Session Y13: Statistical and Nonlinear Physics I

Numerical scheme for predicting the order parameter of the Kuramoto model

The Kuramoto model is a well-studied prototype model for synchronization of a large number $N$ of coupled oscillators. Amongst its most notable feature is a second order phase transition to synchronization, as a function of the ratio between the coupling and the width of the distribution of the independent frequencies. Here we present a numerical scheme for predicting the order parameter for a specified finite set of oscillators. Comparison to direct numerical simulations of the differential equaions show that the scheme works well for large unimodal distributions ($N>1000$). Even for small unimodal populations, the scheme accurately predicts larger values for the order parameter. We then apply this scheme to study the scaling behavior of avalanching (finite jumps in order parameter as a function of external forcing or coupling strength) in the Kuramoto model.   

Dynamics of finite-size networks of coupled oscillators

Mean field models of coupled oscillators do not adequately capture the dynamics of large but finite size networks. For example, the incoherent state of the Kuramoto model of coupled oscillators exhibits marginal modes in mean field theory. We demonstrate that corrections due to finite size effects render these modes stable in the subcritical case, i.e. when the population is not synchronous. This demonstration is facilitated by the construction of a non-equilibrium statistical field theoretic formulation of a generic model of coupled oscillators. This theory is consistent with previous results. In the all-to-all case, the fluctuations in this theory are due completely to finite size corrections, which can be calculated in an expansion in 1/N, where N is the number of oscillators. The N $\rightarrow$ infinity limit of this theory is what is traditionally called mean field theory for the Kuramoto model. We also demonstrate this approach with a system of pulse coupled theta neurons and describe the stability of the population activity.   

Update on the Swinging Atwood's Machine

The Swinging Atwood's machine (SAM) is an Atwood's machine where one of the masses is allowed to swing in a plane. There are only a few integrable examples of mechanical systems, and we show that SAM is completely integrable when the mass ratio is three. We also present an overview of recent results that analyze the dynamics of SAM for other mass ratios using the Painleve analysis and Galois theory, which indicate that SAM is non-integrable for other values of mass ratios.   

Experimental study of a spatiotemporal phase synchronization transition in a 1D-array of nonlocally coupled oscillators

We report the first experimental evidence in hydrodynamics of a phase synchronization transition between 80 nonlocally coupled convective oscillators. The initial pattern corresponds to a spatiotemporal chaotic regime of irregular clusters which becomes unstable by increasing the vertical temperature difference. Further beyond, a robust pattern emerges with two large localized stationary clusters. We show that oscillators belonging to these stationary clusters have become synchronized through a supercritical bifurcation. The antiphase cross-correlations define surfaces of synchronized oscillators, we study how the interaction range between the initially nonlocally coupled oscillators increases as we cross quasi-statically the threshold of this bifurcation.   

The scaling behavior of oscillations arising in delay-coupled optoelectronic devices

We study the effect of asymmetric coupling strength on the onset of oscillations in an experimental system of nonlinear optoelectronic devices with delayed feedback and wide-band bandpass filtering. Specifically, we consider a network consisting of two Mach-Zehnder modulators that are cross-coupled optoelectronically. We find that oscillations appear in the system when the product of the coupling strengths exceeds a critical value. We also find a scaling law that describes how the amplitude of the oscillations depends on the coupling strengths. The observations are in good agreement with predictions from normal form theory.   

Zero-temperature Glauber dynamics of a random-field Isng model on a Bethe lattice

Zero-temperature Glauber dynamics of random-field Ising model has been investigated for some decades as a simple model for understanding intermittent behaviors of complex materials responding to an external field. It has been known that there is a critical point in the parameter space of this model, where an external magenetic field and a dispersion of random field are the parameters of the model. Until now, static properties related to this critical point under quasi-static operation have been studied extensively. However, dynamical behaviors under non quasi-static operation such as quenching, have not been known sufficiently in comparison with the static properties. In this presentation, we derive exactly an evolution equation for an order parameter that describes dynamical behaviors of zero temperature Glauber dynamics of random-field Ising model on a Bethe lattice under a quenched initial condition. By analyzing the obtained evolution equation, we determine the value of a critical exponent that characterizes slow dynamical behavior observed near the critical point.   

Phase-Space Networks of Frustrated spin models

We directly studied the phase spaces of two classical frustrated spin models: the antiferromagnet on triangular lattice and the six-vertex model. Their highly degenerated ground states are mapped as discrete networks such that quantitative network analysis can be applied to phase-space studies. The resulting phase spaces of different models under different boundary conditions share some common features and establish a new class of complex networks with unique topology. We proved that the spectral densities of networks approach the Gaussian distribution at the infinite-size limit. The six-vertex model has a one-to-one correspondence to three-dimensional sphere stacks. This work connects a traditional field (frustrated spin models) and a new field (complex network since 1998), and provides some open questions. Reference: Phys. Rev. E 80 051102 (2009).   

