arXiv subjects
Publications and source records attributed to Karin A. Dahmen.
We exactly prove the no-passing rule in the ground state evolution of the random-field Ising model (RFIM) with monotonically varying external field. In particular, we show that the application of the no-passing rule can speed up the calculation of the zero-temperature equilibrium $M(H)$ curve dramatically.
We present numerical studies of zero-temperature Gaussian random-field Ising model (zt-GRFIM) in both equilibrium and non-equilibrium. We compare the no-passing rule, mean-field exponents and universal quantities in 3D (avalanche critical exponents, fractal dimensions, scaling functions and anisotropy measures) for the equilibrium and non-equilibrium disorder-induced phase transitions. We show compelling evidence that the two transitions belong to the same universality class.
Earthquake phenomenology exhibits a number of power law distributions including the Gutenberg-Richter frequency-size statistics and the Omori law for aftershock decay rates. In search for a basic model that renders correct predictions on long spatio-temporal scales, we discuss results associated with a heterogeneous fault with long range stress-transfer interactions. To better understand earthquake dynamics we focus on faults with Gutenberg-Richter like earthquake statistics and develop two universal scaling functions as a stronger test of the theory against observations than mere scaling exponents that have large error bars. Universal shape profiles contain crucial information on the underlying dynamics in a variety of systems. As in magnetic systems, we find that our analysis for earthquakes provides a good overall agreement between theory and observations, but with a potential discrepancy in one particular universal scaling function for moment-rates. The results reveal interesting connections between the physics of vastly different systems with avalanche noise.
This is a review article of our work on hysteresis, avalanches, and criticality. We provide an extensive introduction to scaling and renormalization--group ideas, and discuss analytical and numerical results for size distributions, correlation functions, magnetization, avalanche durations and average avalanche shapes, and power spectra. We focus here on applications to magnetic Barkhausen noise.
Utilizing the Haar transform, we study the higher order spectral properties of mean field avalanche models, whose avalanche dynamics are described by Poisson statistics at a critical point or critical depinning transition. The Haar transform allows us to obtain a time series of noise powers, $H(f_1,t)$, that gives improved time resolution over the Fourier transform. Using $H(f_1,t)$ we analytically calculate the Haar power spectrum, the real 1.5 spectra, the second spectra, and the real cross second spectra in mean field avalanche models. We verify our theoretical results with the numerical results from a simulation of the T=0 mean field nonequilibrium random field Ising model (RFIM). We also extend our higher order spectra calculation to data obtained from a numerical simulation of the T=0 infinite range RFIM for $d=3$, and experimental data obtained from an amorphous alloy, $Fe_{21}Co_{64}B_{15}$. We compare the results and obtain novel exponents.
We present a novel analytic calculation of the Haar power spectra, and various higher order spectra, of mean field avalanche models. We also compute these spectra from a simulation of the zero-temperature mean field RFIM and infinite range RFIM model for $d = 3$. We compare the results and obtain novel exponents.
Spreading of bacteria in a highly advective, disordered environment is examined. Predictions of super-diffusive spreading for a simplified reaction-diffusion equation are tested. Concentration profiles display anomalous growth and super-diffusive spreading. A perturbation analysis yields a crossover time between diffusive and super-diffusive behavior. The time's dependence on the convection velocity and disorder is tested. Like the simplified equation, the full linear reaction-diffusion equation displays super-diffusive spreading perpendicular to the convection. However, for mean positive growth rates the full nonlinear reaction-diffusion equation produces symmetric spreading with a Fisher wavefront, whereas net negative growth rates cause an asymmetry, with a slower wavefront velocity perpendicular to the convection.
Universal shape profiles in a variety of systems contain crucial information on the underlying dynamics. We develop such shape profiles for earthquakes as a stronger test of theory against observations. The earthquake analysis shows good agreement between theory and observations, with a discrepancy in one universal profile, just as in magnets. The analysis provides a stronger support for the idea that earthquakes are associated with criticality rather than event statistics only. The results point to the existence of deep connections between the physics of avalanches in different systems.
Barkhausen noise as found in magnets is studied both with and without the presence of long-range (LR) demagnetizing fields using the non-equilibrium, zero-temperature random-field Ising model. Two distinct subloop behaviors arise and are shown to be in qualitative agreement with experiments on thin film magnets and soft ferromagnets. With LR fields present subloops resemble a self-organized critical system, while their absence results in subloops that reflect the critical point seen in the saturation loop as the system disorder is changed. In the former case, power law distributions of noise are found in subloops, while in the latter case history-induced critical scaling is studied in avalanche size distributions, spin-flip correlation functions, and finite-size scaling of the second moments of the size distributions. Results are presented for simulations of over 10^8 spins.
We study the dynamics of a viscoelastic medium driven through quenched disorder by expanding about mean field theory in $6-ε$ dimensions. The model exhibits a critical point separating a region where the dynamics is hysteretic with a macroscopic jump between strongly pinned and weakly pinned states, from a region where the sliding state is unique and no jump occurs. The disappearance of the jump at the critical point is described by universal exponents. As suggested in \onlinecite{MMP00}, the model appears to be in the same universality class as the zero-temperature random field Ising model of hysteresis in magnets.
