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 one 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.

Predicting nucleation using machine learning in the Ising model.

We use a convolutional neural network (CNN) and two logistic regression models to predict the probability of nucleation in the two-dimensional Ising model. The three methods successfully predict the probability for the nearest-neighbor Ising model for which classical nucleation is observed. The CNN outperforms the logistic regression models near the spinodal of the long-range Ising model, but the accuracy of its predictions decreases as the quenches approach the spinodal. An occlusion analysis suggests that this decrease is due to the vanishing difference between the density of the nucleating droplet and the background. Our results are consistent with the general conclusion that predictability decreases near a critical point.

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 Liu et al. [K. K. L. Liu et al., preceding paper, Phys. Rev. E 104, 014150 (2021)10.1103/PhysRevE.104.014150] 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 λ. The model is shown numerically to have a phase transition at λ=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, a critical slowing down, and the values of the critical exponents and introduces an energy whose probability satisfies the Boltzmann distribution for λ<1, implying that the system is in thermodynamic equilibrium in the limit that N→∞. We show that the values of the critical exponents obtained by varying λ 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.

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 λ. Our numerical results indicate that the model has a critical point at λ=1 between a phase for λ<1 with economic mobility and exponentially growing wealth of all agents and a nonstationary phase for λ≥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 λ<1. Our estimates of various critical exponents are consistent with a mean-field theory [see W. Klein et al., following paper, Phys. Rev. E 104, 014151 (2021)10.1103/PhysRevE.104.014151]. 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 phase 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 following theoretical paper contribute to our understanding of whether the methods of equilibrium statistical mechanics can be applied to economic systems.

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 system is effectively ergodic because the time-averaged stress on each site converges to the global spatial average. In contrast, below the critical noise, the stress on individual sites becomes trapped in different limit cycles, and the system is not ergodic. To characterize this transition, we use ideas from the study of dynamical systems and compute recurrence plots and the recurrence rate. The order parameter is identified as the recurrence rate averaged over all sites and exhibits a jump at the critical noise. We also use ideas from percolation theory and analyze the clusters of failed sites to find numerical evidence that the transition, when approached from above, can be characterized by exponents that are consistent with hyperscaling.

Prediction in a driven-dissipative system displaying a continuous phase transition using machine learning.

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 predictability decreases as criticality is approached and that prediction is possible only for large, nonscaling events. Our results suggest that earthquake faults that satisfy Gutenberg-Richter scaling are difficult to forecast.

Near-mean-field behavior in the generalized Burridge-Knopoff earthquake model with variable-range stress transfer.

Simple models of earthquake faults are important for understanding the mechanisms for their observed behavior in nature, such as Gutenberg-Richter scaling. Because of the importance of long-range interactions in an elastic medium, we generalize the Burridge-Knopoff slider-block model to include variable range stress transfer. We find that the Burridge-Knopoff model with long-range stress transfer exhibits qualitatively different behavior than the corresponding long-range cellular automata models and the usual Burridge-Knopoff model with nearest-neighbor stress transfer, depending on how quickly the friction force weakens with increasing velocity. Extensive simulations of quasiperiodic characteristic events, mode-switching phenomena, ergodicity, and waiting-time distributions are also discussed. Our results are consistent with the existence of a mean-field critical point and have important implications for our understanding of earthquakes and other driven dissipative systems.

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 are 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 forms 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.

Approaching equilibrium and the distribution of clusters.

We investigate the approach to stable and metastable equilibrium in both nearest-neighbor and long-range Ising models using a cluster representation. The distribution of nucleation times is determined using the Metropolis algorithm and the corresponding phi4 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.

Structure of fluctuations near mean-field critical points and spinodals and its implication for physical processes.

We analyze the structure of fluctuations near critical points and spinodals in mean-field and near-mean-field systems. Unlike systems that are non-mean-field, for which a fluctuation can be represented by a single cluster in a properly chosen percolation model, a fluctuation in mean-field and near-mean-field systems consists of a large number of clusters, which we term fundamental clusters. The structure of the latter and the way that they form fluctuations has important physical consequences for phenomena as diverse as nucleation in supercooled liquids, spinodal decomposition and continuous ordering, and the statistical distribution of earthquakes. The effects due to the fundamental clusters implies that they are physical objects and not only mathematical constructs.

