Session S40: Systems with Large Fluctuations and Strong Correlations II

First-Passage Statistics of Extreme Values

Theoretical concepts from nonequilibrium statistical physics such as scaling and correlations are used to analyze first-passage processes involving extreme values. The focus of this talk is statistics of the running maxima, defined as the largest variable in a sequence of random variables. In particular, the running maxima of multiple independent sequences of stochastic variables are compared. The probability that these maxima remain perfectly ordered decays algebraically with the number of random variables, and the decay exponent characterizing this decay is nontrivial. Exact solutions for the scaling exponents will be discussed for uncorrelated variables as well as Brownian trajectories which are correlated. Relevance of such statistical measures for analysis of empirical data will be discussed as well.   

Emergence of universal statistics from conserved topological features of underlying network dynamics

In this talk I will discuss how universal statistics emerge from conserved topological features of underlying network dynamics. I will indicate how dynamical phase transitions between different network structures also encode universal signatures. I will connect these results with our single-cell experiments on C. crescentus cells.   

Anomalous Dimension in a Two-Species Reaction-Diffusion Model

We consider particles ($A$) diffusing in the presence of traps ($B$), which themselves are diffusing and reacting, i.e. the two-species reaction diffusion model $A+B\to B$ and $B+B\to(0,B)$. We introduce a simulation technique that provides the full probability distribution of particles for a given realization of the trap dynamics. Previous renormalization group analysis predicted that the density of $A$ particles decays as $a~t^{-\theta}$ where $\theta$ is a nontrivial, universal exponent for $d<2$. We compare our results with these predictions, and also demonstrate the scaling of the correlation functions. We discover an anomalous dimension in the particle-particle correlation function, described by $G_{AA}(0)\sim t^\phi$, and we report our measurements for this new exponent.   

Non-equilibrium steady states of stochastic processes with intermittent resetting

Stochastic processes that are randomly reset to an initial condition serve as a showcase to analytically investigate non-equilibrium steady states. Here we study such processes for which the time between the resets is random and drawn from a generic waiting time distribution. We obtain the general solution for the stationary state and quantify the temporal relaxation of the process in terms of its moments. Our results are applied to analyze the efficiency of constrained random search processes. For a fixed mean reset time, we show that the search efficiency can be optimized by adapting the shape of the waiting time distribution.   

Persistent Probability Currents in Non-equilibrium Steady States

For many interesting phenomena in nature, from all life forms to the global climate, the fundamental hypothesis of equilibrium statistical mechanics does not apply. Instead, they are perhaps better characterized by non-equilibrium steady states, evolving with dynamical rules which violate detailed balance. In particular, such dynamics leads to the existence of non-trivial, persistent probability currents - a principal characteristic of non-equilibrium steady states. In turn, they give rise to the notion of 'probability angular momentum'. Observable manifestations of such abstract concepts will be illustrated in two distinct contexts: a heterogeneous nonlinear voter model and our ocean heat content.   

From randomly accelerated particles to L\'{e}vy walks: non-ergodic behavior and aging

For randomly accelerated particles we detected, and were able to analyze in detail (PRL 113, 184101 (2014)), the phenomenon of weak-ergodicity breaking (WEB), i.e. the inequivalence of ensemble- and time-averaged mean-squared displacements (MSD). These results, including their aging time dependence, are relevant for anomalous chaotic diffusion in Hamiltonian systems, for passive tracer transport in turbulent flows, and many other systems showing momentum diffusion. There are, however, several related models, such as the integrated random excursion model, or, space-time correlated L\'{e}vy walks and flights, with similar statistical behavior. We compare the WEB related properties of these models and find surprising differences although, for equivalent parameters, all of them are supposed to lead to the same ensemble-averaged MSD. Our findings are relevant for distinguishing possible models for the anomalous diffusion occurring in experimental situations.   

Disordered confinement and anomalous diffusion

We discuss the effect of disordered confinement on anomalous diffusion. We treat confinement in conjunction with ordinary diffusion and with anomalous diffusions associated with aging and with correlated displacements. In particular, we compute the altered anomalous exponents. Finally, we relate these results to previous work and show that they shed light on the nature of diffusion on percolation clusters.   

