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Arguably, the most fundamental problem in Network Science is finding structure within a complex network. One approach is to partition the nodes into communities that are more densely connected than one expects in a random network. "The" community structure then corresponds to the partition that maximizes Modularity, an objective function that quantifies this idea. Finding the maximizing partition, however, is a computationally difficult, NP-Complete problem. We explore using a recently introduced machine-learning algorithmic scheme to find the structure of benchmark networks. The scheme, known as RenEEL, creates an ensemble of $K$ partitions and updates the ensemble by replacing its worst member with the best of $L$ partitions found by analyzing a simplified network. The updating continues until consensus is achieved within the ensemble. We perform an empirical study of three real-world networks to explore how the Modularity of the consensus partition depends on the values of $K$ and $L$ and relate the results to the extreme value statistics of record-breaking. We find that increasing $K$ is generally more effective than increasing $L$ for finding the best partition., Comment: 12 pages, 4 figures

The evolution of physical systems are often modeled by simple Markovian processes. When settled into stationary states, the probability distributions of such systems are time independent, by definition. However, they do not necessarily fall within the framework of equilibrium statistical mechanics. Instead, they may be non-equilibrium steady states (NESS). One distinguished feature of NESS is the presence of time reversal asymmetry (TRA) and persistent probability current loops. These loops lead naturally to the notion of probability angular momenta, which play a role on the same footing as the noise covariance in stochastic processes. Illustrating with simulations of simple models and physical data, we present ways to detect these signals of TRA, from the subtle to the prominent., Comment: 50 pages, 7 figures, invited submission to Special Issue

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., Comment: 17 pages, 11 figures

We investigate the long-time properties of a dynamic, out-of-equilibrium network of individuals holding one of two opinions in a population consisting of two communities of different sizes. Here, while the agents' opinions are fixed, they have a preferred degree which leads them to endlessly create and delete links. Our evolving network is shaped by homophily/heterophily, which is a form of social interaction by which individuals tend to establish links with others having similar/dissimilar opinions. Using Monte Carlo simulations and a detailed mean-field analysis, we study in detail how the sizes of the communities and the degree of homophily/heterophily affects the network structure. In particular, we show that when the network is subject to enough heterophily, an "overwhelming transition" occurs: individuals of the smaller community are overwhelmed by links from agents of the larger group, and their mean degree greatly exceeds the preferred degree. This and related phenomena are characterized by obtaining the network's total and joint degree distributions, as well as the fraction of links across both communities and that of agents having fewer edges than the preferred degree. We use our mean-field theory to discuss the network's polarization when the group sizes and level of homophily vary., Comment: 24 pages, 10 figures

We consider a dynamic network of individuals that may hold one of two different opinions in a two-party society. As a dynamical model, agents can endlessly create and delete links to satisfy a preferred degree, and the network is shaped by \emph{homophily}, a form of social interaction. Characterized by the parameter $J \in [-1,1]$, the latter plays a role similar to Ising spins: agents create links to others of the same opinion with probability $(1+J)/2$, and delete them with probability $(1-J)/2$. Using Monte Carlo simulations and mean-field theory, we focus on the network structure in the steady state. We study the effects of $J$ on degree distributions and the fraction of cross-party links. While the extreme cases of homophily or heterophily ($J= \pm 1$) are easily understood to result in complete polarization or anti-polarization, intermediate values of $J$ lead to interesting features of the network. Our model exhibits the intriguing feature of an "overwhelming transition" occurring when communities of different sizes are subject to sufficient heterophily: agents of the minority group are oversubscribed and their average degree greatly exceeds that of the majority group. In addition, we introduce an original measure of polarization which displays distinct advantages over the commonly used average edge homogeneity., Comment: The paper includes 12 pages and 9 figures

Unlike typical phase transitions of first and second order, a system displaying the Thouless effect exhibits characteristics of both at the critical point (jumps in the order parameter and anomalously large fluctuations). An $extreme$ Thousless effect was observed in a recently introduced model of social networks consisting of `introverts and extroverts' ($XIE$). We study the fluctuations and correlations of this system using both Monte Carlo simulations and analytic methods based on a self-consistent mean field theory. Due to the symmetries in the model, we derive identities between all independent two point correlations and fluctuations in three quantities (degrees of individuals and the total number of links between the two subgroups) in the stationary state. As simulations confirm these identities, we study only the fluctuations in detail. Though qualitatively similar to those in the 2D Ising model, there are several unusual aspects, due to the extreme Thouless effect. All these anomalous fluctuations can be quantitatively understood with our theory, despite the mean-field aspects in the approximations. In our theory, we frequently encounter the `finite Poisson distribution' (i.e., $x^{n}/n!$ for $n\in \left[ 0,N\right] $ and zero otherwise). Since its properties appear to be quite obscure, we include an Appendix on the details and the related `finite exponential series' $\sum_{0}^{N}x^{n}/n!$. Some simulation studies of joint degree distributions, which provide a different perspective on correlations, have also been carried out.