A Master equation approach to line shape in dissipative systems

The resonant type of experiments, such as ESR and NMR, provide us microscopic informations on materials. In analyzing the experiments, we often assume that the total system is in a factorized (decoupled) state of our relevant system and a heat bath at an initial time, even when the excited state is thermally distributed. However, the {\it whole} material is prepared in an equilibrium state just before application of external field. In such situation, our relevant system has quantum correlation with the heat bath at an initial time, which affects the time evolution of the system in a short time region. This means that we need to extend the linear response theory to include the system-bath correlation at an initial time. In this talk, we propose a new formulation on complex susceptibility which includes the initial correlation between system and bath under the condition that the total system is in an equilibrium state(Phys. Rev. E80 (2009) 021128). In our formulation, we also include frequency shift by system-bath interaction. Applying the obtained formula to spin systems which interact with bosonic reservoir, we find that the effects of initial correlation and frequency shift are reflected in the line shape of complex susceptibility. In the talk, we discuss these effect on one-spin system as well as interacting multiple spin systems.   

Double Percolation Transition in Superconductor/Ferromagnet Nanocomposites

A double percolation transition is identified in a binary network composed of nanoparticles of MgB$_{2}$ superconductor and CrO$_{2}$ half-metallic ferromagnet. Anomalously high-resistance or insulating state, as compared to the conducting or superconducting states in single-component systems of either constituent, is observed between two distinct percolation thresholds. We investigate the scaling behavior near both percolation thresholds, and determine the distinct critical exponents associated with two different types of transitions. This double percolation effect, which is especially pronounced at liquid helium temperatures, is controlled by composite volume fraction and originates from the suppressed interface conduction and tunneling as well as a large geometric disparity between nanoparticles of different species. This sensitivity of the threshold to the geometry is confirmed by replacing CrO$_{2 }$with LSMO particles of different size and shape, which results in significantly different threshold for MgB$_{2.}$   

Switching and control in stochastic double gyres

We consider the problem of stochastic transport in a driven double gyre with time dependent basin boundaries. Noise typically creates almost invariant sets, or meta-stable states. Mixing occurs between these states as a result of stochastic transport. Using stochastic Markov operator theory, we approximate the almost invariant sets and regions with most probable switching. These sets define regions for which control perturbations can facilitate or prevent switching. Switching rates using these control schemes will be derived and compared to numerical simulations. We show how scaling laws change exponentially in the presence of control when compared to the natural switching rate.   

Finite-Size Scaling in Random $K$-SAT Problems

We propose a comprehensive view of threshold behaviors in random $K$-satisfiability ($K$-SAT) problems, in the context of the finite-size scaling (FSS) concept of nonequilibrium absorbing phase transitions using the average SAT (ASAT) algorithm. In particular, we focus on the value of the FSS exponent to characterize the SAT/UNSAT phase transition, which is still debatable. We also discuss the role of the noise (temperature-like) parameter in stochastic local heuristic search algorithms.   

Hybrid Defect Phase Transition: Renormalization Group and Monte Carlo Analysis

For the q-state Potts model with 2 $<$ q $<$= 4 on the square lattice with a defect line, the order parameter on the defect line jumps discontinuously from zero to a nonzero value while the defect energy varies continuously with the temperature at the critical temperature. Monte-Carlo simulations (H. T. Diep, M. Kaufman, Phys Rev E 2009) of the q-state Potts model on a square lattice with a line of defects verify the renormalization group prediction (M. Kaufman, R. B. Griffiths, Phys Rev B 1982) on the occurrence of the hybrid transition on the defect line. This is interesting since for those q values the bulk transition is continuous. This hybrid (continuous - discontinuous) defect transition is induced by the infinite range correlations at the bulk critical point.   

Instantaneous Gelation in Smoluchowski's Coagulation Equation Revisited

We study the solutions of a regularised Smoluchowski coagulation equation with instantaneously gelling kernels. Regularisation is done by introducing a cut-off, $M_{\rm max}$, which physically corresponds to the removal from the system of clusters having mass greater than $M_{\rm max}$. Careful numerical simulations demonstrate that, for monodisperse initial data, the gelation time for $\nu>1$ {\em decreases}, albeit logarithmically slowly, as $M_{\max}$ increases. We thereby clearly demonstrate the instantaneous gelation transition numerically for the first time. The slow dependence on $M_{\rm max}$ explains previous difficulties in characterising the instantaneous gelation transition in simulations and justifies the use of instantaneously gelling kernels as physical models. We also consider solutions with a source of monomers which ultimately reach a stationary state. Approach to the stationary state is non-trivial. Oscillations results from the interplay between the monomer injection and the cut-off which decay very slowly when the cut-off is large.   

Exact results for the Potts model partition function in lattice strips of arbitrary length

The Potts model is of fundamental importance in the study of critical phenomena, especially in two dimensions (2D). Although some relevant quantities such as the critical exponents are known for the ferromagnetic 2D Potts model, the exact partition function in the thermodynamic limit has not been obtained for any lattice in dimensions d$>$1. It is therefore interesting to solve the simpler problem of calculating exactly the partition function for strips of finite width. One way to calculate such function is through the transfer matrix method. We present here a novel approach to obtain the transfer matrices, and show that the procedure has the advantage of being directly applicable to virtually any kind of recursive lattice. We discuss then some new exact results for strips of the kagom\'e and diced lattices.