Bulletin of the American Physical Society
APS March Meeting 2015
Volume 60, Number 1
Monday–Friday, March 2–6, 2015; San Antonio, Texas
Session J42: GSNP Student Speaker Award / Nonlinear Dynamics and Chaos |
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Sponsoring Units: GSNP Chair: Adilson Motter, Northwestern University Room: 214B |
Tuesday, March 3, 2015 2:30PM - 2:42PM |
J42.00001: Phase Transitions on Random Lattices: How Random is Topological Disorder? Hatem Barghathi, Thomas Vojta We study the effects of topological (connectivity) disorder on phase transitions. We identify a broad class of random lattices whose disorder fluctuations decay much faster with increasing length scale than those of generic random systems, yielding a wandering exponent of $\omega=(d-1)/(2d)$ in $d$ dimensions. The stability of clean critical points is thus governed by the criterion $(d+1)\nu > 2$ rather than the usual Harris criterion $d\nu > 2$, making topological disorder less relevant than generic randomness. The Imry-Ma criterion is also modified, allowing first-order transitions to survive in all dimensions $d > 1$. These results explain a host of puzzling violations of the original criteria for equilibrium and nonequilibrium phase transitions on random lattices. We discuss applications, and we illustrate our theory by computer simulations of random Voronoi and other lattices. [Preview Abstract] |
Tuesday, March 3, 2015 2:42PM - 2:54PM |
J42.00002: Shear Jamming in Frictionless Particulate Media Thibault Bertran, Corey S. O'Hern, R.P. Behringer, Bulbul Chakraborty, Mark D. Shattuck We numerically study two-dimensional packings of frictionless bidisperse disks created using compresive and simple shearing protocols. To create jammed packings by compression, we start $N$ particles from random positions and grow their diameters followed by relaxation of particle overlaps using energy minimization. These compressed packings exist over a range of packing fractions $\phi$. As a result, during compression the system may reach a $\phi$ above the minimum value before jamming. If this unjammed packing is then sheared by a strain $\gamma$, it can jam. Using a combination of compression and shearing, we can define jamming protocols as trajectories in the $(\phi, \gamma)$ plane that yield jammed packings. In this plane, we can reach a particular point $(\phi_n, \gamma_n)$ in many ways. We will focus on two protocols: (1) shearing to $\gamma_n$ at $\phi=0$ followed by compression to $\phi_n$ at $\gamma= gamma_n$ and (2) compression to $\phi_n$ at $\gamma=0$ followed by shearing to $\gamma_n$ at $\phi=\phi_n$. For protocol 1, we find that the probability of finding a jammed packing at $\phi$ and $\gamma$, $P(\phi,\gamma)=Q(\phi)$ is indepependent of $\gamma$. For protocol 2, we use a simple theory to deduce $P(\phi,\gamma)$ from $Q(\phi)$. [Preview Abstract] |
Tuesday, March 3, 2015 2:54PM - 3:06PM |
J42.00003: Three Dimensional Characterization of Quantum Vortex Dynamics in Superfluid Helium David Meichle, Daniel Lathrop Vorticity is constrained to line-like topological defects in quantum superfluids, such as liquid Helium below the Lambda transition. We have invented a novel method to disperse fluorescent nanoparticles directly into the superfluid which become trapped on the vortex cores, providing optical tracers. Using a newly constructed multi-camera stereographic microscope, we present data dynamically characterizing vortex reconnections and the subsequent emission of Kelvin waves fully in three dimensions. Statistics of thermally driven counterflow will be compared in 3D to previous measurements in projection. [Preview Abstract] |
Tuesday, March 3, 2015 3:06PM - 3:18PM |
J42.00004: Defect-Stabilized Phases in Extensile Active Nematics Gabriel Redner, Stephen DeCamp, Zvonimir Dogic, Michael Hagan Active nematics are liquid crystals which are driven out of equilibrium by energy-dissipating active stresses. The equilibrium nematic state is unstable in these materials, leading to beautiful and surprising behaviors including the spontaneous generation of topological defect pairs which stream through the system and later annihilate, yielding a complex, seemingly chaotic dynamical steady-state. In this talk, I will describe the emergence of order from this chaos in the form of previously unknown broken-symmetry phases in which the topological defects themselves undergo orientational ordering. We have identified these defect-ordered phases in two realizations of an active nematic: first, a suspension of extensile bundles of microtubules and molecular motor proteins, and second, a computational model of extending hard rods. I will describe the defect-stabilized phases that manifest in these systems, our current understanding of their origins, and discuss whether such phases may be a general feature of extensile active nematics. [Preview Abstract] |
Tuesday, March 3, 2015 3:18PM - 3:30PM |
J42.00005: Jamming Percolation in Three Dimensions Eial Teomy, Antina Ghosh, Yair Shokef We introduce a three-dimensional kinetically-constrained model for jamming and glasses [1], and prove that the fraction of frozen particles is discontinuous at the directed-percolation critical density. In agreement with the accepted scenario for jamming- and glass-transitions, this is a mixed-order transition; the discontinuity is accompanied by diverging length- and time-scales. Because one-dimensional directed-percolation paths comprise the backbone of frozen particles, the unfrozen rattlers may use the third dimension to travel between their cages. Thus the dynamics are diffusive on long-times even above the critical density for jamming. Our new model is a non-trivial extension of the two-dimensional spiral model [2]. \\[4pt] [1] A. Ghosh, E. Teomy, and Y. Shokef, \textit{Europhys. Lett.} \textbf{106}, 16003 (2014).\\[0pt] [2] G. Biroli and C. Toninelli, \textit{Eur. Phys. J. B} \textbf{64}, 567 (2008). [Preview Abstract] |
