within tsallis statistics"

In their recent Letter [Phys. Rev. Lett. 83, 4233 (1999)], Salazar and Toral (ST) study numerically a finite Ising chain with non-integrable interactions decaying like 1/r^(d+sigma) where -d <= sigma <= 0 (like ST, we discuss general dimensionality d). In particular, they explore a presumed connection between non-integrable interactions and Tsalliss non-extensive statistics. We point out that (i) non-integrable interactions provide no more motivation for Tsallis statistics than do integrable interactions, i.e., Gibbs statistics remain meaningful for the non-integrable case, and in fact provide a {\em complete and exact treatment}; and (ii) there are undesirable features of the method ST use to regulate the non-integrable interactions.

Cluster persistence: A discriminating probe of soap froth dynamics

The persistent decay of bubble clusters in coarsening two-dimensional soap froths is measured experimentally as a function of cluster volume fraction. A dramatically stronger decay is observed in comparison to soap froth models and to measurements and calculations of persistence in other systems. The fraction of individual bubbles that contain any persistent area also decays, implying significant bubble motion and suggesting that T1 processes play an important role in froth persistence.

Single-species three-particle reactions in one dimension.

Renormalization group calculations for fluctuation-dominated reaction-diffusion systems are generally in agreement with simulations and exact solutions. However, simulations of the single-species reactions 3A -->(Ø,A,2A) at their upper critical dimension d(c)=1 have found asymptotic densities argued to be inconsistent with renormalization group predictions. We show that this discrepancy is resolved by inclusion of the leading corrections to scaling, which we derive explicitly and show to be universal, a property not shared by the A+A-->(Ø,A) reactions. Finally, we demonstrate that two previous Smoluchowski approaches to this problem reduce, with various corrections, to a single theory which surprisingly yields the same asymptotic density as the renormalization group.

Anomalous velocity distributions in active Brownian suspensions

Large-scale simulations and analytical theory have been combined to obtain the nonequilibrium velocity distribution, f(v), of randomly accelerated particles in suspension. The simulations are based on an event-driven algorithm, generalized to include friction. They reveal strongly anomalous but largely universal distributions, which are independent of volume fraction and collision processes, which suggests a one-particle model should capture all the essential features. We have formulated this one-particle model and solved it analytically in the limit of strong damping, where we find that f(v) decays as 1/v for multiple decades, eventually crossing over to a Gaussian decay for the largest velocities. Many particle simulations and numerical solution of the one-particle model agree for all values of the damping.

Anomalous Dimension in a Two-Species Reaction–Diffusion System

We study a two-species reaction-diffusion system with the reactions

Kac-potential treatment of nonintegrable interactions

A+A\to (0, A)

Cluster Persistence: a Non-Topological Probe of Soap Froth Dynamics

and