The demagnetization curve, or initial magnetization curve, is studied by examining the embedded Barkhausen noise using the non-equilibrium, zero temperature random-field Ising model. The demagnetization curve is found to reflect the critical point seen as the system's disorder is changed. Critical scaling is found for avalanche sizes and the size and number of spanning avalanches. The critical exponents are derived from those related to the saturation loop and subloops. Finally, the behavior in the presence of long range demagnetizing fields is discussed. Results are presented for simulations of up to one million spins.
Crackling noise arises when a system responds to changing external conditions through discrete, impulsive events spanning a broad range of sizes. A wide variety of physical systems exhibiting crackling noise have been studied, from earthquakes on faults to paper crumpling. Because these systems exhibit regular behavior over many decades of sizes, their behavior is likely independent of microscopic and macroscopic details, and progress can be made by the use of very simple models. The fact that simple models and real systems can share the same behavior on a wide range of scales is called universality. We illustrate these ideas using results for our model of crackling noise in magnets, explaining the use of the renormalization group and scaling collapses. This field is still developing: we describe a number of continuing challenges.
We study the pairwise annihilation process $A+A\to$ inert of a number of random walkers, which originally are localized in a small region in space. The size of the colony and the typical distance between particles increases with time and, consequently, the reaction rate goes down. In the long time limit the spatial density profile becomes scale invariant. The mean-field approximation of this scenario bears some surprises. It predicts an upper critical dimension $d_c=2$, with logarithmic corrections at the critical dimension and nontrivial scaling behavior for d<2. Based on an exact solution of the one dimensional system we conjecture that the mean-field {\it exponents} are in fact correct even below the upper critical dimension, down to d=1, while the corresponding scaling function that describes the spatial density profile, changes for d<2 due to the stochastic fluctuations.
In studying the avalanches and noise in a model of hysteresis loops we have developed two relatively straightforward algorithms which have allowed us to study large systems efficiently. Our model is the random-field Ising model at zero temperature, with deterministic albeit random dynamics. The first algorithm, implemented using sorted lists, scales in computer time as O(N log N), and asymptotically uses N (sizeof(double)+ sizeof(int)) bits of memory. The second algorithm, which never generates the random fields, scales in time as O(N \log N) and asymptotically needs storage of only one bit per spin, about 96 times less memory than the first algorithm. We present results for system sizes of up to a billion spins, which can be run on a workstation with 128MB of RAM in a few hours. We also show that important physical questions were resolved only with the largest of these simulations.
We review localization with non-Hermitian time evolution as applied to simple models of population biology with spatially varying growth profiles and convection. Convection leads to a constant imaginary vector potential in the Schroedinger-like operator which appears in linearized growth models. We illustrate the basic ideas by reviewing how convection affects the evolution of a population influenced by a simple square well growth profile. Results from discrete lattice growth models in both one and two dimensions are presented. A set of similarity transformations which lead to exact results for the spectrum and winding numbers of eigenfunctions for random growth rates in one dimension is described in detail. We discuss the influence of boundary conditions, and argue that periodic boundary conditions lead to results which are in fact typical of a broad class of growth problems with convection.
We propose a simple experiment to study delocalization and extinction in inhomogeneous biological systems. The nonlinear steady state for, say, a bacteria colony living on and near a patch of nutrient or favorable illumination (``oasis'') in the presence of a drift term (``wind'') is computed. The bacteria, described by a simple generalization of the Fisher equation, diffuse, divide A -> A+A, die A -> 0, and annihilate A+A -> 0. At high wind velocities all bacteria are blown into an unfavorable region (``desert''), and the colony dies out. At low velocity a steady state concentration survives near the oasis. In between these two regimes there is a critical velocity at which bacteria first survive. If the ``desert'' supports a small nonzero population, this extinction transition is replaced by a delocalization transition with increasing velocity. Predictions for the behavior as a function of wind velocity are made for one and two dimensions.
We present numerical simulations of avalanches and critical phenomena associated with hysteresis loops, modeled using the zero-temperature random-field Ising model. We study the transition between smooth hysteresis loops and loops with a sharp jump in the magnetization, as the disorder in our model is decreased. In a large region near the critical point, we find scaling and critical phenomena, which are well described by the results of an epsilon expansion about six dimensions. We present the results of simulations in 3, 4, and 5 dimensions, with systems with up to a billion spins (1000^3).
Hysteresis, the lag between the force and the response, is often associated with noisy, jerky motion which have recently been called ``avalanches''. The interesting question is why the avalanches come in such a variety of sizes: naively one would expect either all small events or one large one. Power-law distributions are often seen near transitions, or critical points. We study the zero temperature random field Ising model as a model for noise and avalanches in hysteretic systems. Tuning the amount of disorder in the system, we find an ordinary critical point with avalanches on all length scales. We study this critical point in 6-epsilon dimensions, and with simulations in 2, 3, 4, 5, 7, and 9 dimensions with systems as large as 1000^3. The power-law distributions in principle only occur for a special value of the randomness (the critical point), but many decades of scaling occur quite far from this special value: three decades of scaling (a publishable experiment) are obtained a factor of two away from the critical value. Perhaps many of the power laws observed in nature are just systems near their critical points?