Tuesday, March 3, 2015 3:30PM - 3:42PM |
J42.00006: Diffuse globally, compute locally: a cyclist approach to modeling long time robot locomotion Tingnan Zhang, Daniel Goldman, Predrag Cvitanovi\'c To advance autonomous robots we are interested to develop a statistical/dynamical description of diffusive self-propulsion on heterogeneous terrain. We consider a minimal model for such diffusion, the 2-dimensional Lorentz gas, which abstracts the motion of a light, point-like particle bouncing within a large number of heavy scatters (e.g. small robots in a boulder field). We present a precise computation (based on exact periodic orbit theory formula for the diffusion constant) for a periodic triangular Lorentz gas with finite horizon. We formulate a new approach to tiling the plane in terms of three elementary tiling generators which, for the first time, enables use of periodic orbits computed in the fundamental domain (that is, $1/12$ of the hexagonal elementary cell whose translations tile the entire plane). Compared with previous literature, our fundamental domain value of the diffusion constant converges quickly for inter-disk separation/disk radius $>0.2$, with the cycle expansion truncated to only a few hundred periodic orbits of up to $5$ billiard wall bounces. For small inter-disk separations, with periodic orbits up to $6$ bounces, our diffusion constants are close ($<10\%$) to direct numerical simulation estimates and the recent literature probabilistic estimates. [Preview Abstract] |
Tuesday, March 3, 2015 3:42PM - 3:54PM |
J42.00007: Fractal Geometry of Undriven Dissipative Systems Xiaowen Chen, Takashi Nishikawa, Adilson E. Motter Traditional studies of chaos in conservative or driven dissipative systems have established a correspondence between sensitive dependence on initial conditions and fractal basin boundaries. Here, I will present on a new type of chaos due to transient interactions with transient chaotic saddles in undriven dissipative systems. I will show that such systems can develop complicated trajectories, but only exhibit fractality and the Wada property at all scales for specific parameter choices at which the dynamics have a degenerate fixed point. For other parameter choices, the boundaries become simple at sufficiently small but widely different scales across the phase space, despite the observed sensitive dependence on initial conditions. However, such scales are often far below the current computational resolution even for low-dimensional dynamical systems. [Preview Abstract] |
Tuesday, March 3, 2015 3:54PM - 4:06PM |
J42.00008: On Path Attractors, Stochastic Bifurcation and Dephasing In Genetic Networks Davit Potoyan Gene regulatory networks are driven stochastic systems with the noise having two distinct components due to the to birth and death of metabolite molecules and dichotomous nature of gene state switching. Presence of dichotomous gene noise alone has the capacity to significantly perturb the optimal transition paths and steady state probability distributions compared to the macroscopic models and their weak noise approximations. Most importantly dichotomous gene noise can also lead to multimodal distributions due to stochastic bifurcation of the underlying nonlinear dynamical system, which underlies the mechanism of formation of population heterogeneity. In this note we derive approximate path based expression of the time dependent probability of gene circuits which enables deeper exploration of the role of gene noise in formation of epigenetic states and dephasing-like phenomena. [Preview Abstract] |
Tuesday, March 3, 2015 4:06PM - 4:18PM |
J42.00009: Model reduction by manifold boundaries Mark Transtrum Mathemtical models of physical systems can be interpreted as manifolds of predictions embedded in the space of data. For models of complex systems with many parameters, the corresponding model manifold is very high-dimensional but often very thin. This ``low effective dimensionality'' has been described as a hyper-ribbon and is characteristic of systems exhibiting simple, emergent behavior. I discuss a new model reduction method, the manifold boundary approximation method, which constructs a series of models by iteratively approximating the high-dimensional, thin manifold by its boundary. This model reduction method unifies many different model reduction techniques, such as renormalization group and continuum limits, while greatly expanding the domain of tractable models. I demonstrate with a model of a complex signaling network from systems biology. The method produces a series of approximations which reveal how microscopic parameters are systematically ``compressed'' into a few macroscopic degrees of freedom, effectively building a bridge between the microscopic and the macroscopic descriptions. [Preview Abstract] |
Tuesday, March 3, 2015 4:18PM - 4:30PM |
J42.00010: Saturation in coupled oscillators Ahmed Roman, James Hanna We consider a weakly nonlinear system consisting of a resonantly forced oscillator coupled to an unforced oscillator. It has long been known that, for quadratic nonlinearities and a 2:1 resonance between the oscillators, a perturbative solution of the dynamics exhibits a phenomenon known as saturation. At low forcing, the forced oscillator responds, while the unforced oscillator is quiescent. Above a critical value of the forcing, the forced oscillator's steady-state amplitude reaches a plateau, while that of the unforced oscillator increases without bound. We show that, contrary to established folklore, saturation is not unique to quadratically nonlinear systems. We present conditions on the form of the nonlinear couplings and resonance that lead to saturation. Our results elucidate a mechanism for localization or diversion of energy in systems of coupled oscillators, and suggest new approaches for the control or suppression of vibrations in engineered systems. [Preview Abstract] |