Overcoming the slowing down of flat-histogram Monte Carlo simulations: cluster updates and optimized broad-histogram ensembles.

We study the performance of Monte Carlo simulations that sample a broad histogram in energy by determining the mean first-passage time to span the entire energy space of d-dimensional ferromagnetic Ising/Potts models. We first show that flat-histogram Monte Carlo methods with single-spin flip updates such as the Wang-Landau algorithm or the multicanonical method perform suboptimally in comparison to an unbiased Markovian random walk in energy space. For the d = 1, 2, 3 Ising model, the mean first-passage time tau scales with the number of spins N = L(d) as tau proportional N2L(z). The exponent z is found to decrease as the dimensionality d is increased. In the mean-field limit of infinite dimensions we find that z vanishes up to logarithmic corrections. We then demonstrate how the slowdown characterized by z > 0 for finite d can be overcome by two complementary approaches--cluster dynamics in connection with Wang-Landau sampling and the recently developed ensemble optimization technique. Both approaches are found to improve the random walk in energy space so that tau proportional N2 up to logarithmic corrections for the d = 1, 2 Ising model.

Zeros of the partition function and pseudospinodals in long-range Ising models.

The relation between the zeros of the partition function and spinodal critical points in Ising models with long-range interactions is investigated. We find that the spinodal is associated with the zeros of the partition function in four-dimensional complex temperature/magnetic field space. The zeros approach the real temperature/magnetic field plane as the range of interaction increases.

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 the critical exponents for thermodynamic quantities such as the mean magnetization and the susceptibility can be obtained using finite size scaling with the scaling variable equal to N, the number of spins. Surprisingly, the mean value and the most probable value of the magnetization are found to scale differently with N at the critical temperature of the infinite system, and the magnetization probability distribution 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 that hyperscaling can be restored by increasing N while holding the Ginzburg parameter rather than the temperature fixed, or by doing finite size scaling at the pseudocritical temperature where the susceptibility is a maximum for a given value of N. We conclude that finite size scaling for the fully connected Ising model yields different results depending on how the mean-field limit is approached.

Simulation of the Burridge-Knopoff model of earthquakes with variable range stress transfer.

Simple models of earthquake faults are important for understanding the mechanisms for their observed behavior, such as Gutenberg-Richter scaling and the relation between large and small events, which is the basis for various forecasting methods. Although cellular automaton models have been studied extensively in the long-range stress transfer limit, this limit has not been studied for the Burridge-Knopoff model, which includes more realistic friction forces and inertia. We find that the latter model with long-range stress transfer exhibits qualitatively different behavior than both the long-range cellular automaton models and the usual Burridge-Knopoff model with nearest-neighbor springs, depending on the nature of the velocity-weakening friction force. These results have important implications for our understanding of earthquakes and other driven dissipative systems.

Simulations of spinodal nucleation in systems with elastic interactions.

Systems with long-range interactions quenched into a metastable state near the pseudospinodal exhibit nucleation that is qualitatively different from classical nucleation near the coexistence curve. We observe nucleation droplets in Langevin simulations of a two-dimensional model of martensitic transformations and determine that the structure of the nucleating droplet differs from the stable martensite structure. Our results, together with experimental measurements of the phonon dispersion curve, allow us to predict the nature of the droplet. The results have implications for nucleation in many solid-solid transitions and the structure of the final state.

High precision measurement of the static dipole polarizability of cesium.

The cesium 6(2)S(1/2) scalar dipole polarizability alpha(0) has been determined from the time-of-flight of laser cooled and launched cesium atoms traveling through an electric field. We find alpha(0)=6.611+/-0.009 x 10(-39) C m(2)/V=59.42+/-0.08 x 10(-24) cm(3)=401.0+/-0.6a(3)(0). The 0.14% uncertainty is a factor of 14 improvement over the previous measurement. Values for the 6(2)P(1/2) and 6(2)P(3/2) lifetimes and the 6(2)S(1/2) cesium-cesium dispersion coefficient C6 are determined from alpha(0) using the procedure of Derevianko and Porsev [Phys. Rev. A 65, 053403 (2002)]].