Asymptotic Scaling in a Model Class of Anomalous Reaction-Diffusion Equations

Abstract

We analyze asymptotic scaling properties of a model class of anomalous reaction-diffusion (ARD) equations. Numerical experiments show that solutions to these have, for large t, well defined scaling properties. We suggest a general framework to analyze asymptotic symmetry properties; this provides an analytical explanation of the observed asymptotic scaling properties for the considered ARD equations.     

Minimal path on the hierarchical diamond lattice

We consider the minimal paths on a hierarchical diamond lattice, where bonds are assigned a random weight. Depending on the initial distribution of weights, we find all possible asymptotic scaling properties. The different cases found are the small-disorder case, the analog of Lévy's distributions with a power-law decay at-, and finally a limit of large disorder which can be identified as a percolation problem. The asymptotic shape of the stable distributions of weights of the minimal path are obtained, as well as their scaling properties. As a side result, we obtain the asymptotic form of the distribution of effective percolation thresholds for finite-size hierarchical lattices.     

Roughness scaling and sensitivity to initial conditions in a symmetric restricted ballistic deposition model

In this work, we introduce a restricted ballistic deposition model with symmetric growth rules that favors the formation of local finite slopes. It is the simplest model which, even without including a diffusive relaxation mode of the interface, leads to a macroscopic groove instability. By employing a finite-size scaling of numerical simulation data, we determine the scaling behavior of the surface structure grown over a one-dimensional substrate of linear size L. We found that the surface profile develops a macroscopic groove with the asymptotic surface width scaling as , with . The early-time dynamics is governed by the scaling law , with . We further investigate the sensitivity to initial conditions of the present model by applying damage spreading techniques. We find that the early-time distance between two initially close surface configurations grows in a ballistic fashion as , but a slower Brownian-like scaling () sets up for evolution times much larger than a characteristic time scale . Received 26 May 2000     

Reaction-limited sintering in nearly saturated environments

We study the shape and growth rate of necks between sintered spheres with dissolution–precipitation dynamics in the reaction-limited regime. We determine the critical shape that separates those initial neck shapes that can sinter from those that necessarily dissolve, as well as the asymptotic evolving shape of sinters far from the critical shape. We compare our results with past results for the asymptotic neck shape in closely related but more complicated models of surface dynamics; in particular, we confirm a scaling conjecture, originally due to Kuczinsky. Finally, we consider the relevance of this problem to the diagenesis of sedimentary rocks and other applications.     

Asymptotic analysis of the lattice Boltzmann equation

In this article we analyze the lattice Boltzmann equation (LBE) by using the asymptotic expansion technique. We first relate the LBE to the finite discrete-velocity model (FDVM) of the Boltzmann equation with the diffusive scaling. The analysis of this model directly leads to the incompressible Navier–Stokes equations, as opposed to the compressible Navier–Stokes equations obtained by the Chapman–Enskog analysis with convective scaling. We also apply the asymptotic analysis directly to the fully discrete LBE, as opposed to the usual practice of analyzing a continuous equation obtained through the Taylor-expansion of the LBE. This leads to a consistency analysis which provides order-by-order information about the numerical solution of the LBE. The asymptotic technique enables us to analyze the structure of the leading order errors and the accuracy of numerically derived quantities, such as vorticity. It also justifies the use of Richardson’s extrapolation method. As an example, a two-dimensional Taylor-vortex flow is used to validate our analysis. The numerical results agree very well with our analytic predictions.     

A spatial finite size scaling method for calculation of critical parameters for weakly bound states

We present a finite size scaling approach to calculate critical parameters and exponents of quantum few-body system for which a bound state energy becomes absorbed or degenerates with the continuum. The scaling is done by introducing a cutoff radius for the potential. The asymptotic behavior for large values of the cutoff parameter is determined by the exact critical parameters.     

