arXiv subjects

Ronald Dickman

Publications and source records attributed to Ronald Dickman.

At least 19 recordsLinked to original sources

arXiv2026

Mean-Field Theory for the Three-State Active Lattice Gas Model

We develop a mean-field description including spatial structure for a simplified version of the three-state active matter model studied by Venzel et al. (Phys. Rev. E 110, 014109 (2024)). The resulting triangular lattice of coupled nonlinear differential equations are integrated numerically using a fourth-order Runge-Kutta scheme. Starting from various ordered initial configurations, we probe the stability of the corresponding stationary states, revealing the presence of various high-density ordered structures in the density(\r{ho})-noise(η) plane. The results are compared with Monte Carlo simulations of the simplified model, yielding, in certain cases, unexpected transitions between ordered configuration types.

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arXiv2024

Three-state active lattice gas: a discrete Vicseklike model with excluded volume

We study a discrete-space model of active matter with excluded volume. Particles are restricted to the sites of a triangular lattice, and can assume one of three orientations. Varying the density and noise intensity, Monte Carlo simulations reveal a variety of spatial patterns. Ordered states occur in the form of condensed structures, which (away from the full occupancy limit) coexist with a low-density vapor. The condensed structures feature low particle mobility, particularly those that wrap the system via the periodic boundaries. As the noise intensity is increased, dense structures give way to a disordered phase. We characterize the parameter values associated with the condensed phases and perform a detailed study of the order-disorder transition at (1) full occupation and (2) at a density of 0.1. In the former case, the model possesses the same symmetry as the three-state Potts model and exhibits a continuous phase transition, as expected, with critical exponents consistent with those of the associated Potts model. In the low-density case, the transition is clearly discontinuous, with strong dependence of the final state upon the initial configuration, hysteresis,and nonmonotonic dependence of the Binder cumulant upon noise intensity.

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arXiv2022

Percolation in two-species antagonistic random sequential adsorption in two dimensions

We consider two-species random sequential adsorption (RSA) in which species A and B adsorb randomly on a lattice with the restriction that opposite species cannot occupy nearest-neighbor sites. When the probability $x_A$ of choosing an A particle for an adsorption trial reaches a critical value $0.626441(1)$, the A species percolates and/or the blocked sites X (those with at least one A and one B nearest neighbor) percolate. Analysis of the size-distribution exponent $τ$, the wrapping probabilities, and the excess cluster number shows that the percolation transition is consistent with that of ordinary percolation. We obtain an exact result for the low $x_B = 1 - x_A$ jamming behavior: $θ_A = 1 - x_B +b_2 x_B^2+\mathcal{O}(x_B^3)$, $θ_B = x_B/(z+1)+\mathcal{O}(x_B^2)$ for a $z$-coordinated lattice, where $θ_A$ and $θ_B$ are respectively the saturation coverages of species A and B. We also show how differences between wrapping probabilities of A and X clusters, as well as differences in the number of A and X clusters, can be used to find the transition point accurately. For the one-dimensional case a three-site approximation appears to provide exact results for the coverages.

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arXiv2021

Contact processes on fragmented domains

Motivated by recent findings of enhanced species survival when fragmented habitats are reconnected through narrow strips of land [S. Pimm, and C. N. Jenkins, Am. Sci. {\bf 107}(3), 162 (2019).], we study the effect of a corridor connecting two square regions on the survival time of three models exhibiting extinction/survival phase transitions: the basic contact process (CP), the diffusive contact process, and the two-species symbiotic contact process (2SCP). We find that connecting the regions generally increases the survival time for $λ\ge λ_c$, where $λ$ is the reproduction rate and $λ_c$ its critical value. The enhancement of the survival time increases with $λ$, and is largest in the 2SCP.

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arXiv2021

Microemulsions in the driven Widom-Rowlinson lattice gas

An investigation of the two-dimensional Widom-Rowlinson lattice gas under an applied drive uncovered a remarkable non-equilibrium steady state in which uniform stripes (reminiscent of an equilibrium lamellar phase) form perpendicular to the drive direction [R. Dickman and R. K. P. Zia, Phys. Rev. E 97, 062126 (2018)]. Here we study this model at low particle densities in two and three dimensions, where we find a disordered phase with a characteristic length scale (a "microemulsion") along the drive direction. We develop a continuum theory of this disordered phase to derive a coarse-grained field-theoretic action for the non-equilibrium dynamics. The action has the form of two coupled driven diffusive systems with different characteristic velocities, generated by an interplay between the particle repulsion and the drive. We then show how fluctuation corrections in the field theory may generate the characteristic features of the microemulsion phase, including a peak in the static structure factor corresponding to the characteristic length scale. This work lays the foundation for understanding the stripe phenomenon more generally.