Though the Boltzmann-Gibbs framework of equilibrium statistical mechanics has been successful in many arenas, it is clearly inadequate for describing many interesting natural phenomena driven far from equilibrium. The simplest step towards that goal is a better understanding of nonequilibrium steady-states (NESS). Here we focus on one of the distinctive features of NESS, persistent probability currents, and their manifestations in our climate system. We consider the natural variability of the steady-state climate system, which can be approximated as a NESS. These currents must form closed loops, which are odd under time reversal, providing the crucial difference between systems in thermal equilibrium and NESS. Seeking manifestations of such current loops leads us naturally to the notion of probability angular momentum and oscillations in the space of observables. Specifically, we will relate this concept to the asymmetric part of certain time-dependent correlation functions. Applying this approach, we propose that these current loops give rise to preferred spatio-temporal patterns of natural climate variability that take the form of climate oscillations such as the El-Ni\~{n}o Southern Oscillation (ENSO) and the Madden-Julien Oscillation (MJO). In the space of climate indices, we observe persistent currents and define a new diagnostic for these currents: the probability angular momentum. Using the observed climatic time series of ENSO and MJO, we compute both the averages and the distributions of the probability angular momentum. These results are in good agreement with the analysis from a linear Gaussian model. We propose that, in addition to being a new quantification of climate oscillations across models and observations, the probability angular momentum provides a meaningful characterization for all statistical systems in NESS., Comment: Journal of Statistical Physics (2019)

If a system undergoing phase transitions exhibits some characteristics of both first and second order, it is said to be of 'mixed order' or to display the Thouless effect. Such a transition is present in a simple model of a dynamic social network, in which $N_{I/E}$ extreme introverts/extroverts always cut/add random links. In particular, simulations showed that $\left\langle f\right\rangle $, the average fraction of cross-links between the two groups (which serves as an 'order parameter' here), jumps dramatically when $\Delta \equiv N_{I}-N_{E}$ crosses the 'critical point' $\Delta _{c}=0$, as in typical first order transitions. Yet, at criticality, there is no phase co-existence, but the fluctuations of $f$ are much larger than in typical second order transitions. Indeed, it was conjectured that, in the thermodynamic limit, both the jump and the fluctuations become maximal, so that the system is said to display an 'extreme Thouless effect.' While earlier theories are partially successful, we provide a mean-field like approach that accounts for all known simulation data and validates the conjecture. Moreover, for the critical system $N_{I}=N_{E}=L$, an analytic expression for the mesa-like stationary distribution, $P\left( f\right) $, shows that it is essentially flat in a range $\left[ f_{0},1-f_{0}\right] $, with $f_0 \ll 1$. Numerical evaluations of $f_{0}$ provides excellent agreement with simulation data for $L\lesssim 2000$. For large $L$, we find $f_{0}\rightarrow \sqrt{\left( \ln L^2 \right) /L}$ , though this behavior begins to set in only for $L>10^{100}$. For accessible values of $L$, we provide a transcendental equation for an approximate $f_{0}$ which is better than $\sim$1% down to $L=100$. We conjecture how this approach might be used to attack other systems displaying an extreme Thouless effect., Comment: 6 pages, 4 figures

When three species compete cyclically in a well-mixed, stochastic system of $N$ individuals, extinction is known to typically occur at times scaling as the system size $N$. This happens, for example, in rock-paper-scissors games or conserved Lotka-Volterra models in which every pair of individuals can interact on a complete graph. Here we show that if the competing individuals also have a "social temperament" to be either introverted or extroverted, leading them to cut or add links respectively, then long-living state in which all species coexist can occur when both introverts and extroverts are present. These states are non-equilibrium quasi-steady states, maintained by a subtle balance between species competition and network dynamcis. Remarkably, much of the phenomena is embodied in a mean-field description. However, an intuitive understanding of why diversity stabilizes the co-evolving node and link dynamics remains an open issue., Comment: 8 pages, 4 figures

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., Comment: 39 pages, 18 figures

Motivated by the intriguing behavior displayed in a dynamic network that models a population of extreme introverts and extroverts (XIE), we consider the spectral properties of ensembles of random split graph adjacency matrices. We discover that, in general, a gap emerges in the bulk spectrum between -1 and 0 that contains a single eigenvalue. An analytic expression for the bulk distribution is derived and verified with numerical analysis. We also examine their relation to chiral ensembles, which are associated with bipartite graphs., Comment: 17 pages, 6 figures

