Extreme Thouless effect in a minimal model of dynamic social networks.

In common descriptions of phase transitions, first-order transitions are characterized by discontinuous jumps in the order parameter and normal fluctuations, while second-order transitions are associated with no jumps and anomalous fluctuations. Outside this paradigm are systems exhibiting "mixed-order" transitions displaying a mixture of these characteristics. When the jump is maximal and the fluctuations range over the entire range of allowed values, the behavior has been coined an "extreme Thouless effect." Here we report findings of such a phenomenon in the context of dynamic, social networks. Defined by minimal rules of evolution, it describes a population of extreme introverts and extroverts, who prefer to have contacts with, respectively, no one or everyone. From the dynamics, we derive an exact distribution of microstates in the stationary state. With only two control parameters, N(I,E) (the number of each subgroup), we study collective variables of interest, e.g., X, the total number of I-E links, and the degree distributions. Using simulations and mean-field theory, we provide evidence that this system displays an extreme Thouless effect. Specifically, the fraction X/(N(I)N(E)) jumps from 0 to 1 (in the thermodynamic limit) when N(I) crosses N(E), while all values appear with equal probability at N(I)=N(E).

Epidemic spreading on preferred degree adaptive networks.

We study the standard SIS model of epidemic spreading on networks where individuals have a fluctuating number of connections around a preferred degree κ. Using very simple rules for forming such preferred degree networks, we find some unusual statistical properties not found in familiar Erdös-Rényi or scale free networks. By letting κ depend on the fraction of infected individuals, we model the behavioral changes in response to how the extent of the epidemic is perceived. In our models, the behavioral adaptations can be either 'blind' or 'selective'--depending on whether a node adapts by cutting or adding links to randomly chosen partners or selectively, based on the state of the partner. For a frozen preferred network, we find that the infection threshold follows the heterogeneous mean field result λ(c)/µ = <κ>/<κ2> and the phase diagram matches the predictions of the annealed adjacency matrix (AAM) approach. With 'blind' adaptations, although the epidemic threshold remains unchanged, the infection level is substantially affected, depending on the details of the adaptation. The 'selective' adaptive SIS models are most interesting. Both the threshold and the level of infection changes, controlled not only by how the adaptations are implemented but also how often the nodes cut/add links (compared to the time scales of the epidemic spreading). A simple mean field theory is presented for the selective adaptations which capture the qualitative and some of the quantitative features of the infection phase diagram.

Opinion formation on adaptive networks with intensive average degree.

We study the evolution of binary opinions on a simple adaptive network of N nodes. At each time step, a randomly selected node updates its state ("opinion") according to the majority opinion of the nodes that it is linked to; subsequently, all links are reassigned with probability p̃ (q̃) if they connect nodes with equal (opposite) opinions. In contrast to earlier work, we ensure that the average connectivity ("degree") of each node is independent of the system size ("intensive"), by choosing p̃ and q̃ to be of O(1/N). Using simulations and analytic arguments, we determine the final steady states and the relaxation into these states for different system sizes. We find two absorbing states, characterized by perfect consensus, and one metastable state, characterized by a population split evenly between the two opinions. The relaxation time of this state grows exponentially with the number of nodes, N. A second metastable state, found in the earlier studies, is no longer observed.

Competition between multiple totally asymmetric simple exclusion processes for a finite pool of resources.

Using Monte Carlo simulations and a domain-wall theory, we discuss the effect of coupling several totally asymmetric simple exclusion processes (TASEPs) to a finite reservoir of particles. This simple model mimics directed biological transport processes in the presence of finite resources such as protein synthesis limited by a finite pool of ribosomes. If all TASEPs have equal length, we find behavior which is analogous to a single TASEP coupled to a finite pool. For the more generic case of chains with different lengths, several unanticipated regimes emerge. A generalized domain-wall theory captures our findings in good agreement with simulation results.

Opinion dynamics on an adaptive random network.

