Searcharxiv⌕ Search

arXiv subjects

Robin C. Ball

Publications and source records attributed to Robin C. Ball.

27 records · Page 2Linked to original sources
arXiv2004

Off-lattice Noise Reduced Diffusion-limited Aggregation in Three Dimensions

Using off-lattice noise reduction it is possible to estimate the asymptotic properties of diffusion-limited aggregation clusters grown in three dimensions with greater accuracy than would otherwise be possible. The fractal dimension of these aggregates is found to be 2.50 +/- 0.01, in agreement with earlier studies, and the asymptotic value of the relative penetration depth is 0.122 +/- 0.002. The multipole powers of the growth measure also exhibit universal asymptotes. The fixed point noise reduction is estimated to be ε= 0.0035 meaning that large clusters can be identified with a low noise regime. The slowest correction to scaling exponents are measured for a number of properties of the clusters, and the exponent for the relative penetration depth and quadrupole moment are found to be significantly different from each other. The relative penetration depth exhibits the slowest correction to scaling of all quantities, which is consistent with a theoretical result derived in two dimensions.

cond-mat.stat-mech↗
arXiv2004

Growth by Random Walker Sampling, and Scaling of the Dielectric Breakdown Model

Random walkers absorbing on a boundary sample the Harmonic Measure linearly and independently: we discuss how the recurrence times between impacts enable non-linear moments of the measure to be estimated. From this we derive a new technique to simulate Dielectric Breakdown Model growth which is governed nonlinearly by the Harmonic Measure. Recurrence times are shown to be accurate and effective in probing the multifractal growth measure of diffusion limited aggregation. For the Dielectric Breakdown Model our new technique grows large clusters efficiently and we are led to significantly revise earlier exponent estimates. Previous results by two conformal mapping techniques were less converged than expected, and in particular a recent theoretical suggestion of superuniversality is firmly refuted.

cond-mat.stat-mech↗
arXiv2003

Diffusion-limited aggregation in channel geometry

We performed extensive numerical simulation of diffusion-limited aggregation in two dimensional channel geometry. Contrary to earlier claims, the measured fractal dimension D = 1.712 +- 0.002 and its leading correction to scaling are the same as in the radial case. The average cluster, defined as the average conformal map, is similar but not identical to Saffman-Taylor fingers.

cond-mat.stat-mech↗
arXiv2003

Characterisation of the probabilistic travelling salesman problem

We show that Stochastic Annealing can be successfully applied to gain new results on the Probabilistic Traveling Salesman Problem (PTSP). The probabilistic "traveling salesman" must decide on an a priori order in which to visit n cities (randomly distributed over a unit square) before learning that some cities can be omitted. We find the optimized average length of the pruned tour follows E(\bar{L}_{pruned}) = \sqrt{np} (0.872-0.105p) f(np) where p is the probability of a city needing to be visited, and f(np) -> 1 as np -> infinity. The average length of the a priori tour (before omitting any cities) is found to follow E(L_{a priori}) =\sqrt{n/p}β(p) where β(p)=1/(1.25-0.82 ln(p)) is measured for 0.05 < p < 0.6. Scaling arguments and indirect measurements suggest that β(p) tends towards a constant for p<0.03. Our stochastic annealing algorithm is based on limited sampling of the pruned tour lengths, exploiting the sampling error to provide the analogue of thermal fluctuations in simulated (thermal) annealing. The method has general application to the optimization of functions whose cost to evaluate rises with the precision required.

physics.comp-ph↗
arXiv2003

Stochastic Annealing

We demonstrate that is it possible to simulate a system in thermal equilibrium even when the energy cannot be evaluated exactly, provided the error distribution is known. This leads to an effective optimisation strategy for problems where the evaluation of each design can only be sampled statistically.

cond-mat.stat-mech↗
arXiv2002

Role of surface waves on the relation between crack speed and the work of fracture

We show that the delivery of fracture work to the tip of an advancing planar crack is strongly reduced by surface phonon emission, leading to forbidden ranges of crack speed. The emission can be interpreted through dispersion of the group velocity, and Rayleigh and Love branches contribute as well as other high frequency branches of the surface wave dispersion relations. We also show that the energy release rate which enters the Griffith criterion for the crack advance can be described as the product of the continuum solution with a function that only depends on the lattice geometry and describes the lattice influence on the phonon emission. Simulations are performed using a new finite element model for simulating elasticity and fractures. The model, built to allow fast and very large three-dimensional simulations, is applied to the simplified case of two dimensional samples.

cond-mat.mtrl-sci↗
arXiv2002

Off-lattice noise reduction and the ultimate scaling of DLA in two dimensions

Off-lattice DLA clusters grown with different levels of noise reduction are found to be consistent with a simple fractal fixed point. Cluster shapes and their ensemble variation exhibit a dominant slowest correction to scaling, and this also accounts for the apparent ``multiscaling'' in the DLA mass distribution. We interpret the correction to scaling in terms of renormalized noise. The limiting value of this variable is strikingly small and is dominated by fluctuations in cluster shape. Earlier claims of anomalous scaling in DLA were misled by the slow approach to this small fixed point value.

cond-mat.stat-mech↗
arXiv1998

Dynamics of Granular Stratification

Spontaneous stratification in granular mixtures has been recently reported by H. A. Makse et al. [Nature 386, 379 (1997)]. Here we study experimentally the dynamical processes leading to spontaneous stratification. Using a high-speed video camera, we study a rapid flow regime where the rolling grains size segregate during the avalanche. We characterize the dynamical process of stratification by measuring all relevant quantities: the velocity of the rolling grains, the velocity of the kink, the wavelength of the layers, the rate of collision between rolling and static grains, and all the angles of repose characterizing the mixture. The wavelength of the layers behaves linearly with the thickness of the layer of rolling grains (i.e., with the flow rate), in agreement with theoretical predictions. The velocity profile of the grains in the rolling phase is a linear function of the position of the grains along the moving layer. We also find that the speed of the upward-moving kink has the same value as the mean speed of the downward-moving grains. We measure the shape and size of the kink, as well as the profiles of the rolling and static phases of grains, and find agreement with recent theoretical predictions.

cond-mat.stat-mech↗