We study the dynamics of the out-of-equilibrium nonlinear q-voter model with two types of susceptible voters and zealots, introduced in [EPL 113, 48001 (2016)]. In this model, each individual supports one of two parties and is either a susceptible voter of type $q_1$ or $q_2$, or is an inflexible zealot. At each time step, a $q_i$-susceptible voter ($i = 1,2$) consults a group of $q_i$ neighbors and adopts their opinion if all group members agree, while zealots are inflexible and never change their opinion. This model violates detailed balance whenever $q_1 \neq q_2$ and is characterized by two distinct regimes of low and high density of zealotry. Here, by combining analytical and numerical methods, we investigate the non-equilibrium stationary state of the system in terms of its probability distribution, non-vanishing currents and unequal-time two-point correlation functions. We also study the switching times properties of the model by exploiting an approximate mapping onto the model of [Phys. Rev. E 92, 012803 (2015)] that satisfies the detailed balance, and also outline some properties of the model near criticality., Comment: 17 pages, 12 figures

Many interesting phenomena in nature are described by stochastic processes with irreversible dynamics. To model these phenomena, we focus on a master equation or a Fokker-Planck equation with rates which violate detailed balance. When the system settles in a stationary state, it will be a nonequilibrium steady state (NESS), with time independent probability distribution as well as persistent probability current loops. The observable consequences of the latter are explored. In particular, cyclic behavior of some form must be present: some are prominent and manifest, while others are more obscure and subtle. We present a theoretical framework to analyze such properties, introducing the notion of "probability angular momentum" and its distribution. Using several examples, we illustrate the manifest and subtle categories and how best to distinguish between them. These techniques can be applied to reveal the NESS nature of a wide range of systems in a large variety of areas. We illustrate with one application: variability of ocean heat content in our climate system., Comment: 4 pages, 2 figures

We consider a toy model of interacting extrovert and introvert agents introduced earlier by Liu et al [Europhys. Lett. {\bf 100} (2012) 66007]. The number of extroverts, and introverts is $N$ each. At each time step, we select an agent at random, and allow her to modify her state. If an extrovert is selected, she adds a link at random to an unconnected introvert. If an introvert is selected, she removes one of her links. The set of $N^2$ links evolves in time, and may be considered as a set of Ising spins on an $N \times N$ square-grid with single-spin-flip dynamics. This dynamics satisfies detailed balance condition, and the probability of different spin configurations in the steady state can be determined exactly. The effective hamiltonian has long-range multi-spin couplings that depend on the row and column sums of spins. If the relative bias of choosing an extrovert over an introvert is varied, this system undergoes a phase transition from a state with very few links to one in which most links are occupied. We show that the behavior of the system can be determined exactly in the limit of large $N$. The behavior of large fluctuations in the total numer of links near the phase transition is determined. We also discuss two variations, called egalitarian and elitist agents, when the agents preferentially add or delete links to their least/ most-connected neighbor. These shows interesting cooperative behavior., Comment: Minor correction in one equation

We introduce an heterogeneous nonlinear $q$-voter model with zealots and two types of susceptible voters, and study its non-equilibrium properties when the population is finite and well mixed. In this two-opinion model, each individual supports one of two parties and is either a zealot or a susceptible voter of type $q_1$ or $q_2$. While here zealots never change their opinion, a $q_i$-susceptible voter ($i=1,2$) consults a group of $q_i$ neighbors at each time step, and adopts their opinion if all group members agree. We show that this model violates the detailed balance whenever $q_1 \neq q_2$ and has surprisingly rich properties. Here, we focus on the characterization of the model's non-equilibrium stationary state (NESS) in terms of its probability distribution and currents in the distinct regimes of low and high density of zealotry. We unveil the NESS properties in each of these phases by computing the opinion distribution and the circulation of probability currents, as well as the two-point correlation functions at unequal times (formally related to a "probability angular momentum"). Our analytical calculations obtained in the realm of a linear Gaussian approximation are compared with numerical results., Comment: 6 pages, 2 figures, supplementary material and movie available at https://dx.doi.org/10.6084/m9.figshare.2060595

Spatial patterning can be crucially important for understanding the behavior of interacting populations. Here we investigate a simple model of parasite and host populations in which parasites are random walkers that must come into contact with a host in order to reproduce. We focus on the spatial arrangement of parasites around a single host, and we derive using analytics and numerical simulations the necessary conditions placed on the parasite fecundity and lifetime for the populations long-term survival. We also show that the parasite population can be pushed to extinction by a large drift velocity, but, counterintuitively, a small drift velocity generally increases the parasite population., Comment: 6 pages, 6 figures

