Computational model of a vector-mediated epidemic.

We discuss a lattice model of vector-mediated transmission of a disease to illustrate how simulations can be applied in epidemiology. The population consists of two species, human hosts and vectors, which contract the disease from one another. Hosts are sedentary, while vectors (mosquitoes) diffuse in space. Examples of such diseases are malaria, dengue fever, and Pierce's disease in vineyards. The model exhibits a phase transition between an absorbing (infection free) phase and an active one as parameters such as infection rates and vector density are varied.

Steady-state thermodynamics: Description equivalence and violation of reservoir independence.

For stochastic lattice models in spatially uniform nonequilibrium steady states, an effective thermodynamic temperature T and chemical potential µ can be defined via coexistence with heat and particle reservoirs. We verify that the probability distribution P_{N} for the number of particles in the driven lattice gas with nearest-neighbor exclusion in contact with a particle reservoir with dimensionless chemical potential µ^{*} possesses a large-deviation form in the thermodynamic limit. This implies that the thermodynamic properties determined in isolation (fixed particle number representation) and in contact with a particle reservoir (fixed dimensionless chemical potential representation) are equal. We refer to this as description equivalence. This finding motivates investigation of whether the effective intensive parameters so obtained depend on the nature of the exchange between system and reservoir. For example, a stochastic particle reservoir is usually taken to insert or remove a single particle in each exchange, but one may also consider a reservoir that inserts or removes a pair of particles in each event. In equilibrium, equivalence of pair and single-particle reservoirs is guaranteed by the canonical form of the probability distribution on configuration space. Remarkably, this equivalence is violated in nonequilibrium steady states, limiting the generality of steady-state thermodynamics based on intensive variables.

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}+O(x_{B}^{3}), Î¸_{B}=x_{B}/(z+1)+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 percolation transition point accurately.

Discontinuous phase transition in a dimer lattice gas.

I study a dimer model on the square lattice with nearest neighbor exclusion as the only interaction. Detailed simulations using tomographic entropic sampling show that as the chemical potential is varied, there is a strongly discontinuous phase transition, at which the particle density jumps by about 18% of its maximum value, 1/4. The transition is accompanied by the onset of orientational order, to an arrangement corresponding to the {1/2, 0, 1/2} structure identified by Phares et al. [Physica B 409, 1096 (2011)] in a dimer model with finite repulsion at fixed density. Using finite-size scaling and Binder's cumulant, the expected scaling behavior at a discontinuous transition is verified in detail. The discontinuous transition can be understood qualitatively given that the model possesses eight equivalent maximum-density configurations, so that its coarse-grained description corresponds to that of the q = 8 Potts model.

Data-Driven Models of Efficient Chromatic Coding in the Outer Retina.

Recent experimental work on zebrafish has shown the in vivo activity of photoreceptors and horizontal cells (HCs) as a function of the stimulus spectrum, highlighting the appearance of chromatic-opponent signals at their first synaptic connection. Altogether with the observed lack of excitatory intercone connections, these findings suggest that the mechanism yielding early color opponency in zebrafish is dominated by inhibitory feedback. We propose a neuronal population model based on zebrafish retinal circuitry to investigate whether networks with predominantly inhibitory feedback are more advantageous in encoding chromatic information than networks with mixed excitatory and inhibitory mechanisms. We show that networks with dominant inhibitory feedback exhibit a unique and reliable encoding of chromatic information. In contrast, this property is not guaranteed in networks with strong excitatory intercone connections, exhibiting bistability. These findings provide a theoretical explanation for the absence of excitatory intercone couplings in zebrafish color circuits. In addition, our study shows that these networks, with only one type of horizontal cell, are suitable to encode most of the variance from the zebrafish environment. However, at least two successive layers of inhibitory neurons are needed to reach the optimum. Finally, we contrast the encoding performance of networks with different opsin sensitivities, showing an improvement of only 13% compared with zebrafish, suggesting that the zebrafish retina is adapted to encode color information from its habitat efficiently.

Hidden role of mutations in the evolutionary process.

