arXiv subjects

W. Klein

Publications and source records attributed to W. Klein.

At least 19 recordsLinked to original sources

arXiv2022

Cluster Scaling and Critical Points: A Cautionary Tale

Many systems in nature are conjectured to exist at a critical point, including the brain and earthquake faults. The primary reason for this conjecture is that the distribution of clusters (avalanches of firing neurons in the brain or regions of slip in earthquake faults) can be described by a power law. Because there are other mechanisms such as $1/f$ noise that can produce power laws, other criteria that the cluster critical exponents must satisfy can be used to conclude whether or not the observed power law behavior indicates an underlying critical point rather than an alternate mechanism. We show how a possible misinterpretation of the cluster scaling data can lead to incorrectly conclude that the measured critical exponents do not satisfy these criteria. Examples of the possible misinterpretation of the data for one-dimensional random site percolation and the one-dimensional Ising model are presented. We stress that the interpretation of a power law cluster distribution indicating the presence of a critical point is subtle, and its misinterpretation might lead to the abandonment of a promising area of research.

cond-mat.stat-mech↗

arXiv2021

Simulation of a generalized asset exchange model with economic growth and wealth distribution

The agent-based Yard-Sale model of wealth inequality is generalized to incorporate exponential economic growth and its distribution. The distribution of economic growth is nonuniform and is determined by the wealth of each agent and a parameter $\lambda$. Our numerical results indicate that the model has a critical point at $\lambda=1$ between a phase for $\lambda < 1$ with economic mobility and exponentially growing wealth of all agents and a non-stationary phase for $\lambda \geq 1$ with wealth condensation and no mobility. We define the energy of the system and show that the system can be considered to be in thermodynamic equilibrium for $\lambda < 1$. Our estimates of various critical exponents are consistent with a mean-field theory (see following paper). The exponents do not obey the usual scaling laws unless a combination of parameters that we refer to as the Ginzburg parameter is held fixed as the transition is approached. The model illustrates that both poorer and richer agents benefit from economic growth if its distribution does not favor the richer agents too strongly. This work and the accompanying theory paper contribute to understanding whether the methods of equilibrium statistical mechanics can be applied to economic systems.

cond-mat.stat-mech↗

arXiv2021

Mean-field theory of an asset exchange model with economic growth and wealth distribution

We develop a mean-field theory of the growth, exchange and distribution (GED) model introduced by Kang et al. (preceding paper) that accurately describes the phase transition in the limit that the number of agents $N$ approaches infinity. The GED model is a generalization of the Yard-Sale model in which the additional wealth added by economic growth is nonuniformly distributed to the agents according to their wealth in a way determined by the parameter $\lambda$. The model was shown numerically to have a phase transition at $\lambda=1$ and be characterized by critical exponents and critical slowing down. Our mean-field treatment of the GED model correctly predicts the existence of the phase transition, critical slowing down, the values of the critical exponents, and introduces an energy whose probability satisfies the Boltzmann distribution for $\lambda < 1$, implying that the system is in thermodynamic equilibrium in the limit that $N \to \infty$. We show that the values of the critical exponents obtained by varying $\lambda$ for a fixed value of $N$ do not satisfy the usual scaling laws, but do satisfy scaling if a combination of parameters, which we refer to as the Ginzburg parameter, is much greater than one and is held constant. We discuss possible implications of our results for understanding economic systems and the subtle nature of the mean-field limit in systems with both additive and multiplicative noise.

cond-mat.stat-mech↗

arXiv2019

Scaling of causal neural avalanches in a neutral model

Neural avalanches are collective firings of neurons that exhibit emergent scale-free behavior. Understanding the nature and distribution of these avalanches is an important element in understanding how the brain functions. We study a model of neural avalanches for which the dynamics are governed by neutral theory. The neural avalanches are defined using causal connections between the firing neurons. We analyze the scaling of causal neural avalanches as the critical point is approached from the absorbing phase. By using cluster analysis tools from percolation theory, we characterize the critical properties of the neural avalanches. We identify the tuning parameters consistent with experiments. The scaling hypothesis provides a unified explanation of the power laws which characterize the critical point. The critical exponents characterizing the avalanche distributions and divergence of the response functions are consistent with the predictions of the scaling hypothesis. We use a universal scaling function for the avalanche profile to find that the firing rates for avalanches of different durations show data collapse after appropriate rescaling. We also find data collapse for the avalanche distribution functions, which is stronger evidence of criticality than just the existence of power laws. Critical slowing-down and power law relaxation of avalanches is observed as the system is tuned to its critical point. We discuss how our results motivate future empirical studies of criticality in the brain.

physics.bio-ph↗

arXiv2019

Prediction in a driven-dissipative system displaying a continuous phase transition