Since their inception about a decade ago, dynamic networks which adapt to the state of the nodes have attracted much attention. One simple case of such an adaptive dynamics is a model of social networks in which individuals are typically comfortable with a certain number of contacts, i.e., preferred degrees. This paper is partly a review of earlier work of single homogeneous systems and ones with two interacting networks, and partly a presentation of some new results. In general, the dynamics does not obey detailed balance and the stationary distributions are not known analytically. A particular limit of the latter is a system of extreme introverts and extroverts - the XIE model. Remarkably, in this case, the detailed balance condition is satisfied, the exact distribution and an effective Hamiltonian can be found explicitly. Further, the model exhibits a phase transition in which the total number of links in the system - a macroscopically interesting quantity, displays an extreme Thouless effect. We show that in the limit of large populations and away from the transition, the model reduces to one with non-interacting agents of the majority subgroup. We determine the nature of fluctuations near the transition. We also introduce variants of the model where the agents show preferential attachment or detachment. There are significant changes to the degree distributions in the steady state, some of which can be understood by theoretical arguments and some remain to be explored. Many intriguing questions are posed, providing some food for thought and avenues for future research., Comment: 38 pages, 18 figures (some with multiple parts), to appear in JSTAT

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/\left( N_{I}N_{E}\right) $ jumps from $0$ to $1$ (in the thermodynamic limit) when $N_{I}$ crosses $N_{E}$, while all values appear with equal probability at $N_{I}=N_{E}$., Comment: arXiv admin note: substantial text overlap with arXiv:1408.5421

The statistical mechanics of a one-dimensional Ising model in thermal equilibrium is well-established, textbook material. Yet, when driven far from equilibrium by coupling two sectors to two baths at different temperatures, it exhibits remarkable phenomena, including an unexpected 'freezing by heating.' These phenomena are explored through systematic numerical simulations. Our study reveals complicated relaxation processes as well as a crossover between two very different steady-state regimes., Comment: 31 pages, 17 figures, accepted for publication in the Physical Review E

Motivated by fundamental issues in non-equilibrium statistical mechanics (NESM), we study the venerable susceptible-infected (SIS) model of disease spreading in an idealized, simple setting. Using Monte Carlo and analytic techniques, we consider a fully connected, uni-directional network of odd number of nodes, each having an equal number of in- and out-degrees. With the standard SIS dynamics at high infection rates, this system settles into an active non-equilibrium steady state. We find the exact probability distribution and explore its implications for NESM, such as the presence of persistent probability currents.

In a recent work \cite{LiuJoladSchZia13}, we introduced dynamic networks with preferred degrees and presented simulation and analytic studies of a single, homogeneous system as well as two interacting networks. Here, we extend these studies to a wider range of parameter space, in a more systematic fashion. Though the interaction we introduced seems simple and intuitive, it produced dramatically different behavior in the single- and two-network systems. Specifically, partitioning the single network into two identical sectors, we find the cross-link distribution to be a sharply peaked Gaussian. In stark contrast, we find a very broad and flat plateau in the case of two interacting identical networks. A sound understanding of this phenomenon remains elusive. Exploring more asymmetric interacting networks, we discover a kind of `universal behavior' for systems in which the `introverts' (nodes with smaller preferred degree) are far outnumbered. Remarkably, an approximation scheme for their degree distribution can be formulated, leading to very successful predictions., Comment: Accepted by JSTAT

In response to increasing calls for the reform of the undergraduate science curriculum for life science majors and pre-medical students (Bio2010, Scientific Foundations for Future Physicians, Vision & Change), an interdisciplinary team has created NEXUS/Physics: a repurposing of an introductory physics curriculum for the life sciences. The curriculum interacts strongly and supportively with introductory biology and chemistry courses taken by life sciences students, with the goal of helping students build general, multi-discipline scientific competencies. In order to do this, our two-semester NEXUS/Physics course sequence is positioned as a second year course so students will have had some exposure to basic concepts in biology and chemistry. NEXUS/Physics stresses interdisciplinary examples and the content differs markedly from traditional introductory physics to facilitate this. It extends the discussion of energy to include interatomic potentials and chemical reactions, the discussion of thermodynamics to include enthalpy and Gibbs free energy, and includes a serious discussion of random vs. coherent motion including diffusion. The development of instructional materials is coordinated with careful education research. Both the new content and the results of the research are described in a series of papers for which this paper serves as an overview and context., Comment: 12 pages

