Session L44: Focus Session: Systems far from Equilibrium II

Extremes in systems with linear and nonlinear memory by the return-interval approach

The occurrence of extreme events above a certain threshold $Q$ in time series can be characterized by their return intervals $r_i$. Here we review recent work on the distribution $P_Q(r)$ of the return intervals and their correlation properties (i) in systems with linear long-term memory and (ii) in systems with non-linear memory. Examples for (i) are temperature records, examples for (ii) are financial records. The distribution of the return intervals is an important quantity in risk estimation since it enables one to calculate the probability that an extreme event occurs in the next period of time. We discuss the different functional forms of $P_Q(r)$ that range from simple exponential (random systems) to stretched exponentials (systems with long-term memory) and q-exponentials (systems with non-linear memory). We show that both linear and non-linear memory lead to long-term memory in the return intervals, which then results in a clustering of the extreme events. Both the distribution of the return intervals and their correlation properties can be used as a test bed for computer models.   

Rare Event Extinction on Stochastic Networks

We consider the problem of extinction processes on random networks with a given structure. For sufficiently large well-mixed populations, the process of extinction of one or more state variable components occurs in the tail of the quasi-stationary probability distribution, thereby making it a rare event. Here we show how to extend the theory of large deviations to random networks to predict extinction times. In particular, we use the theory to find the most probable path leading to extinction. We apply the methodology to epidemic models and discover how mean extinction times scale with epidemiological and network parameters in Erdos-Renyi networks. The results are shown to compare quite well with Monte Carlo simulations of the network in predicting both the most probable paths to extinction and mean extinction times.   

Revealing non-Gaussian noise through noise-induced switching in a parametric oscillator

Rates of noise-induced switching between coexisting states of dynamical systems exponentially strongly depend on the noise characteristics. We use the related sensitivity to reveal the deviation of the noise from Gaussian. We study a parametrically driven nonlinear oscillator, which has two stable states of forced vibrations at half the frequency $\omega_F$ of the driving field. The states are identical, except that they are shifted in phase by $\pi$. Noise causes switching between the states. A stationary noise leads to a stationary distribution over the states. If the noise is Gaussian and coordinate-independent, the probability densities of noise pulses of the opposite polarities are the same. As a result, the state populations are equal. The difference of the state populations is an indication of non-Gaussian noise. We illustrate the effect for a sinusoidal signal at frequency $\omega_F/2$ modulated by Poisson-distributed pulses. We show, theoretically and through the experiment with a micro-electro-mechanical system, that the population difference is highly sensitive to the rate and amplitude of the pulses and displays a characteristic nonmonotonic dependence on the pulse rate. The theory is in quantitative agreement with the experiment.   

Dependence of mean switching times on relative noise intensity in fast-slow dynamical systems

Recently, we used a geometric minimum action method to analytically and numerically study the dependence of most probable escape paths (MPEPs) on relative noise intensities in a generic two-dimensional fast-slow dynamical system [1]. In this talk, we apply and extend this approach to study MPEPs and associated mean switching times in a quadratic integrate-and-fire model of single neuron dynamics as a function of relative noise intensity in the two dynamical variables. Here, one variable is associated with the membrane potential while the second is associated with membrane ionic permeability. Noise-induced switching times correspond to the rate at which spontaneous action potentials occur. The fast-slow nature of this system allows us to derive analytical expressions for both the MPEPs and mean switching times as functions of the relative noise intensity. Derived expressions are found to be in good agreement with both computational minimization of the geometric action as well as direct simulation of the underlying stochastic differential equations. [1] P. H. Dannenberg, J. C. Neu, and S. W. Teitsworth, Phys. Rev. Lett. 113, 020601 (2014).   