Multifractality of Brownian motion near absorbing polymers

We characterize the multifractal behavior of Brownian motion in the vicinity of an absorbing star polymer. We map the problem to an O(M)-symmetric phi(4)-field theory relating higher moments of the Laplacian field of Brownian motion to corresponding composite operators. The resulting spectra of scaling dimensions of these operators display the convexity properties that are necessarily found for multifractal scaling but unusual for power of field operators in field theory. Using a field-theoretic renormalization group approach we obtain the multifractal spectrum for absorption at the core of a polymer star as an asymptotic series. We evaluate these series using resummation techniques.     

Stiff polymers, foams, and fiber networks

We study the elasticity of fibrous materials composed of generalized stiff polymers. It is shown that, in contrast to cellular foam-like structures, affine strain fields are generically unstable. Instead, a subtle interplay between the architecture of the network and the elastic properties of its building blocks leads to intriguing mechanical properties with intermediate asymptotic scaling regimes. We present exhaustive numerical studies based on a finite element method complemented by scaling arguments.     

Multifractal behaviour of n-simplex lattice

We study the asymptotic behaviour of resistance scaling and fluctuation of resistance that give rise to flicker noise in an n-simplex lattice. We propose a simple method to calculate the resistance scaling and give a closed-form formula to calculate the exponent, β _L, associated with resistance scaling, for any n. Using current cumulant method we calculate the exact noise exponent for n-simplex lattices.     

Multiple scaling regimes in simple aging models

We investigate aging in glassy systems based on a simple model, where a point in configuration space performs thermally activated jumps between the minima of a random energy landscape. The model allows us to show explicitly a subaging behavior and multiple scaling regimes for the correlation function. Both the exponents characterizing the scaling of the different relaxation times with the waiting time and those characterizing the asymptotic decay of the scaling functions are obtained analytically by invoking a "partial equilibrium" concept.     

The non-linear sigma model: 3-loop renormalization and lattice scaling

We present a perturbative calculation of the first non-universal terms of the renormalization group functions for the non-linear σ-model regularized on a lattice, for a choice of different lattice actions. Such terms, hitherto unknown, fix the leading corrections to universal asymptotic scaling. However, by comparing with Monte Carlo simulations, one sees that the actual deviations from universal scaling are due to non-perturbative effects, and not to perturbative analytic corrections. The latter are nonetheless relevant to determine the ratios of scaling operators to Λ-parameters.     

Critical properties of hadronic matter in thermodynamical models

The temperature T₀ in certain thermodynamical models for strongly interacting systems taken as the critical point yields directly the critical exponents for the specific heat and compressibility. We discuss the implications of thermodynamical scaling using various asymptotic conditions.     

Scaling and decay in periodically driven scattering systems

We investigate irregular scattering in a periodically driven Hamiltonian system of one degree of freedom. The potential is asymptotically attracting, so there exist parabolically escaping scattering orbits, i.e. orbits with asymptotic energy E(out)=0. The scattering functions (i.e. the asymptotic out-variables as functions of an asymptotic in-variable) show a characteristic algebraic scaling in the vicinity of these orbits. This behavior is explained by asymptotic properties of the interaction. As a consequence, the number N(Deltat) of temporarily bound particles decays algebraically with the delay time Deltat, although no KAM scenario can be found in phase space. On the other hand, we find the number N(n) of temporarily bound particles to decay exponentially with the number n of zeros of x(t).     

Asymptotic analysis of the spatial discretization of radiation absorption and re-emission in Implicit Monte Carlo

We perform an asymptotic analysis of the spatial discretization of radiation absorption and re-emission in Implicit Monte Carlo (IMC), a Monte Carlo technique for simulating nonlinear radiative transfer. Specifically, we examine the approximation of absorption and re-emission by a spatially continuous artificial-scattering process and either a piecewise-constant or piecewise-linear emission source within each spatial cell. We consider three asymptotic scalings representing (i) a time step that resolves the mean-free time, (ii) a Courant limit on the time-step size, and (iii) a fixed time step that does not depend on any asymptotic scaling. For the piecewise-constant approximation, we show that only the third scaling results in a valid discretization of the proper diffusion equation, which implies that IMC may generate inaccurate solutions with optically large spatial cells if time steps are refined. However, we also demonstrate that, for a certain class of problems, the piecewise-linear approximation yields an appropriate discretized diffusion equation under all three scalings. We therefore expect IMC to produce accurate solutions for a wider range of time-step sizes when the piecewise-linear instead of piecewise-constant discretization is employed. We demonstrate the validity of our analysis with a set of numerical examples.     