We study a simple model of dynamic networks, characterized by a set preferred degree, $\kappa$. Each node with degree $k$ attempts to maintain its $\kappa$ and will add (cut) a link with probability $w(k;\kappa)$ ($1-w(k;\kappa)$). As a starting point, we consider a homogeneous population, where each node has the same $\kappa$, and examine several forms of $w(k;\kappa)$, inspired by Fermi-Dirac functions. Using Monte Carlo simulations, we find the degree distribution in steady state. In contrast to the well-known Erd\H{o}s-R\'{e}nyi network, our degree distribution is not a Poisson distribution; yet its behavior can be understood by an approximate theory. Next, we introduce a second preferred degree network and couple it to the first by establishing a controllable fraction of inter-group links. For this model, we find both understandable and puzzling features. Generalizing the prediction for the homogeneous population, we are able to explain the total degree distributions well, but not the intra- or inter-group degree distributions. When monitoring the total number of inter-group links, $X$, we find very surprising behavior. $X$ explores almost the full range between its maximum and minimum allowed values, resulting in a flat steady-state distribution, reminiscent of a simple random walk confined between two walls. Both simulation results and analytic approaches will be discussed., Comment: Accepted by JSTAT

In an accelerated exclusion process (AEP), each particle can "hop" to its adjacent site if empty as well as "kick" the frontmost particle when joining a cluster of size $\ell \leq \ell_\text{max}$. With various choices of the interaction range, $\ell_\text{max}$, we find that the steady state of AEP can be found in a homogeneous phase with augmented currents (AC) or a segregated phase with holes moving at unit velocity (UV). Here we present a detailed study on the emergence of the novel phases, from two perspectives: the AEP and a mass transport process (MTP). In the latter picture, the system in the UV phase is composed of a condensate in coexistence with a fluid, while the transition from AC to UV can be regarded as condensation. Using Monte Carlo simulations, exact results for special cases, and analytic methods in a mean field approach (within the MTP), we focus on steady state currents and cluster sizes. Excellent agreement between data and theory is found, providing an insightful picture for understanding this model system., Comment: 13 pages, 8 figures

Using Monte Carlo and analytic techniques, we study a minimal dynamic network involving two populations of nodes, characterized by different preferred degrees. Reminiscent of introverts and extroverts in a population, one set of nodes, labeled \textit{introverts} ($I$), prefers fewer contacts (a lower degree) than the other, labeled \textit{extroverts} ($E$). As a starting point, we consider an \textit{extreme} case, in which an $I$ simply cuts one of its links at random when chosen for updating, while an $E$ adds a link to a random unconnected individual (node). The model has only two control parameters, namely, the number of nodes in each group, $N_{I}$ and $N_{E}$). In the steady state, only the number of crosslinks between the two groups fluctuates, with remarkable properties: Its average ($X$) remains very close to 0 for all $N_{I}>N_{E}$ or near its maximum ($\mathcal{N}\equiv N_{I}N_{E}$) if $N_{I}

We study the stochastic evolution of four species in cyclic competition in a well mixed environment. In systems composed of a finite number $N$ of particles these simple interaction rules result in a rich variety of extinction scenarios, from single species domination to coexistence between non-interacting species. Using exact results and numerical simulations we discuss the temporal evolution of the system for different values of $N$, for different values of the reaction rates, as well as for different initial conditions. As expected, the stochastic evolution is found to closely follow the mean-field result for large $N$, with notable deviations appearing in proximity of extinction events. Different ways of characterizing and predicting extinction events are discussed., Comment: 19 pages, 6 figures, submitted to J. Stat. Mech

The resources in a cell are finite, which implies that the various components of the cell must compete for resources. One such resource is the ribosomes used during translation to create proteins. Motivated by this example, we explore this competition by connecting two totally asymmetric simple exclusion processes (TASEPs) to a finite pool of particles. Expanding on our previous work, we focus on the effects on the density and current of having different entry and exit rates., Comment: 15 pages, 9 figures, v2: minor revisions, v3: additional reference & minor corrections

We aim to clarify the relationship between interacting three-species models and the two-species Lotka-Volterra (LV) model. We utilize mean-field theory and Monte Carlo simulations on two-dimensional square lattices to explore the temporal evolution characteristics of two different interacting three-species predator-prey systems: (1) a cyclic rock-paper-scissors (RPS) model with conserved total particle number but strongly asymmetric reaction rates that lets the system evolve towards one corner of configuration space; (2) a hierarchical food chain where an additional intermediate species is inserted between the predator and prey in the LV model. For model variant (1), we demonstrate that the evolutionary properties of both minority species in the steady state of this stochastic spatial three-species corner RPS model are well approximated by the LV system, with its emerging characteristic features of localized population clustering, persistent oscillatory dynamics, correlated spatio-temporal patterns, and fitness enhancement through quenched spatial disorder in the predation rates. In contrast, we could not identify any regime where the hierarchical model (2) would reduce to the two-species LV system. In the presence of pair exchange processes, the system remains essentially well-mixed, and we generally find the Monte Carlo simulation results for the spatially extended model (2) to be consistent with the predictions from the corresponding mean-field rate equations. If spreading occurs only through nearest-neighbor hopping, small population clusters emerge; yet the requirement of an intermediate species cluster obviously disrupts spatio-temporal correlations between predator and prey, and correspondingly eliminates many of the intriguing fluctuation phenomena that characterize the stochastic spatial LV system., Comment: 13 pages, 4 figures; to appear in Eur. Phys. J. B (2012)