Prediction in complex systems at criticality is believed to be very difficult, if not impossible. Of particular interest is whether earthquakes, whose distribution follows a power law (Gutenberg-Richter) distribution, are in principle unpredictable. We study the predictability of event sizes in the Olmai-Feder-Christensen model at different proximities to criticality using a convolutional neural network. The distribution of event sizes satisfies a power law with a cutoff for large events. We find that prediction decreases as criticality is approached and that prediction is possible only for large, non-scaling events. Our results suggest that earthquake faults that satisfy Gutenberg-Richter scaling are difficult to forecast.

cond-mat.stat-mech↗

arXiv2019

Novel effective ergodicity breaking phase transition in a driven-dissipative system

We show that the Olami-Feder-Christensen model exhibits an effective ergodicity breaking transition as the noise is varied. Above the critical noise, the average stress on each site converges to the global average. Below the critical noise, the stress on individual sites becomes trapped in different limit cycles. We use ideas from the study of dynamical systems and compute recurrence plots and the recurrence rate. We identify the order parameter as the recurrence rate averaged over all sites and find numerical evidence that the transition can be characterized by exponents that are consistent with hyperscaling.

nlin.CD↗

arXiv2014

Anomalous mean-field behavior of the fully connected Ising model

Although the fully connected Ising model does not have a length scale, we show that its critical exponents can be found using finite size scaling with the scaling variable equal to N, the number of spins. We find that at the critical temperature of the infinite system the mean value and the most probable value of the magnetization scale differently with N, and the probability distribution of the magnetization is not a Gaussian, even for large N. Similar results inconsistent with the usual understanding of mean-field theory are found at the spinodal. We relate these results to the breakdown of hyperscaling and show how hyperscaling can be restored by increasing N while holding the Ginzburg parameter rather than the temperature fixed.

cond-mat.stat-mech↗

arXiv2013

Liquid to solid nucleation via onion structure droplets

We study homogeneous nucleation from a deeply quenched metastable liquid to a spatially modulated phase. We find, for a general class of density functional theories, that the universally favored nucleating droplet in dimensions $d \geq 3$ is spherically symmetric with radial modulations resembling the layers of an onion. The existence of this droplet has important implications for systems with effective long-range interactions, and potentially applies to polymers, plasmas, and metals.

cond-mat.stat-mech↗

arXiv2013

The Effect of Growth On Equality in Models of the Economy

We investigate the relation between economic growth and equality in a modified version of the agent-based asset exchange model (AEM). The modified model is a driven system that for a range of parameter space is effectively ergodic in the limit of an infinite system. We find that the belief that "a rising tide lifts all boats" does not always apply, but the effect of growth on the wealth distribution depends on the nature of the growth. In particular, we find that the rate of growth, the way the growth is distributed, and the percentage of wealth exchange determine the degree of equality. We find strong numerical evidence that there is a phase transition in the modified model, and for a part of parameter space the modified AEM acts like a geometric random walk.

q-fin.GN↗

arXiv2013

Effect of Dilution on Spinodals and Pseudospinodals

We investigate the effect of quenched dilution on the critical and spinodal points in the infinite range (mean-field) and long-range (near-mean-field) Ising model. We find that unlike the short-range Ising model, the effect of the dilution is not simply related to the divergence of the specific heat, i.e., the Harris criterion. We also find the mean-field behavior differs from that of $d=4$ (upper critical dimension) nearest neighbor model at the critical point. These results are an important first step for understanding the effect of the spinodal on the nucleation process as well as the properties of the metastable state in systems of considerable interest in material science, geophysics, and econophysics which in general have defects.

cond-mat.stat-mech↗

arXiv2011

Scaling of Earthquake Models with Inhomogeneous Stress Dissipation

Natural earthquake fault systems are highly non-homogeneous. The inhomogeneities occur be- cause the earth is made of a variety of materials which hold and dissipate stress differently. In this work, we study scaling in earthquake fault models which are variations of the Olami-Feder- Christensen (OFC) and Rundle-Jackson-Brown (RJB) models. We use the scaling to explore the effect of spatial inhomogeneities due to damage and inhomogeneous stress dissipation in the earthquake-fault-like systems when the stress transfer range is long, but not necessarily longer than the length scale associated with the inhomogeneities of the system. We find that the scaling depends not only on the amount of damage, but also on the spatial distribution of that damage.

physics.geo-ph↗

arXiv2010

Gutenberg-Richter Scaling - A New Paradigm

We introduce a new model for an earthquake fault system that is composed of non-interacting simple lattice models with different levels of damage denoted by $q$. The undamaged lattice models ($q=0$) have Gutenberg-Richter scaling with a cumulative exponent $β=1/2$, whereas the damaged models do not have well defined scaling. However, if we consider the "fault system" consisting of all models, damaged and undamaged, we get excellent scaling with the exponent depending on the relative frequency with which faults with a particular amount of damage occur in the fault system. This paradigm combines the idea that Gutenberg-Richter scaling is associated with an underlying critical point with the notion that the structure of a fault system also affects the statistical distribution of earthquakes. In addition, it provides a framework in which the variation, from one tectonic region to another, of the scaling exponent, or $b$-value, can be understood.

