Nonequilibrium phase ordering with a global conservation law

Socioeconomic agents as active matter in nonequilibrium Sakoda-Schelling models

How robust are socioeconomic agent-based models with respect to the details of the agents' decision rule? We tackle this question by considering an occupation model in the spirit of the Sakoda-Schelling model, historically introduced to shed light on segregation dynamics among human groups. For a large class of utility functions and decision rules, we pinpoint the nonequilibrium nature of the agent dynamics, while recovering an equilibrium-like phase separation phenomenology. Within the mean-field approximation we show how the model can be mapped, to some extent, onto an active matter field description. Finally, we consider nonreciprocal interactions between two populations and show how they can lead to nonsteady macroscopic behavior. We believe our approach provides a unifying framework to further study geography-dependent agent-based models, notably paving the way for joint consideration of population and price dynamics within a field theoretic approach.

Oscillations in an array of bistable microelectrodes coupled through a globally conserved quantity

The dynamical behavior of an array of microelectrodes is investigated under controlled current conditions during CO electrooxidation, a bistable electrochemical reaction with an S-shaped negative differential resistance (S-NDR) current-potential curve. Under these conditions, the total current constitutes a globally conserved quantity, thus coupling all microelectrodes globally. Upon increasing the total current, the microelectrodes activate one by one, with a single microelectrode being on its intermediate S-NDR current branch and the other ones being either on their passive or their active branches. When a few coupled microelectrodes are activated, the electrochemical system exhibits spontaneous potential oscillations. Mathematical analysis shows that oscillations arise already in a two group approximation of the dynamics, the two groups consisting of 1 electrode and n - 1 electrodes with n ≥ 3, respectively, with each group being described by a single evolution equation. In this minimal representation, oscillations occur when the single electrode is on the intermediate branch and the larger group is on the active branch.

Complex phase ordering of the one-dimensional Heisenberg model with conserved order parameter

We study the phase-ordering kinetics of the one-dimensional Heisenberg model with conserved order parameter by means of scaling arguments and numerical simulations. We find a rich dynamical pattern with a regime characterized by two distinct growing lengths. Spins are found to be coplanar over regions of a typical size LV(t), while inside these regions smooth rotations associated to a smaller length LC(t) are observed. Two different and coexisting ordering mechanisms are associated to these lengths, leading to different growth laws LV(t) approximately t1/3 and LC(t) approximately t1/4 violating dynamical scaling.

Far-from-equilibrium Ostwald ripening in electrostatically driven granular powders

We report an experimental study of cluster size distributions in electrostatically driven granular submonolayers. The cluster size distribution in this far-from-equilibrium process exhibits dynamic scaling behavior characteristic of the (nearly equilibrium) Ostwald ripening, controlled by the attachment and detachment of the "gas" particles. The scaled size distribution, however, is different from the classical Wagner distribution obtained in the limit of a vanishingly small area fraction of the clusters. A much better agreement is found with the theory of Phys. Rev. E 65, 046117 (2002)] which accounts for the cluster merger.

Dynamic phase separation: from coarsening to turbulence via structure formation

We investigate some new two-dimensional evolution models belonging to the class of convective Cahn-Hilliard models: (i) a local model with a scalar order parameter, (ii) a nonlocal model with a scalar order parameter, and (iii) a model with a vector order parameter. These models are applicable to phase-separating system where concentration gradients cause hydrodynamic motion due to buoyancy or Marangoni effect. The numerical study of the models shows transition from coarsening, typical of Cahn-Hilliard systems, to spatiotemporally irregular behavior (turbulence), typical of the Kuramoto-Sivashinsky equation, which is obtained in the limit of very strong driving. The transition occurs not in a straightforward way, but through the formation of spatial patterns that emerge for intermediate values of the driving intensity. As in driven one-dimensional models studied before, the mere presence of the driving force, however small, breaks the symmetry between the two separating phases, as well as increases the coarsening rate. With increasing driving, coarsening stops. The dynamics is generally irregular at strong driving, but exhibits specific structural features.

Phase ordering with a global conservation law: Ostwald ripening and coalescence

Globally conserved phase ordering dynamics is investigated in systems with short range correlations at t=0. A Ginzburg-Landau equation with a global conservation law is employed as the phase field model. The conditions are found under which the sharp-interface limit of this equation is reducible to the area-preserving motion by curvature. Numerical simulations show that, for both critical and off-critical quench, the equal-time pair correlation function exhibits dynamic scaling, and the characteristic coarsening length obeys l(t) approximately t(1/2). For the critical quench, our results are in excellent agreement with earlier results. For off-critical quench (Ostwald ripening) we investigate the dynamics of the size distribution function of the minority phase domains. The simulations show that, at large times, this distribution function has a self-similar form with growth exponent 1/2. The scaled distribution, however, strongly differs from the classical Wagner distribution. We attribute this difference to coalescence of domains. A theory of Ostwald ripening is developed that takes into account binary coalescence events. The theoretical scaled distribution function agrees well with that obtained in the simulations.

Normal scaling in globally conserved interface-controlled coarsening of fractal clusters

We find that globally conserved interface-controlled coarsening of diffusion-limited aggregates exhibits dynamic scale invariance (DSI) and normal scaling. This is demonstrated by a numerical solution of the Ginzburg-Landau equation with a global conservation law. The general sharp-interface limit of this equation is introduced and reduced to volume preserving motion by mean curvature. A simple example of globally conserved interface-controlled coarsening system: the sublimation/deposition dynamics of a solid and its vapor in a small closed vessel, is presented in detail. The results of the numerical simulations show that the scaled form of the correlation function has a power-law tail accommodating the fractal initial condition. The coarsening length exhibits normal dynamic scaling. A decrease of the cluster radius with time, predicted by DSI, is observed. The difference between global and local conservation is discussed.

Area-preserving dynamics of a long slender finger by curvature: a test case for globally conserved phase ordering

A long and slender finger can serve as a simple "test bed" for different phase-ordering models. In this work, the globally conserved, interface-controlled dynamics of a long finger is investigated, analytically and numerically, in two dimensions. An important limit is considered when the finger dynamics is reducible to area-preserving motion by curvature. A free boundary problem for the finger shape is formulated. An asymptotic perturbation theory is developed that uses the finger aspect ratio as a small parameter. The leading-order approximation is a modification of the Mullins finger (a well-known analytic solution) whose width is allowed to slowly vary with time. This time dependence is described, in the leading order, by an exponential law with the characteristic time proportional to the (constant) finger area. The subleading terms of the asymptotic theory are also calculated. Finally, the finger dynamics is investigated numerically, employing the Ginzburg-Landau equation with a global conservation law. The theory is in very good agreement with the numerical solution.