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arXiv2021

Phase diagram and critical properties of a two-dimensional associating lattice gas

We revisit the associating lattice gas~(ALG) introduced by Henriques \textit{et al.} [PRE 71, 031504 (2005)] in its symmetric version. In this model, defined on the triangular lattice, interaction between molecules occupying nearest-neighbor sites depends on their relative orientation, mimicking the formation of hydrogen bonds in network-forming fluids. Although all previous studies of this model agree that it has a disordered fluid (DF), a low-density liquid (LDL) and a high-density liquid (HDL) phase, quite different forms have been reported for its phase diagram. Here, we present a thorough investigation of its phase behavior using both transfer matrix calculations and Monte Carlo (MC) simulations, along with finite-size scaling extrapolations. Results in striking agreement are found using these methods. The critical point associated with the DF-HDL transition at full occupancy, dentified by Furlan and coworkers [Phys.~Rev.~E~{\bf 100}, 022109 (2019)] is shown to be one terminus of a {\it critical line} separating these phases. In opposition to previous simulation studies, we find that the transition between the DF and LDL phases is always discontinuous, similar to the LDL-HDL transition. The associated coexistence lines meet at the point where the DF-HDL critical line ends, making it critical-end-point. Overall, the form of the phase diagram observed in our simulations is very similar to that found in the exact solution of the model on a Husimi lattice. Our results confirm that, despite the existence of some waterlike anomalies in this model, it is unable to reproduce key features of the phase behavior of liquid water.

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arXiv2021

Population dynamics in the triplet annihilation model with a mutating reproduction rate

I study a population model in which the reproduction rate lambda is inherited with mutation, favoring fast reproducers in the short term, but conflicting with a process that eliminates agglomerations of individuals. The model is a variant of the triplet annihilation model introduced several decades ago [R. Dickman, Phys. Rev. B~{\bf 40}, 7005 (1989)] in which organisms ("particles") reproduce and diffuse on a lattice, subject to annihilation when (and only when) occupying three consecutive sites. For diffusion rates below a certain value, the population possesses two "survival strategies": (i) rare reproduction (0 < lambda < lambda_{c,1}), in which a low density of diffusing particles renders triplets exceedingly rare, and (ii) frequent reproduction (lambda > lambda_{c,2}). For lambda between lambda_{c,1} and lambda_{c,2} there is no active steady state. In the rare-reproduction regime, a mutating $λ$ leads to stochastic boom-and-bust cycles in which the reproduction rate fluctuates upward in certain regions, only to lead to extinction as the local value of lambda becomes excessive. The global population can nevertheless survive due to the presence of other regions, with reproduction rates that have yet to drift upward.

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arXiv2020

Jamming and percolation of dimers in restricted-valence random sequential adsorption

Restricted-valence random sequential adsorption~(RSA) is studied in its pure and disordered versions, on the square and triangular lattices. For the simplest case~(pure on the square lattice) we prove the absence of percolation for maximum valence $V_{\rm max}=2$. In other cases, Monte Carlo simulations are used to investigate the percolation threshold, universality class, and jamming limit. Our results reveal a continuous transition for the majority of the cases studied. The percolation threshold is computed through finite-size scaling analysis of seven properties; its value increases with the average valency. Scaling plots and data-collapse analyses show that the transition belongs to the standard percolation universality class even in disordered cases

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arXiv2020

Multirange Ising model on the square lattice

We study the Ising model on $\mathbb{Z}^{2}$ and show, via numerical simulation, that allowing interactions between spins separated by distances $1$ and $m$ (two ranges), the critical temperature, $ T_c (m) $, converges monotonically to the critical temperature of the Ising model on $\mathbb{Z}^4$ as $ m \to \infty $. Only interactions between spins located in directions parallel to each coordinate axis are considered. We also simulated the model with interactions between spins at distances of $ 1 $, $ m $ and $ u $ (three ranges), with $ u $ a multiple of $ m $; in this case our results indicate that $ T_c(m, u) $ converges to the critical temperature of the model on $ \mathbb{Z}^6$. For percolation, analogous results were proven for the critical probability $p_c$ [B. N. B. de Lima, R. P. Sanchis and R. W. C. Silva, Stochastic Process. Appl. {\bf 121}, 2043 (2011)].