cond-mat.stat-mech↗

arXiv2010

A damage-mechanics model for fracture nucleation and propagation

In this paper a composite model for earthquake rupture initiation and propagation is proposed. The model includes aspects of damage mechanics, fiber-bundle models, and slider-block models. An array of elements is introduced in analogy to the fibers of a fiber bundle. Time to failure for each element is specified from a Poisson distribution. The hazard rate is assumed to have a power-law dependence on stress. When an element fails it is removed, the stress on a failed element is redistributed uniformly to a specified number of neighboring elements in a given range of interaction. Damage is defined to be the fraction of elements that have failed. Time to failure and modes of rupture propagation are determined as a function of the hazard-rate exponent and the range of interaction.

physics.geo-ph↗

arXiv2009

A Cellular Automaton Model of Damage

We investigate the role of equilibrium methods and stress transfer range in describing the process of damage. We find that equilibrium approaches are not applicable to the description of damage and the catastrophic failure mechanism if the stress transfer is short ranged. In the long range limit, equilibrium methods apply only if the healing mechanism associated with ruptured elements is instantaneous. Furthermore we find that the nature of the catastrophic failure depends strongly on the stress transfer range. Long range transfer systems have a failure mechanism that resembles nucleation. In short range stress transfer systems, the catastrophic failure is a continuous process that, in some respects, resembles a critical point.

cond-mat.stat-mech↗

arXiv2008

Early time kinetics of systems with spatial symmetry breaking

In this paper we present a study of the early stages of unstable state evolution of systems with spatial symmetry changes. In contrast to the early time linear theory of unstable evolution described by Cahn, Hilliard, and Cook, we develop a generalized theory that predicts two distinct stages of the early evolution for symmetry breaking phase transitions. In the first stage the dynamics is dominated by symmetry preserving evolution. In the second stage, which shares some characteristics with the Cahn-Hilliard-Cook theory, noise driven fluctuations break the symmetry of the initial phase on a time scale which is large compared to the first stage for systems with long interaction ranges. To test the theory we present the results of numerical simulations of the initial evolution of a long-range antiferromagnetic Ising model quenched into an unstable region. We investigate two types of symmetry breaking transitions in this system: disorder-to-order and order-to-order transitions. For the order-to-order case, the Fourier modes evolve as a linear combination of exponentially growing or decaying terms with different time scales.

cond-mat.stat-mech↗

arXiv2008

Beyond Cahn-Hilliard-Cook: Early time behavior of symmetry breaking phase transition kinetics

We extend the early time ordering theory of Cahn, Hilliard, and Cook (CHC) so that our generalized theory applies to solid-to-solid transitions. Our theory involves spatial symmetry breaking (the initial phase contains a symmetry not present in the final phase). The predictions of our generalization differ from those of the CHC theory in two important ways: exponential growth does not begin immediately following the quench, and the objects that grow exponentially are not necessarily Fourier modes. Our theory is consistent with simulation results for the long-range antiferromagnetic Ising model.

cond-mat.stat-mech↗

arXiv2007

Approaching equilibrium and the distribution of clusters

We investigate the approach to stable and metastable equilibrium in Ising models using a cluster representation. The distribution of nucleation times is determined using the Metropolis algorithm and the corresponding $ϕ^{4}$ model using Langevin dynamics. We find that the nucleation rate is suppressed at early times even after global variables such as the magnetization and energy have apparently reached their time independent values. The mean number of clusters whose size is comparable to the size of the nucleating droplet becomes time independent at about the same time that the nucleation rate reaches its constant value. We also find subtle structural differences between the nucleating droplets formed before and after apparent metastable equilibrium has been established.

cond-mat.stat-mech↗

arXiv2007

Homogeneous and heterogeneous nucleation of Lennard-Jones liquids

The homogeneous and heterogeneous nucleation of a Lennard-Jones liquid is investigated using the umbrella sampling method. The free energy cost of forming a nucleating droplet is determined as a function of the quench depth, and the saddle point nature of the droplets is verified using an intervention technique. The structure and symmetry of the nucleating droplets is found for a range of temperatures. We find that for deep quenches the nucleating droplets become more anisotropic and diffuse with no well defined core or surface. The environment of the nucleating droplets form randomly stacked hexagonal planes. This behavior is consistent with a spinodal nucleation interpretation. We also find that the free energy barrier for heterogeneous nucleation is a minimum when the lattice spacing of the impurity equals the lattice spacing of the equilibrium crystalline phase. If the lattice spacing of the impurity is different, the crystal grows into the bulk instead of wetting the impurity.

cond-mat.stat-mech↗