We revisit the classical model for voter dynamics in a two-party system with two basic modifications. In contrast to the original voter model studied in regular lattices, we implement the opinion formation process in a random network of agents in which interactions are no longer restricted by geographical distance. In addition, we incorporate the rapidly changing nature of the interpersonal relations in the model. At each time step, agents can update their relationships. This update is determined by their own opinion, and by their preference to make connections with individuals sharing the same opinion, or rather with opponents. In this way, the network is built in an adaptive manner, in the sense that its structure is correlated and evolves with the dynamics of the agents. The simplicity of the model allows us to examine several issues analytically. We establish criteria to determine whether consensus or polarization will be the outcome of the dynamics and on what time scales these states will be reached. In finite systems consensus is typical, while in infinite systems a disordered metastable state can emerge and persist for infinitely long time before consensus is reached.

Power spectra of the total occupancy in the totally asymmetric simple exclusion process.

As a solvable and broadly applicable model system, the totally asymmetric exclusion process enjoys iconic status in the theory of nonequilibrium phase transitions. Here, we focus on the time dependence of the total number of particles on a 1-dimensional open lattice and its power spectrum. Using both Monte Carlo simulations and analytic methods, we explore its behavior in different characteristic regimes. In the maximal current phase and on the coexistence line (between high and low density phases), the power spectrum displays algebraic decay, with exponents -1.62 and -2.00, respectively. Deep within the high and low density phases, we find pronounced oscillations, which damp into power laws. This behavior can be understood in terms of driven biased diffusion with conserved noise in the bulk.

Coarsening of "clouds" and dynamic scaling in a far-from-equilibrium model system.

A two-dimensional lattice gas of two species, driven in opposite directions by an external force, undergoes a jamming transition if the filling fraction is sufficiently high. Using Monte Carlo simulations, we investigate the growth of these jams (''clouds''), as the system approaches a nonequilibrium steady state from a disordered initial state. We monitor the dynamic structure factor S(k{x},k{y};t) and find that the k{x}=0 component exhibits dynamic scaling, of the form S(0,k{y};t)=t;{beta}S[over](k{y}t;{alpha}) . Over a significant range of times, we observe excellent data collapse with alpha=12 and beta=1 . The effects of varying filling fraction and driving force are discussed.

Inhomogeneous exclusion processes with extended objects: the effect of defect locations.

We study the effects of local inhomogeneities, i.e., slow sites of hopping rate q<1, in a totally asymmetric simple exclusion process for particles of size l>or=1 (in units of the lattice spacing). We compare the simulation results of l=1 and l>1 and notice that the existence of local defects has qualitatively similar effects on the steady state. We focus on the stationary current as well as the density profiles. If there is only a single slow site in the system, we observe a significant dependence of the current on the location of the slow site for both l=1 and l>1 cases. When two slow sites are introduced, more intriguing phenomena emerge, e.g., dramatic decreases in the current when the two are close together. In addition, we study the asymptotic behavior when q-->0. We also explore the associated density profiles and compare our findings to an earlier study using a simple mean-field theory. We then outline the biological significance of these effects.

Strongly anisotropic roughness in surfaces driven by an oblique particle flux.

Using field theoretic renormalization, an MBE-type growth process with an obliquely incident influx of atoms is examined. The projection of the beam on the substrate plane selects a "parallel" direction, with rotational invariance restricted to the transverse directions. Depending on the behavior of an effective anisotropic surface tension, a line of second-order transitions is identified, as well as a line of potentially first-order transitions, joined by a multicritical point. Near the second-order transitions and the multicritical point, the surface roughness is strongly anisotropic. Four different roughness exponents are introduced and computed, describing the surface in different directions, in real or momentum space. The results presented challenge an earlier study of the multicritical point.

Steady states of a nonequilibrium lattice gas.

We present a Monte Carlo study of a lattice gas driven out of equilibrium by a local hopping bias. Sites can be empty or occupied by one of two types of particles, which are distinguished by their response to the hopping bias. All particles interact via excluded volume and a nearest-neighbor attractive force. The main result is a phase diagram with three phases: a homogeneous phase and two distinct ordered phases. Continuous boundaries separate the homogeneous phase from the ordered phases, and a first-order line separates the two ordered phases. The three lines merge in a nonequilibrium bicritical point.

Exact dynamics of a reaction-diffusion model with spatially alternating rates.