Noise Induced Switching and Extinction in Systems with Delay

We consider the rates of noise-induced switching between multiple attractors of dynamical systems with delay, and the rates of noise-induced extinction in delayed systems modeling population dynamics. In the weak noise limit, the rates of inter-attractor switching and extinction are exponentially small. To logarithmic accuracy, the formulation of the rates is reduced to variational problems, which give the most probable paths followed in both switching or extinction dynamics. We show that the equations for the most probable paths are acausal and formulate the appropriate boundary conditions. Explicit general results of the rates are obtained for small delay compared to the relaxation rate, and verified using a direct variational method to find the rates. We find that the analytical results agree well with the numerical simulations for both switching and extinction rates.   

Probability current loops in non-equilibrium steady states and statistical properties of angular momenta in configuration space

Unlike systems in thermal equilibrium, steady probability current loops persist in non-equilibrium stationary states. One of the consequences is that, in the space of two or more observable quantities ($q_{i}$), the average ``angular momentum'' ($\left\langle L_{ij}\right\rangle \equiv \left\langle q_{i}\times \dot{q}_{j}\right\rangle $) is typically non-trivial. In addition, the full distribution of $L$ often display remarkable properties. We will provide a general framework for the study of $% L$, as well as specific examples -- in the context of both exactly solvable models (based on linear Langevin equations with additive white noise) and physical data of global ocean heat content.   

Experimental Observation of Dynamic Phase Transitions in uniaxial Co-Films

We studied the time dependent magnetic reversal behavior of uniaxial films in the vicinity of the dynamic phase transition (DPT) as a function of the period P and bias H$_{b}$ of an oscillating magnetic field. For our experiments, we have used Co-films with in-plane orientation of the uniaxial magneto-crystalline anisotropy axis to avoid complications due to long-range magneto-static interactions. Correspondingly, we have grown 30 nm thick Co-films that exhibit (10-10) surface orientation by means of suitable growth sequences and deposition conditions [1]. For the dynamic field response experiments, we utilized a home-built high-sensitivity magneto-optical Kerr effect set-up, which allowed for real-time low-noise hysteresis loop measurements with P as small as 0.580 ms. Our experiments reveal in addition to the DPT at a critical period P$_{c}$, the occurrence of transient dynamic behavior for P \textless P$_{c}$ [2]. Our data are consistently explained by a phase line at H$_{b} \quad =$ 0 for P \textless P$_{c}$, which causes a first order phase transition in between two antiparallel dynamic order states, thus indicating far-reaching similarities of the DPT to equilibrium phase transitions [2]. However, we also observe anomalies, such as unusual fluctuation pattern in the P-H$_{b}$ plane, which might be related to the recently suggested occurrence of dynamically ``dead'' surfaces [3]. References: [1] O. Idigoras et al., J. Appl. Phys. \textbf{115, }083912 (2014); [2] A. Berger et al., Phys. Rev. Lett. \textbf{111}, 190602 (2013); [3] H. Park and M. Pleimling, Phys. Rev. Lett. \textbf{109}, 175703 (2012).   

Effects of magnetic field quenches on the relaxation dynamics of vortex lines in disordered type-II superconductors

Understanding the relaxation dynamics of vortex matter in disordered type-II superconductors from experimentally realizable initial conditions may improve material characterization and optimization for technological applications. We model magnetic flux lines in the London limit as interacting directed elastic lines subject to uncorrelated point-like or extended columnar pinning centers. We employ a Langevin Molecular Dynamics algorithm to simulate the vortex dynamics. We analyze the vortex relaxation kinetics following sudden magnetic field changes by instantaneously adding or removing lines from the system at random. By studying two-time correlation functions such as the mean-square displacement and height autocorrelation function, as well as one-time observables such as the ratio of pinned line elements and radius of gyration, we disentangle the effects of the competing repulsive vortex interaction and pinning and we compare the distinct relaxation properties due to the type of disorder. We discovered some universal features regardless of the type of quench and the presence of vortex interactions, and others that are dependent on the type of disorder and the system's initial conditions.   