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arXiv2019

Synchronization of Discrete Oscillators on Ring Lattices and Small-World Networks

A lattice of three-state stochastic phase-coupled oscillators introduced by Wood it et al. exhibits a phase transition at a critical value of the coupling parameter $a$, leading to stable global oscillations (GO). On a complete graph, upon further increase in $a$, the model exhibits an infinite-period (IP) phase transition, at which collective oscillations cease and discrete rotational ($C_3$) symmetry is broken. In the case of large negative values of the coupling, Escaff et al. discovered the stability of travelling-wave states with no global synchronization but with local order. Here, we verify the IP phase in systems with long-range coupling but of lower connectivity than a complete graph and show that even for large positive coupling, the system sometimes fails to reach global order. The ensuing travelling-wave state appears to be a metastable configuration whose birth and decay (into the previously described phases) are associated with the initial conditions and fluctuations.

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arXiv2019

Uphill migration in coupled driven particle systems

In particle systems subject to a nonuniform drive, particle migration is observed from the driven to the non--driven region and vice--versa, depending on details of the hopping dynamics, leading to apparent violations of Fick's law and of steady--state thermodynamics. We propose and discuss a very basic model in the framework of independent random walkers on a pair of rings, one of which features biased hopping rates, in which this phenomenon is observed and fully explained.

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arXiv2019

Order-disorder transition in a two-dimensional associating lattice gas

We study an associating lattice gas (ALG) using Monte Carlo simulation and solutions on Husimi lattices. In this model, the molecules have an orientational degree of freedom and the interactions depend on the relative orientations of nearest-neighbor molecules. We focus on the transition between the high-density liquid (HDL) phase and the isotropic gas phase in the limit of full occupancy ($μ\to \infty$). Simulation results show a continuous phase transition at $τ_c=k_BT_c/γ=0.4763(1)$ (where $-γ$ is the bond energy) between the low-temperature HDL phase, with a non-vanishing mean orientation of the molecules, and the high-temperature isotropic phase. Results for critical exponents and the Binder cumulant indicate that the transition belongs to the three-state Potts model universality class, even though the ALG Hamiltonian does not have the full permutation symmetry of the Potts model. In contrast with simulation, the Husimi lattice results furnish a discontinuous phase transition, characterized by a discontinuity of the nematic order parameter. The transition temperatures ($τ_c=0.51403$ and $0.51207$ for trees built with triangles and hexagons, respectively) are slightly higher than the one found via simulation. Since the Husimi lattice studies show that the ALG phase diagram features a discontinuous gas-HDL line for finite $μ$, three possible scenarios arise. The first is that in the limit $μ\to \infty$ the first-order line ends in a critical point; the second is a change in the nature of the transition at some finite chemical potential; the third is that the entire line is one of continuous phase transitions. Results from other ALG models and the fact that mean-field approximations show a discontinuous phase transition for the three-state Potts model (known to possess a continuous transition) lends some weight to the third alternative.

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arXiv2018

Continuously variable spreading exponents in the absorbing Nagel-Schreckenberg model

I study the critical behavior of a traffic model with an absorbing state. The model is a variant of the Nagel-Schreckenberg (NS) model, in which drivers do not decelerate if their speed is smaller than their headway, the number of empty sites between them and the car ahead. This makes the free-flow state (i.e., all vehicles traveling at the maximum speed, v_{max}, and with all headways greater than v_{max}) {\it absorbing}; such states are possible for for densities rho smaller than a critical value rho_c = 1/(v_{max} + 2). Drivers with nonzero velocity, and with headway equal to velocity, decelerate with probability p. This {\it absorbing Nagel-Schreckenberg} (ANS) model, introduced in [Phys. Rev. E {\bf 95}, 022106 (2017)], exhibits a line of continuous absorbing-state phase transitions in the rho-p plane. Here I study the propagation of activity from a localized seed, and find that the active cluster is compact, as is the active region at long times, starting from uniformly distributed activity. The critical exponents delta (governing the decay of the survival probability) and eta (governing the growth of activity) {\it vary continuously} along the critical line. The exponents satisfy a hyperscaling relation associated with compact growth.

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arXiv2018

Driven Widom-Rowlinson lattice gas

In the Widom-Rowlinson lattice gas, two particle species (A, B) diffuse freely via particle-hole exchange, subject to both on-site exclusion and prohibition of A-B nearest-neighbor pairs. As an athermal system, the overall densities are the only control parameters. As the densities increase, an entropically driven phase transition occurs, leading to ordered states with A- and B-rich domains separated by hole-rich interfaces. Using Monte Carlo simulations, we analyze the effect of imposing a drive on this system, biasing particle moves along one direction. Our study parallels that for a driven Ising lattice gas -- the Katz-Lebowitz-Spohn (KLS) model, which displays atypical collective behavior, e.g., structure factors with discontinuity singularities and ordered states with domains only parallel to the drive. Here, other novel features emerge, including structure factors with kink singularities (best fitted to |q|), maxima at non-vanishing wavevector values, oscillating correlation functions, and ordering into multiple striped domains perpendicular to the drive, with a preferred wavelength depending on density and drive intensity. Moreover, the (hole-rich) interfaces between the domains are statistically rough (whether driven or not), in sharp contrast with those in the KLS model, in which the drive suppresses interfacial roughness. Defining a novel order parameter (to account for the emergence of multistripe states), we map out the phase diagram in the density-drive plane and present preliminary evidence for a critical phase in this driven lattice gas.