We present the exact solution for the full dynamics of a nonequilibrium spin chain and its dual reaction-diffusion model, for arbitrary initial conditions. The spin chain is driven out of equilibrium by coupling alternating spins to two thermal baths at different temperatures. In the reaction-diffusion model, this translates into spatially alternating rates for particle creation and annihilation, and even negative "temperatures" have a perfectly natural interpretation. Observables of interest include the magnetization, the particle density, and all correlation functions for both models. Two generic types of time dependence are found: if both temperatures are positive, the magnetization, density, and correlation functions decay exponentially to their steady-state values. In contrast, if one of the temperatures is negative, damped oscillations are observed in all quantities. They can be traced to a subtle competition of pair creation and annihilation on the two sublattices. We comment on the limitations of mean-field theory and propose an experimental realization of our model in certain conjugated polymers and linear chain compounds.

Anomalous nucleation far from equilibrium.

We present precision Monte Carlo data and analytic arguments for an asymmetric exclusion process, involving two species of particles driven in opposite directions on a 2xL lattice. To resolve a stark discrepancy between earlier simulation data and an analytic conjecture, we argue that the presence of a single macroscopic cluster is an intermediate stage of a complex nucleation process: in smaller systems, this cluster is destabilized while larger systems form multiple clusters. Both limits lead to exponential cluster size distributions, controlled by very different length scales.

Driven diffusive systems: how steady states depend on dynamics.

In contrast to equilibrium systems, nonequilibrium steady states depend explicitly on the underlying dynamics. Using Monte Carlo simulations with Metropolis, Glauber, and heat bath rates, we illustrate this expectation for an Ising lattice gas, driven far from equilibrium by an "electric" field. While heat bath and Glauber rates generate essentially identical data for structure factors and two-point correlations, Metropolis rates give noticeably weaker correlations, as if the "effective" temperature were higher in the latter case. We also measure energy histograms and define a simple ratio which is exactly known and closely related to the Boltzmann factor for the equilibrium case. For the driven system, the ratio probes a thermodynamic derivative which is found to be dependent on dynamics.

Stationary correlations for a far-from-equilibrium spin chain.

A kinetic one-dimensional Ising model on a ring evolves according to a generalization of Glauber rates, such that spins at even (odd) lattice sites experience a temperature T(e) (T(o)). Detailed balance is violated so that the spin chain settles into a nonequilibrium stationary state, characterized by multiple interactions of increasing range and spin order. We derive the equations of motion for arbitrary correlation functions and solve them to obtain an exact representation of the steady state. Two nontrivial amplitudes reflect the sublattice symmetries; otherwise, correlations decay exponentially, modulo the periodicity of the ring. In the long-chain limit, they factorize into products of two-point functions, in precise analogy to the equilibrium Ising chain. The exact solution confirms the expectation, based on simulations and renormalization group arguments, that the long-time, long-distance behavior of this two-temperature model is Ising-like, in spite of the apparent complexity of the stationary distribution.

Microscopic kinetics and time-dependent structure factors.

The time evolution of structure factors (SF) in the disordering process of an initially phase-separated lattice depends crucially on the microscopic disordering mechanism, such as Kawasaki dynamics (KD) or vacancy-mediated disordering (VMD). Monte Carlo simulations show unexpected "dips" in the SFs. A phenomenological model is introduced to explain the dips in the odd SFs, and an analytical solution of KD is derived, in excellent agreement with simulations. The presence (absence) of dips in the even SFs for VMD (KD) marks a significant but not yet understood difference of the two dynamics.

Field-induced vacancy localization in a driven lattice gas: scaling of steady states

With the help of Monte Carlo simulations and a mean-field theory, we investigate the ordered steady-state structures resulting from the motion of a single vacancy on a periodic lattice which is filled with two species of oppositely "charged" particles. An external field biases particle-vacancy exchanges according to the particle's charge, subject to an excluded volume constraint. The steady state exhibits charge segregation, and the vacancy is localized at one of the two characteristic interfaces. Charge and hole density profiles, an appropriate order parameter, and the interfacial regions themselves exhibit characteristic scaling properties with system size and field strength. The lattice spacing is found to play a significant role within the mean-field theory.

Viability of competing field theories for the driven lattice gas

It has recently been suggested that the driven lattice gas should be described by an alternate field theory in the limit of infinite drive. We review the original and the alternate field theory, invoking several well-documented key features of the microscopics. Since the alternate field theory fails to reproduce these characteristics, we argue that it cannot serve as a viable description of the driven lattice gas. Recent results, for the critical exponents associated with this theory, are reanalyzed and shown to be incorrect.