Pinning time statistics for vortex lines in disordered environments

We study the pinning dynamics of magnetic flux (vortex) lines in a disordered type-II superconductor. Using numerical simulations of a directed elastic line model, we extract the pinning time distributions of vortex line segments. We compare different model implementations for the disorder in the surrounding medium: discrete, localized pinning potential wells that are either attractive and repulsive or purely attractive, and whose strengths are drawn from a Gaussian distribution; as well as continuous Gaussian random potential landscapes. We find that both schemes yield power law distributions in the pinned phase as predicted by extreme-event statistics, yet they differ significantly in their effective scaling exponents and their short-time behavior.   

Dead magnetic surface layers near dynamic phase transitions

We have performed a detailed study of the dynamic phase transition (DPT) for a magnetic layer system with surfaces subjected to an oscillatory external magnetic field in mean field approximation (MFA). Specifically, we focused our study on bulk-terminated surfaces, $i.e.$, we deal with multilayer systems that have the same exchange coupling strength between nearest neighbors everywhere, including at the surface. We are able to reproduce within the MFA the absence of a surface phase transition at the bulk critical point that was previously reported by Tauscher \textit{et al.} utilizing Monte-Carlo simulations [1]. In addition, we observe that the DPT is also absent or at least severely suppressed for several layers below the surface, which exhibit susceptibility peaks that are four orders of magnitude smaller than the one corresponding to the bulk. Most importantly, we identify the reason for this ``dead'' surface behavior. The oscillatory magnetization M(t) response to the external magnetic field is not synchronous in between the surface and the bulk near the DPT. This lack of correlation prevents the layers from sufficiently supporting each other's dynamic ordering, so that the surface and the layers close to it cannot follow the bulk DPT.\\[4pt] [1] K. Tauscher \textit{et al.} Phys. Rev. E \textbf{89}, 022121 (2014).   

Spherical surface growth models

We study several surface growth models obtained by treating and replacing the non-linear term in the noisy Burgers equation or the KPZ equation by a mean spherical condition. We want to explore the consequences of such constraints on the Edwards-Wilkinson (EW) interface. In those exactly solvable models, one has to solve the spherical conditions and then we can derive the two-time quantities (the correlation function and the response function). Therefore, we have access to the non-equilibrium exponents and compare them to those of the EW model or the KPZ model.   

Failure of Steady State Thermodynamics

To be useful, steady state thermodynamics (SST) must be self-consistent and have predictive value. Consistency of SST was recently verified for driven lattice gases under global weak exchange. Here, I verify consistency of SST under local (pointwise) exchange, but only in the limit of a vanishing exchange rate; for a finite exchange rate the coexisting systems have different chemical potentials. I consider the lattice gas with nearest-neighbor exclusion on the square lattice, with nearest-neighbor hopping (NNE dynamics), and with hopping to both nearest and next-nearest neighbors (NNE2 dynamics). I show that SST does not predict the coexisting densities under a nonuniform drive, or in the presence of a nonuniform density provoked by a hard wall or nonuniform transition rates. The steady state chemical potential profile is, moreover, nonuniform at coexistence, contrary to the basic principles of thermodynamics. Finally, I discuss examples of a pair of systems possessing {\it identical steady states}, but which {\it do not coexist} when placed in contact. These results cast serious doubt on the consistency and predictive value of SST.   

Self-organization in non-equilibrium systems

The question about why complex systems self-organize to reach more efficient and robust states is still without a satisfactory answer. We approach it from a physics perspective, where energy gradients lead to change in the structure of systems to ensure the most efficient energy transport. This approach stems from fundamental variational principles in physics, such as the principle of least action, which determine the motion of particles. We compare energy transport through a cell which has random motion of its molecules, and a cell which can form convection cells. We examine the sign of change of entropy, and the action needed for the motion inside those systems. The system in which convective motion occurs, reduces the time for energy transmission, compared to random motion. For more complex systems, this convection cells become a network of transport channels, for the purpose of obeying the equations of motion in this geometry. Those transport networks are an essential feature of complex systems in biology, ecology, economy and society in general. This approach can help explain some of the features of those transport networks, and how they correlate with the level of complexity of systems.