Non-equilibrium dynamics of the complex Ginzburg-Landau equation

The complex Ginzburg-Landau equation combines the quantum many-particle nonlinear Schrödinger equation with the time-dependent Ginzburg-Landau equation or model A relaxational dynamics. It arises in quite diverse contexts that include spontaneous pattern formation out of equilibrium, chemical oscillations, multi-mode lasers, thermal convection in binary fluids, cyclic population dynamics, and driven-dissipative Bose-Einstein condensates. Indeed, the complex Ginzburg-Landau equation exhibits a remarkably rich phase diagram with intriguing dynamics. We employ detailed numerical studies as well as analytical tools such as the perturbative renormalization group and the spherical model limit to study the non-equilibrium coarsening and critical aging scaling for the complex Ginzburg-Landau equation following quenches from an initial disordered configuration to either one of the ordered phases or the critical point.   

Multiscale phenomena and crossover in fluctuations in non-equilibrium systems

Fluctuations in multiscale phenomena in natural systems, e. g., Earth's magnetosphere, exhibit crossover behavior in the scaling exponents. These exponents represent the nature of correlation in the system and the crossover shows the presence of more than one type of correlation. An accurate characterization of the crossover behavior is thus needed for a better understanding of the inherent correlations in the system. A multi-step process is developed for accurate computation of the crossover behavior. First the detrended fluctuation analysis is used to remove the trends in the data and the scaling exponents are computed. The crossover point is then computed by a Hyperbolic regression technique, with no prior assumptions. The time series data of the magnetic field variations in the Earth's magnetosphere is analyzed with these techniques and yields a crossover behavior with a time scale of ~ 4 hrs. A Langevin model of the magnetospheric dynamics yields an excellent fit to the crossover in the scaling exponents and thus provide a model of the non-equilibrium system.   

Ligand binding kinetics in surface plasmon resonance devices: A Monte Carlo simulation analysis

Surface plasmon resonance (SPR) chips are widely used to measure association and dissociation rates for the binding kinetics between two species of chemicals, e.g., cell receptors and ligands. It is commonly assumed that ligands are spatially well mixed in the SPR region, and hence a mean-field rate equation description is appropriate. This approximation however ignores the spatial fluctuations as well as temporal correlations induced by multiple local rebinding events, which become prominent for slow diffusion rates. We report detailed Monte Carlo simulations of ligand binding kinetics in an SPR cell subject to laminar flow. We extract the binding and dissociation rates by means of the techniques frequently employed in experimental analysis that are motivated by the mean-field approximation. We find major discrepancies in a wide parameter regime between the thus extracted rates and the input simulation values. These results underscore the crucial quantitative importance of spatio-temporal correlations in binary reaction kinetics in SPR cell geometries.   

Thermodynamic and Information Entropy in Electroconvection

Transitions in driven systems often produce wild fluctuations that can be both detrimental and beneficial. Our fundamental understanding of these transients is inadequate to permit optimal interactions with systems ranging from biology, to energy generation, to finance. Here we report on experiments performed in electroconvecting liquid crystals where we abruptly change the electrical forcing across the sample from a state below defect turbulence into a state of defect turbulence. We simultaneously measure the electrical power flow through the liquid crystal as well as image the structure in the sample. These measurements enable us to simultaneously track the evolution of the thermodynamic and information entropies. Our experiments demonstrate that there are strong correlations between the fluctuations in these two entropic measures however they are not exact. We will discuss these discrepancies as well as the relevance of large transient fluctuations in non-equilibrium transitions in general.   

Low dissipation in non-equilibrium control: sampling the ensemble of efficient protocols.

Designing schemes to efficiently control fluctuating, non-equilibrium systems is problem of fundamental importance and tremendous practical interest. A number of optimization techniques have proven fruitful in the pursuit of optimal control, but these approaches focus on the singular goal of finding the exact, optimal protocol. Here, we investigate the diversity of protocols that achieve low dissipation with a Monte Carlo path sampling algorithm. Akin to Boltzmann weighting configurations in Metropolis Monte Carlo, each protocol is exponentially biased by its mean dissipation. We show that the ensemble of low dissipation protocols can be sampled exactly in the Gaussian limit and that the method continues to robustly generate low dissipation protocols, even as the external control drives the system far from equilibrium.   

Geometry of dissipative evolution equations

The modeling of continuum dissipative evolution equations remains a challenge and is primarily based on phenomenological constitutive relations. In this talk we present some connections between the geometry of dissipative gradient flows, the principle of maximum entropy production, large deviation principles for stochastically augmented evolution equations and fluctuation-dissipation relations.