Unlike equilibrium statistical mechanics, with its well-established foundations, a similar widely-accepted framework for non-equilibrium statistical mechanics (NESM) remains elusive. Here, we review some of the many recent activities on NESM, focusing on some of the fundamental issues and general aspects. Using the language of stochastic Markov processes, we emphasize general properties of the evolution of configurational probabilities, as described by master equations. Of particular interest are systems in which the dynamics violate detailed balance, since such systems serve to model a wide variety of phenomena in nature. We next review two distinct approaches for investigating such problems. One approach focuses on models sufficiently simple to allow us to find exact, analytic, non-trivial results. We provide detailed mathematical analyses of a one-dimensional continuous-time lattice gas, the totally asymmetric exclusion process (TASEP). It is regarded as a paradigmatic model for NESM, much like the role the Ising model played for equilibrium statistical mechanics. It is also the starting point for the second approach, which attempts to include more realistic ingredients in order to be more applicable to systems in nature. Restricting ourselves to the area of biophysics and cellular biology, we review a number of models that are relevant for transport phenomena. Successes and limitations of these simple models are also highlighted., Comment: 72 pages, 18 figures, Accepted to: Reports on Progress in Physics

We study the standard SIS model of epidemic spreading on networks where individuals have a fluctuating number of connections around a preferred degree $\kappa $. Using very simple rules for forming such preferred degree networks, we find some unusual statistical properties not found in familiar Erd\H{o}s-R\'{e}nyi or scale free networks. By letting $\kappa $ 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 $\lambda_{c}/\mu =/$ 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., Comment: 21 pages, 7 figures

The phenomenon of protein synthesis has been modeled in terms of totally asymmetric simple exclusion processes (TASEP) since 1968. In this article, we provide a tutorial of the biological and mathematical aspects of this approach. We also summarize several new results, concerned with limited resources in the cell and simple estimates for the current (protein production rate) of a TASEP with inhomogeneous hopping rates, reflecting the characteristics of real genes., Comment: 25 pages, 7 figures

Population dynamics in systems composed of cyclically competing species has been of increasing interest recently. Here, we investigate a system with four or more species. Using mean field theory, we study in detail the trajectories in configuration space of the population fractions. We discover a variety of orbits, shaped like saddles, spirals, and straight lines. Many of their properties are found explicitly. Most remarkably, we identify a collective variable which evolves simply as an exponential: $\mathcal{Q}% \propto e^{\lambda t}$, where $\lambda$ is a function of the reaction rates. It provides information on the state of the system for late times (as well as for $t\rightarrow -\infty $). We discuss implications of these results for the evolution of a finite, stochastic system. A generalization to an arbitrary number of cyclically competing species yields valuable insights into universal properties of such systems., Comment: 15 pages, 5 figures, submitted to Physical Review E

We consider a system of $N$ individuals consisting of $S$ species that interact pairwise: $x_m+x_\ell \rightarrow 2x_m\,\,$ with arbitrary probabilities $p_m^\ell $. With no spatial structure, the master equation yields a simple set of rate equations in a mean field approximation, the focus of this note. Generalizing recent findings of cyclically competing three- and four-species models, we cast these equations in an appealingly simple form. As a result, many general properties of such systems are readily discovered, e.g., the major difference between even and odd $S$ cases. Further, we find the criteria for the existence of (subspaces of) fixed points and collective variables which evolve trivially (exponentially or invariant). These apparently distinct aspects can be traced to the null space associated with the interaction matrix, $p_m^\ell $. Related to the left- and right- zero-eigenvectors, these appear to be "dual" facets of the dynamics. We also remark on how the standard Lotka-Volterra equations (which include birth/death terms) can be regarded as a special limit of a pairwise interacting system.

Generalizing the cyclically competing three-species model (often referred to as the rock-paper-scissors game), we consider a simple system of population dynamics without spatial structures that involves four species. Unlike the previous model, the four form alliance pairs which resemble partnership in the game of Bridge. In a finite system with discrete stochastic dynamics, all but 4 of the absorbing states consist of coexistence of a partner-pair. From a master equation, we derive a set of mean field equations of evolution. This approach predicts complex time dependence of the system and that the surviving partner-pair is the one with the larger product of their strengths (rates of consumption). Simulations typically confirm these scenarios. Beyond that, much richer behavior is revealed, including complicated extinction probabilities and non-trivial distributions of the population ratio in the surviving pair. These discoveries naturally raise a number of intriguing questions, which in turn suggests a variety of future avenues of research, especially for more realistic models of multispecies competition in nature., Comment: 6 pages, 4 figures, to appear in EPL