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arXiv2017

The advantage of being slow: the quasi-neutral contact process

According to the competitive exclusion principle, in a finite ecosystem, extinction occurs naturally when two or more species compete for the same resources. An important question that arises is: when coexistence is not possible, which mechanisms confer an advantage to a given species against the other(s)? In general, it is expected that the species with the higher reproductive/death ratio will win the competition, but other mechanisms, such as asymmetry in interspecific competition or unequal diffusion rates, have been found to change this scenario dramatically. In this work, we examine competitive advantage in the context of quasi-neutral population models, including stochastic models with spatial structure as well as macroscopic (mean-field) descriptions. We employ a two-species contact process in which the "biological clock" of one species is a factor of $α$ slower than that of the other species. Our results provide new insights into how stochasticity and competition interact to determine extinction in finite spatial systems. We find that a species with a slower biological clock has an advantage if resources are limited, winning the competition against a species with a faster clock, in relatively small systems. Periodic or stochastic environmental variations also favor the slower species, even in much larger systems.

q-bio.PE↗

arXiv2017

Activated Random Walkers: Facts, Conjectures and Challenges

We study a particle system with hopping (random walk) dynamics on the integer lattice $\mathbb Z^d$. The particles can exist in two states, active or inactive (sleeping); only the former can hop. The dynamics conserves the number of particles; there is no limit on the number of particles at a given site. Isolated active particles fall asleep at rate $λ> 0$, and then remain asleep until joined by another particle at the same site. The state in which all particles are inactive is absorbing. Whether activity continues at long times depends on the relation between the particle density $ζ$ and the sleeping rate $λ$. We discuss the general case, and then, for the one-dimensional totally asymmetric case, study the phase transition between an active phase (for sufficiently large particle densities and/or small $λ$) and an absorbing one. We also present arguments regarding the asymptotic mean hopping velocity in the active phase, the rate of fixation in the absorbing phase, and survival of the infinite system at criticality. Using mean-field theory and Monte Carlo simulation, we locate the phase boundary. The phase transition appears to be continuous in both the symmetric and asymmetric versions of the process, but the critical behavior is very different. The former case is characterized by simple integer or rational values for critical exponents ($β= 1$, for example), and the phase diagram is in accord with the prediction of mean-field theory. We present evidence that the symmetric version belongs to the universality class of conserved stochastic sandpiles, also known as conserved directed percolation. Simulations also reveal an interesting transient phenomenon of damped oscillations in the activity density.

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arXiv2016

A traffic model with an absorbing-state phase transition

We consider a modified Nagel-Schreckenberg (NS) model in which drivers do not decelerate if their speed is smaller than the headway (number of empty sites to the car ahead). (In the original NS model, such a reduction in speed occurs with probability $p$, independent of the headway, as long as the current speed is greater than zero.) In the modified model the free-flow state (with all vehicles traveling at the maximum speed, $v_{max}$) is {\it absorbing} for densities $ρ$ smaller than a critical value $ρ_c = 1/(v_{max} + 2)$. The phase diagram in the $ρ- p$ plane is reentrant: for densities in the range $ρ_{c,<} < ρ< ρ_c$, both small and large values of $p$ favor free flow, while for intermediate values, a nonzero fraction of vehicles have speeds $< v_{max}$. In addition to representing a more realistic description of driving behavior, this change leads to a better understanding of the phase transition in the original model. Our results suggest an unexpected connection between traffic models and stochastic sandpiles.

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arXiv2016

Spatio-temporal generalization of the Harris criterion and its application to diffusive disorder

We investigate how a clean continuous phase transition is affected by spatio-temporal disorder, i.e., by an external perturbation that fluctuates in both space and time. We derive a generalization of the Harris criterion for the stability of the clean critical behavior in terms of the space-time correlation function of the external perturbation. As an application, we consider diffusive disorder, i.e, an external perturbation governed by diffusive dynamics, and its effects on a variety of equilibrium and nonequilibrium critical points. We also discuss the relation between diffusive disorder and diffusive dynamical degrees of freedom on the example of model C of the Hohenberg-Halperin classification, and we comment on Griffiths singularities.

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