Mutations not only alter allele frequencies in a genetic pool but may also determine the fate of an evolutionary process. Here we study which allele fixes in a one-step, one-way model including the wild type and two adaptive mutations. We study the effect of the four basic evolutionary mechanisms-genetic drift, natural selection, mutation, and gene flow-on mutant fixation and its kinetics. Determining which allele is more likely to fix is not simply a question of comparing fitnesses and mutation rates. For instance, if the allele of interest is less fit than the other, then not only must it have a greater mutation rate, but also its mutation rate must exceed a specific threshold for it to prevail. We find exact expressions for such conditions. Our conclusions are based on the mathematical description of two extreme but important regimes, as well as on simulations.

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 nonequilibrium 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)10.1103/PhysRevE.97.062126]. 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 nonequilibrium 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.

Multirange Ising model on the square lattice.

We study the Ising model on the square lattice (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 Z^{4} as m→∞. 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 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, Stochast. Process. Appl. 121, 2043 (2011)STOPB70304-414910.1016/j.spa.2011.05.009].

Steady-state entropy: A proposal based on thermodynamic integration.

Defining an entropy function out of equilibrium is an outstanding challenge. For stochastic lattice models in spatially uniform nonequilibrium steady states, definitions of temperature T and chemical potential µ have been verified using coexistence with heat and particle reservoirs. For an appropriate choice of exchange rates, T and µ satisfy the zeroth law, marking an important step in the development of steady-state thermodynamics. These results suggest that an associated steady-state entropy S_{th} be constructed via thermodynamic integration, using relations such as (∂S/∂E)_{V,N}=1/T, ensuring that derivatives of S_{th} with respect to energy and particle number yield the expected intensive parameters. We determine via direct calculation the stationary nonequilibrium probability distribution of the driven lattice gas with nearest-neighbor exclusion, the Katz-Lebowitz-Spohn driven lattice gas, and a two-temperature Ising model so that we may evaluate the Shannon entropy S_{S} as well as S_{th} defined above. Although the two entropies are identical in equilibrium (as expected), they differ out of equilibrium; for small values of the drive, D, we find |S_{S}-S_{th}|∝D^{2}, as expected on the basis of symmetry. We verify that S_{th} is not a state function: changes ΔS_{th} depend not only on the initial and final points, but also on the path in parameter space. The inequivalence of S_{S} and S_{th} implies that derivatives of S_{S} are not predictive of coexistence. In other words, a nonequilibrium steady state is not determined by maximizing the Shannon entropy. Our results cast doubt on the possibility of defining a state function that plays the role of a thermodynamic entropy for nonequilibrium steady states.

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

We study an associating lattice gas (ALG) using Monte Carlo simulation on the triangular lattice and semianalytical 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, mimicking the formation of hydrogen bonds. We focus on the transition between the high-density liquid (HDL) phase and the isotropic phase in the limit of full occupancy, corresponding to chemical potential µâ†’∞, which has not yet been studied systematically. Simulations yield a continuous phase transition at τ_{c}=k_{B}T_{c}/γ=0.4763(1) (where -γ is the bond energy) between the low-temperature HDL phase, with a nonvanishing 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 analyses 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 that found via simulation. Since the Husimi lattice studies show that the ALG phase diagram features a discontinuous isotropic-HDL line for finite µ, three possible scenarios arise for the triangular lattice. The first is that in the limit µâ†’∞ 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.

Absorbing-state phase transitions: exact solutions of small systems.

I derive precise results for absorbing-state phase transitions using exact (numerically determined) quasistationary (QS) probability distributions for small systems. Analysis of the contact process on rings of 23 or fewer sites yields critical properties (control parameter, order-parameter ratios, and critical exponents z and betanu_(perpendicular)) with an accuracy of 0.06% or better; precise results are also obtained for the pair contact process. The QS distribution yields insights on the statistical entropy of these models. Preliminary application to a model in the stochastic sandpile class is also described.

Contact process on a Voronoi triangulation.

We study the continuous absorbing-state phase transition in the contact process on the Voronoi-Delaunay lattice. The Voronoi construction is a natural way to introduce quenched coordination disorder in lattice models. We simulate the disordered system using the quasistationary simulation method and determine its critical exponents and moment ratios. Our results suggest that the critical behavior of the disordered system is unchanged with respect to that on a regular lattice, i.e., that of directed percolation.