Tuesday, March 3, 2015 4:30PM - 4:42PM |
J42.00011: Regular and Chaotic Motion of a Piecewise Smooth Bouncer Cameron Langer, Bruce Miller The dynamical properties of a particle in a gravitational field colliding with a rigid wall moving with piecewise constant velocity are studied. The linear nature of the wall's motion permits further analytical investigation than is possible for the system's sinusoidal counterpart. We consider three distinct collision models: elastic, and inelastic with either a constant or velocity dependent restitution coefficient. We confirm the existence of unbounded orbits (Fermi acceleration) in the elastic model, and find regular and chaotic behavior in the inelastic cases. We also examine trajectories wherein the particle experiences an infinite number of collisions in a finite time i.e., the phenomenon of inelastic collapse. We address these ``sticking solutions'' and their relation to both the overall dynamics and the phenomenon of self-reanimating chaos. Additionally, we investigate the long-term behavior of the system as a function of both initial conditions and parameter values. We find novel bifurcation phenomena not seen in the sinusoidal model. Our investigations reveal that, although the piecewise linear bouncer is a simplified version of the sinusoidal model, it captures essential features of the latter and also exhibits behavior unique to the discontinuous dynamics. [Preview Abstract] |
(Author Not Attending)
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J42.00012: Nonlinear Dynamics and Thermodynamics of a One-Dimensional Plasma in Simulation Pankaj Kumar, Bruce Miller We report on the results of a simulation study of the nonlinear dynamics and the thermodynamics of a single-component one-dimensional plasma with periodic boundary conditions. For a system of the plasma with three particles, we plot the Poincare maps and calculate the largest Lyapunov exponents. The results indicate that the three-particle system exhibits interesting dynamics with the phase-space containing periodic, quasiperiodic, as well as chaotic regions for different initial conditions. Special periodic orbits have been identified and their stabilities have been examined for the three-particle system. The behavior of the system in the thermodynamic limit has been simulated using large versions of the system and the dependences of the pressure, the coupling strength and the largest Lyapunov exponent on the average per-particle kinetic energy are presented. The results of the thermodynamic-limit simulations indicate that the net pressure is equal to the kinetic pressure for all temperatures and there is no phase transition. [Preview Abstract] |
Tuesday, March 3, 2015 4:54PM - 5:06PM |
J42.00013: Evolution of a One-dimensional, Two Component, Universe Yui Shiozawa, Bruce Miller, Jean-Louis Rouet While the universe we observe today exhibits local filament-like structures, with stellar clusters and large voids between them, the primordial universe is believed to have been nearly homogeneous with slight variations in matter density. To understand how the observed hierarchical structure was formed, researchers have developed a one-dimensional analogue of the universe that can simulate the evolution of a large number of matter particles. Investigations to date demonstrate that this model reveals structure formation that shares essential features with the three-dimensional observations. In the present work, we have expanded on this concept to include two species of matter, specifically dark matter and luminous matter. In our simulation, luminous matter is treated in a way that loses energy in interaction with itself. The results of the simulations clearly show the formation of a Cantor set like multifractal pattern over time in configuration space as well as in phase space. In contrast with most earlier studies, mass-oriented methods for computing the multifractal dimensions were performed on various subsets of the matter distribution in order to understand the bottom-up structure formation. [Preview Abstract] |
Tuesday, March 3, 2015 5:06PM - 5:18PM |
J42.00014: Cosmology in One Dimension: Fractal Dimensions from Mass Oriented Partitions Bruce Miller, Jean-Louis Rouet, Yui Shiozawa The distribution of visible matter in the universe has its origin in the weak fluctuations of density that existed at the epoch of recombination. The hierarchical distribution of the present universe, with its galaxies, clusters and super-clusters of galaxies indicates the absence of a natural length scale. Numerical simulations of a one-dimensional system permit us to precisely follow the evolution starting with an initial perturbation in the Hubble flow. The limitation to one dimension removes the necessity to make approximations in calculating the gravitational field and the system dynamics. It is then possible to accurately follow the trajectories of particles for a long time. The simulations show the emergence of a self-similar hierarchical structure in both the phase space and the configuration space and invites the implementation of a multifractal analysis. Here we apply four different methods for computing generalized fractal dimensions $D_q$ of the distribution of particles in configuration space. We first employ the conventional methods based on partitions of equal size and then less familiar methods based on partitions of equal mass. We show that the latter are superior for computing generalized dimensions for indices $q<-1$ which characterize regions of low density. [Preview Abstract] |
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