Emergent phenomena and fluctuations in cooperative systems

We explore the role of cooperativity and large deviations on a set of fundamental non-equilibrium many-body systems. In the cooperative asymmetric exclusion process, particles hop to the right at a constant rate only when the right neighboring site is vacant and hop at a faster rate when the left neighbor is occupied. In this model, a host of new heterogeneous density profile evolutions arise, including inverted shock waves and continuous compression waves. Cooperativity also drives the growth of complex networks via preferential attachment, where well-connected nodes are more likely to attract future connections. We introduce the mechanism of hindered redirection and show that it leads to network evolution by sublinear preferential attachment. We further show that no local growth rule can recreate superlinear preferential attachment. We also introduce enhanced redirection and show that the rule leads to networks with three unusual properties: (i) many macrohubs -- nodes whose degree is a finite fraction of the number of nodes in the network, (ii) a non-extensive degree distribution, and (iii) large fluctuations between different realizations of the growth process. We next examine large deviations in the diffusive capture model, where N diffusing predators initially all located at L 'chase' a diffusing prey initially at x<L. The prey survives if it reaches a haven at the origin without meeting any predator. We reduce the stochastic movement of the many predators to a deterministic trajectory of a single effective predator. Using optimized Monte Carlo techniques, we simulate up to 10^500 predators to confirm our analytic prediction that the prey survival probability S ~ N^-z^2, where z=x/L. Last, we quantify `survival of the scarcer' in two-species competition. In this model, individuals of two distinct species reproduce and engage in both intra-species and inter-species competition. Here a well-mixed population typically reaches a quasi steady state. We show that in this quasi-steady state the situation may arise where species A is less abundant than B but rare fluctuations make it more likely that species B first becomes extinct

Dynamic Singularities in Cooperative Exclusion

We investigate cooperative exclusion, in which the particle velocity can be an increasing function of the density. Within a hydrodynamic theory, an initial density upsteps and downsteps can evolve into: (a) shock waves, (b) continuous compression or rarefaction waves, or (c) a mixture of shocks and continuous waves. These unusual phenomena arise because of an inflection point in the current versus density relation. This anomaly leads to a group velocity that can either be an increasing or a decreasing function of the density on either side of these wave singularities.Comment: 4 pages, 4 figures, 2 column revtex 4-1 format; version 2: substantially rewritten and put in IOP format, mail results unchanged; version 3: minor changes, final version for publication in JSTA

Can a Lamb Reach a Haven Before Being Eaten by Diffusing Lions?

We study the survival of a single diffusing lamb on the positive half line in the presence of N diffusing lions that all start at the same position L to the right of the lamb and a haven at x=0. If the lamb reaches this haven before meeting any lion, the lamb survives. We investigate the survival probability of the lamb, S_N(x,L), as a function of N and the respective initial positions of the lamb and the lions, x and L. We determine S_N(x,L) analytically for the special cases of N=1 and N--->oo. For large but finite N, we determine the unusual asymptotic form whose leading behavior is S_N(z)\simN^{-z^2}, with z=x/L. Simulations of the capture process very slowly converge to this asymptotic prediction as N reaches 10^{500}.Comment: 13 pages, 6 figures, IOP format; v2: small changes in response to referee and editor comment