Continuum limit of a hierarchical SU(2) lattice gauge theory in 4 dimensions

We study nonperturbative renormalizability of ad=4 hierarchical SU(2) gauge model that realizes Migdal's recursion relation as an exact renormalization group transformation. A continuum limit of effective actions is shown to exist as the scaling limit, both for initial Wilson and heat kernel actions. These limit effective actions exhibit ultraviolet asymptotic freedom and provide a strictly positive string tension.     

Unstable state dynamics: Treatment of colored noise

The diffusion of a particle set near an unstable point in a bistable potential is considered. The scaling theory of fluctuations proposed originally for onedimensional systems driven by Gaussian white noise is extended to arbitrary dimensions. The merits and drawbacks of the scaling theory are discussed by taking a model problem in one dimension. It is shown in passing that the saddle point approximation enables one to get analytic expressions for various moments of the stochastic process. The two different methods to include asymptotic fluctuations-which are absent in the usual scaling solution-are shown to be equivalent. An alternate way of including asymptotic fluctuations is attempted by solving the associated Fokker-Planck equation using the Fer formula. The reason for the failure of this method is traced. After this, it is argued that the unified scaling theory should be applicable for treatment of colored noise as well, for the scaling assumption is independent of the statistical property of the driving noise. Explicit Monte Carlo simulation of a model onedimensional system driven by exponentially correlated Gaussian noise is performed and compared with the scaling solution to bolster this point. The agreement is very good.     

A JUSTIFICATION FOR THE SCALING OF THE THOM-SYSTEM

In the present paper we discuss the critical behavior of Thornsystem using Catastrophe Theory. The universal critical asymptotic form of the family of free energy functions for Thomsystem with one order parameter and two field parameters is obtained. The expressions of critical exponents, the scaling laws, and the scaling hypotheses are all derived from this universal asymptotic form.     

Diffractive dissociation and the early KNO scaling

The assumption of asymptotic validity of early KNO scaling combined with the existence of diffraction is shown to result in strong restrictions on the asymptotic multiplicity distributions.     

Dynamical scaling: the two-dimensional XY model following a quench

To sensitively test scaling in the two-dimensional XY model quenched from high temperatures into the ordered phase, we study the difference between measured correlations and the (scaling) results of a Gaussian-closure approximation. We also directly compare various length scales. All of our results are consistent with dynamical scaling and an asymptotic growth law L approximately (t/ln[t/t(0)])(1/2), though with a time scale t(0) that depends on the length scale in question. We then reconstruct correlations from the minimal-energy configuration consistent with the vortex positions, and find them significantly different from the "natural" correlations - though both scale with L. This indicates that both topological (vortex) and nontopological "spin-wave" contributions to correlations are relevant arbitrarily late after the quench. We also present a consistent definition of dynamical scaling applicable more generally, and emphasize how to generalize our approach to other quenched systems where dynamical scaling is in question. Our approach directly applies to planar liquid-crystal systems.     

Exact scaling behavior of partially convex vesicles

We solve analytically for the perimeter-area generating functions for two models of vesicles. While from the solution of the first model, staircase polygons, one can easily extract the asymptotic scaling behavior, the exact solution of the second, column-convex polygons, is difficult to analyze. This leads us to apply a recently developed method for deriving the scaling behavior indirectly, utilizing a set of nonlinear differential equations. One result of this work is a nontrivial confirmation of the scaling/universality hypothesis.