We consider a system in a non-equilibrium steady state by joining two semi-infinite Ising chains coupled to thermal reservoirs with {\em different} temperatures, $T$ and $T^{\prime}$. To compute the energy flux from the hot bath through our system into the cold bath, we exploit Glauber heat-bath dynamics to derive an exact equation for the two-spin correlations, which we solve for $T^{\prime}=\infty$ and arbitrary $T$. We find that, in the $T'=\infty$ sector, the in-flux occurs only at the first spin. In the $T<\infty$ sector (sites $x=1,2,...$), the out-flux shows a non-trivial profile: $F(x)$. Far from the junction of the two chains, $F(x)$ decays as $e^{-x/\xi}$, where $\xi$ is twice the correlation length of the {\em equilibrium} Ising chain. As $T\rightarrow 0$, this decay crosses over to a power law ($x^{-3}$) and resembles a "critical" system. Simulations affirm our analytic results., Comment: 6 pages, 4 figures, submitted to EPL

We study the dynamical properties of a finite dynamical network composed of two interacting populations, namely; extrovert ($a$) and introvert ($b$). In our model, each group is characterized by its size ($N_a$ and $N_b$) and preferred degree ($\kappa_a$ and $\kappa_b\ll\kappa_a$). The network dynamics is governed by the competing microscopic rules of each population that consist of the creation and destruction of links. Starting from an unconnected network, we give a detailed analysis of the mean field approach which is compared to Monte Carlo simulation data. The time evolution of the restricted degrees $\moyenne{k_{bb}}$ and $\moyenne{k_{ab}}$ presents three time regimes and a non monotonic behavior well captured by our theory. Surprisingly, when the population size are equal $N_a=N_b$, the ratio of the restricted degree $\theta_0=\moyenne{k_{ab}}/\moyenne{k_{bb}}$ appears to be an integer in the asymptotic limits of the three time regimes. For early times (defined by $t

Ubiquitous in nature, convection cells are a clear signature of systems out-of-equilibrium. Typically, they are driven by external forces, like gravity (in combination with temperature gradients) or shear. In this article, we show the existence of such cells in possibly the simplest system, one that involves only a temperature gradient. In particular, we consider an Ising lattice gas on a square lattice, in contact with two thermal reservoirs, one at infinite temperature and another at $T$. When this system settles into a non-equilibrium stationary state, many interesting phenomena exist. One of these is the emergence of convection cells, driven by spontaneous symmetry breaking when $T$ is set below the critical temperature., Comment: published version, 2 figures, 5 pages

Motivated by a series of experiments that revealed a temperature dependence of the dynamic scaling regime of growing surfaces, we investigate theoretically how a nonequilibrium growth process reacts to a sudden change of system parameters. We discuss quenches between correlated regimes through exact expressions derived from the stochastic Edwards-Wilkinson equation with a variable diffusion constant. Our study reveals that a sudden change of the diffusion constant leads to remarkable changes in the surface roughness. Different dynamic regimes, characterized by a power-law or by an exponential relaxation, are identified, and a dynamic phase diagram is constructed. We conclude that growth processes provide one of the rare instances where quenches between correlated regimes yield a power-law relaxation., Comment: 8 pages, 5 figures, to appear in Phys. Rev. E

When Lenz proposed a simple model for phase transitions in magnetism, he couldn't have imagined that the "Ising model" was to become a jewel in field of equilibrium statistical mechanics. Its role spans the spectrum, from a good pedagogical example to a universality class in critical phenomena. A quarter century ago, Katz, Lebowitz and Spohn found a similar treasure. By introducing a seemingly trivial modification to the Ising lattice gas, they took it into the vast realms of non-equilibrium statistical mechanics. An abundant variety of unexpected behavior emerged and caught many of us by surprise. We present a brief review of some of the new insights garnered and some of the outstanding puzzles, as well as speculate on the model's role in the future of non-equilibrium statistical physics., Comment: 3 figures. Proceedings of 100th Statistical Mechanics Meeting, Rutgers, NJ (December, 2008)

To synthesize proteins in a cell, an mRNA has to work with a finite pool of ribosomes. When this constraint is included in the modeling by a totally asymmetric simple exclusion process (TASEP), non-trivial consequences emerge. Here, we consider its effects on the power spectrum of the total occupancy, through Monte Carlo simulations and analytical methods. New features, such as dramatic suppressions at low frequencies, are discovered. We formulate a theory based on a linearized Langevin equation with discrete space and time. The good agreement between its predictions and simulation results provides some insight into the effects of finite resoures on a TASEP., Comment: 4 pages, 2 figures v2: formatting changes

