A Lattice Gas Coupled to Two Thermal Reservoirs: Monte Carlo and Field Theoretic Studies

We investigate the collective behavior of an Ising lattice gas, driven to non-equilibrium steady states by being coupled to {\em two} thermal baths. Monte Carlo methods are applied to a two-dimensional system in which one of the baths is fixed at infinite temperature. Both generic long range correlations in the disordered state and critical poperties near the second order transition are measured. Anisotropic scaling, a key feature near criticality, is used to extract $T_{c}$ and some critical exponents. On the theoretical front, a continuum theory, in the spirit of Landau-Ginzburg, is presented. Being a renormalizable theory, its predictions can be computed by standard methods of $\epsilon$-expansions and found to be consistent with simulation data. In particular, the critical behavior of this system belongs to a universality class which is quite {\em different} from the uniformly driven Ising model.Comment: 21 pages, 15 figure

Getting More from Pushing Less: Negative Specific Heat and Conductivity in Non-equilibrium Steady States

For students familiar with equilibrium statistical mechanics, the notion of a positive specific heat, being intimately related to the idea of stability, is both intuitively reasonable and mathematically provable. However, for system in non-equilibrium stationary states, coupled to more than one energy reservoir (e.g., thermal bath), negative specific heat is entirely possible. In this paper, we present a ``minimal'' system displaying this phenomenon. Being in contact with two thermal baths at different temperatures, the (internal) energy of this system may increase when a thermostat is turned down. In another context, a similar phenomenon is negative conductivity, where a current may increase by decreasing the drive (e.g., an external electric field). The counter-intuitive behavior in both processes may be described as `` getting more from pushing less.'' The crucial ingredients for this phenomenon and the elements needed for a ``minimal'' system are also presented.Comment: 14 pages, 3 figures, accepted for publication in American Journal of Physic

Exact results for the extreme Thouless effect in a model of network dynamics

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 figure

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 $X/\left( N_{I}N_{E}\right)$ 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:1408.542

Fluctuations and correlations in population models with age structure

We study the population profile in a simple discrete time model of population dynamics. Our model, which is closely related to certain ``bit-string'' models of evolution, incorporates competition for resources via a population dependent death probability, as well as a variable reproduction probability for each individual as a function of age. We first solve for the steady-state of the model in mean field theory, before developing analytic techniques to compute Gaussian fluctuation corrections around the mean field fixed point. Our computations are found to be in good agreement with Monte-Carlo simulations. Finally we discuss how similar methods may be applied to fluctuations in continuous time population models.Comment: 4 page

Roughening transition, surface tension and equilibrium droplet shapes in a two-dimensional Ising system

The exact surface tension for all angles and temperatures is given for the two-dimensional square Ising system with anisotropic nearest-neighbour interactions. Using this in the Wulff construction, droplet shapes are computed and illustrated. Letting temperature approach zero allows explicit study of the roughening transition in this model. Results are compared with those of the solid-on-solid approximation