Improving Wang-Landau sampling with adaptive windows.

Wang-Landau sampling (WLS) of large systems requires dividing the energy range into "windows" and joining the results of simulations in each window. The resulting density of states (and associated thermodynamic functions) is shown to suffer from boundary effects in simulations of lattice polymers and the five-state Potts model. Here, we implement WLS using adaptive windows. Instead of defining fixed energy windows (or windows in the energy-magnetization plane for the Potts model), the boundary positions depend on the set of energy values on which the histogram is flat at a given stage of the simulation. Shifting the windows each time the modification factor f is reduced, we eliminate border effects that arise in simulations using fixed windows. Adaptive windows extend significantly the range of system sizes that may be studied reliably using WLS.

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 interesting features emerge, including structure factors with kink singularities (best fitted to |q|), maxima at nonvanishing wave-vector 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 an order parameter that accounts 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.

Colloids in a periodic potential: driven lattice gas in continuous space.

Motivated by recent studies of colloidal particles in optical-tweezer arrays, we study a two-dimensional model of a colloidal suspension in a periodic potential. The particles tend to stay near potential minima, approximating a lattice gas. The interparticle interaction, a sum of Yukawa terms, features short-range repulsion and attraction at somewhat larger separations, such that two particles cannot occupy the same potential well, but occupation of adjacent cells is energetically favored. Monte Carlo simulation reveals that the equilibrium system exhibits condensation, as in the Ising model or lattice gas with conserved magnetization; the transition appears to be continuous at one-half occupancy. We study the effect of biased hopping, favoring motion along one lattice direction, as might be generated by a steady flow relative to the potential array. This system is found to exhibit features of the driven lattice gas: the interface is oriented along the drive, and appears to be smooth. A weak drive facilitates ordering of the particles into high- and low-density regions, while stronger bias tends to destroy order, and leads to very large energy fluctuations. We also study ordering in a moving periodic potential. Our results suggest possible realizations of equilibrium and driven lattice gases in a colloidal suspension subject to an optical tweezer array.

Dynamic stability in random and scale-free B-lymphocyte networks.

One of the most intriguing features of the immune system is regulation: a limited response when perturbed repeatedly. We propose a minimal network model for immune regulation in a lymphocyte network containing two types of elements: B lymphocytes and ligands that bind to their receptors. Effective interactions between B cells, mediated by other components of the immune system can be excitatory or inhibitory. In our model, B cell clones and ligand species are represented by nodes, and interactions by links. We expect that, as in many complex systems, the connectivity distribution is broad, motivating study of the model on a scale-free network; for comparison we study the same dynamics on a random graph. We characterize the dynamics of the model and its response to perturbations. Our model reproduces several key features of immune system dynamics: regulation (saturation of response), and more rapid response upon repeated perturbation with the same agents. Our results suggest that a scale-free network of interactions contributes to the regulation and dynamics of the immune system.

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.

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 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

Critical exponents for the restricted sandpile.

I report large-scale Monte Carlo studies of a one-dimensional height-restricted stochastic sandpile using the quasistationary simulation method. Results for systems of up to 50 000 sites yield estimates for critical exponents that differ significantly from those obtained using smaller systems, but are consistent with recent predictions derived from a Langevin equation for stochastic sandpiles [Ramasco, Phys. Rev. E 69, 045105(R) (2004)]. This suggests that apparent violations of universality in one-dimensional sandpiles are due to strong corrections to scaling and finite-size effects.

Moment ratios for the pair-contact process with diffusion.

We study the continuous absorbing-state phase transition in the one-dimensional pair contact process with diffusion (PCPD). In previous studies [Dickman and de Menezes, Phys. Rev. E 66, 045101(R) (2002)], the critical point moment ratios of the order parameter showed anomalous behavior, growing with system size rather than taking universal values, as expected. Using the quasistationary simulation method we determine the moments of the order parameter up to fourth order at the critical point, in systems of up to 40,960 sites. Due to strong finite-size effects, the ratios converge only for large system sizes. Moment ratios and associated order-parameter histograms are compared with those of directed percolation. We also report an improved estimate [pc=0.077092(1)] for the location of the critical point in the nondiffusive pair contact process.