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 new regimes emerge. A generalized domain wall theory captures our findings in good agreement with simulation results., Comment: 14 pages, 13 figures, v2: minor changes

We study the dynamics of density fluctuations in the steady state of a non-equilibrium system, the Zero-Range Process on a ring lattice. Measuring the time series of the total number of particles in a \emph{segment} of the lattice, we find remarkable structures in the associated power spectra, namely, two distinct components of damped-oscillations. The essential origin of both components is shown in a simple pedagogical model. Using a more sophisticated theory, with an effective drift-diffusion equation governing the stochastic evolution of the local particle density, we provide reasonably good fits to the simulation results. The effects of altering various parameters are explored in detail. Avenues for improving this theory and deeper understanding of the role of particle interactions are indicated., Comment: 21 pages, 15 figures

We revisit a totally asymmetric simple exclusion process (TASEP) with open boundaries and a global constraint on the total number of particles [Adams, et. al. 2008 J. Stat. Mech. P06009]. In this model, the entry rate of particles into the lattice depends on the number available in the reservoir. Thus, the total occupation on the lattice feeds back into its filling process. Although a simple domain wall theory provided reasonably good predictions for Monte Carlo simulation results for certain quantities, it did not account for the fluctuations of this feedback. We generalize the previous study and find dramatically improved predictions for, e.g., the density profile on the lattice and provide a better understanding of the phenomenon of "shock localization.", Comment: 11 pages, 3 figures, v2: Minor changes

We study a totally asymmetric simple exclusion process (TASEP) with one defect site, hopping rate $q<1$, near the system boundary. Regarding our system as a pair of uniform TASEP's coupled through the defect, we study various methods to match a \emph{finite} TASEP and an \emph{infinite} one across a common boundary. Several approximation schemes are investigated. Utilizing the finite segment mean-field (FSMF) method, we set up a framework for computing the steady state current $J$ as a function of the entry rate $% \alpha $ and $q$. For the case where the defect is located at the entry site, we obtain an analytical expression for $J(\alpha, q) $ which is in good agreement with Monte Carlo simulation results. When the defect is located deeper in the bulk, we refined the scheme of MacDonald, et.al. [Biopolymers, \textbf{6}, 1 (1968)] and find reasonably good fits to the density profiles before the defect site. We discuss the strengths and limitations of each method, as well as possible avenues for further studies., Comment: 16 pages, 4 figures

The Legendre transform is an important tool in theoretical physics, playing a critical role in classical mechanics, statistical mechanics, and thermodynamics. Yet, in typical undergraduate or graduate courses, the power of motivation and elegance of the method are often missing, unlike the treatments frequently enjoyed by Fourier transforms. We review and modify the presentation of Legendre transforms in a way that explicates the formal mathematics, resulting in manifestly symmetric equations, thereby clarifying the structure of the transform algebraically and geometrically. Then we bring in the physics to motivate the transform as a way of choosing independent variables that are more easily controlled. We demonstrate how the Legendre transform arises naturally from statistical mechanics and show how the use of dimensionless thermodynamic potentials leads to more natural and symmetric relations., Comment: 11 pages, 3 figures

The totally asymmetric simple exclusion process (TASEP) is a well studied example of far-from-equilibrium dynamics. Here, we consider a TASEP with open boundaries but impose a global constraint on the total number of particles. In other words, the boundary reservoirs and the system must share a finite supply of particles. Using simulations and analytic arguments, we obtain the average particle density and current of the system, as a function of the boundary rates and the total number of particles. Our findings are relevant to biological transport problems if the availability of molecular motors becomes a rate-limiting factor., Comment: 14 pages, 7 figures, uses iopart12.clo and iopart.cls

We study the effects of local inhomogeneities, i.e., slow sites of hopping rate $q<1$, in a totally asymmetric simple exclusion process (TASEP) for particles of size $\ell \geq 1$ (in units of the lattice spacing). We compare the simulation results of $\ell =1$ and $\ell >1$ 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 \emph{location} of the slow site for both $\ell =1$ and $\ell >1$ 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 $q\to 0$. 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., Comment: 14 pages; 14 figures

We propose an exactly solvable model for the dynamics of voters in a two-party system. The opinion formation process is modeled on a random network of agents. The dynamical nature of interpersonal relations is also reflected in the model, as the connections in the network evolve with the dynamics of the voters. In the infinite time limit, an exact solution predicts the emergence of consensus, for arbitrary initial conditions. However, before consensus is reached, two different metastable states can persist for exponentially long times. One state reflects a perfect balancing of opinions, the other reflects a completely static situation. An estimate of the associated lifetimes suggests that lack of consensus is typical for large systems., Comment: 4 pages, 6 figures, submitted to Phys. Rev. Lett