Session D52: General Statistical and Nonlinear Physics and GSNP Student Prize Talks

Dynamic phase transition in the classical anisotropic XY model on a square lattice

Ginzburg-Landau analysis of the anisotropic XY model in a spatially homogeneous oscillating magnetic field on a square lattice suggests the existence of several dynamical phases - Ising symmetry restoring order (Ising SRO), Ising symmetry breaking order (SBO), XY symmetry restoring order (XY SRO), and XY symmetry breaking order (XY SBO). We investigate the presence of these phases and the dynamic phase transition (DPT) between these phases using classical Monte Carlo simulation techniques. We explore the system for a range of values for the external field amplitude, field frequency, and anisotropy parameter. Utilizing the period-averaged magnetization (in both the x- and y- component) as the dynamic order parameter we confirm the presence of multiple DPT transitions in the model. We also construct the probability density histograms of the dynamic order parameter to validate the existence of the four DPT phases.   

Non-universal local critical exponents at a non-equilibrium phase transition

We study the dynamic phase transition in the two-dimensional semi-infinite kinetic Ising model in an oscillating field. We focus on the critical regime where the competition between the half-period of the oscillating field $t_{1/2}$ and the metastable lifetime $\langle \tau \rangle$ is most pronounced. We focus on layer-dependent quantities, such as the period-averaged magnetization per layer $Q(z)$ and the layer susceptibility $\chi_Q(z)$, and determine surface critical exponents through finite size scaling. We find that the values of these non-equilibrium exponents are non-universal as they depend on the strength of the surface couplings. Results for the three-dimensional model are also briefly discussed and compared to the two-dimensional case.   

Extremal Optimization for p-Spin Models

It was shown recently that finding ground states in the 3-spin model on a 2d dimensional triangular lattice poses an NP-hard problem [1]. We use the extremal optimization (EO) heuristic [2] to explore ground state energies and finite-size scaling corrections [3]. EO predicts the thermodynamic ground state energy with high accuracy, based on the observation that finite size corrections appear to decay purely with system size. Just as found in 3-spin models on $r$-regular graphs, there are no noticeable anomalous corrections to these energies. Interestingly, the results are sufficiently accurate to detect alternating patters in the energies when the lattice size $L$ is divisible by 6. Although ground states seem very prolific and might seem easy to obtain with simple greedy algorithms, our tests show significant improvement in the data with EO. \\[4pt] [1] PRE 83 (2011) 046709,\hfil\break [2] PRL 86 (2001) 5211,\hfil\break [3] S. Boettcher and S. Falkner (in preparation).   

A novel Kinetic Monte Carlo algorithm for Non-Equilibrium Simulations

We have developed an off-lattice kinetic Monte Carlo simulation scheme for reaction-diffusion problems in soft matter systems. The definition of transition probabilities in the Monte Carlo scheme are taken identical to the transition rates in a renormalized master equation of the diffusion process and match that of the Glauber dynamics of Ising model. Our scheme provides several advantages over the Brownian dynamics technique for non-equilibrium simulations. Since particle displacements are accepted/rejected in a Monte Carlo fashion as opposed to moving particles following a stochastic equation of motion, nonphysical movements (e.g., violation of a hard core assumption) are not possible (these moves have zero acceptance). Further, the absence of a stochastic ``noise'' term resolves the computational difficulties associated with generating statistically independent trajectories with definitive mean properties. Finally, since the timestep is independent of the magnitude of the interaction forces, much longer time-steps can be employed than Brownian dynamics. We discuss the applications of this scheme for dynamic self-assembly of photo-switchable nanoparticles and dynamical problems in polymeric systems.   

How an aggregation process helps us understanding solid fragmentation

We report on an experiment intended to understand the fragmentation of a ring composed of cohesive solid spheres (magnets in dipolar interaction). At initial time, the ring is forced to expand radially and the spheres separate from each other. Because of the dipolar attractive force between the spheres, their uniform angular distribution is unstable and the spheres aggregate with each other to form fragments. We record the full dynamics of the spheres assembly and we show that the final fragment size distribution is the signature of the aggregation process giving birth to it. In particular, we introduce a Weber number $We$, based on the radial velocity of the ring, the density of the spheres and their magnetization. We find that the final mean fragment size scales like $We^{-1/3}$ and that the standard deviation of the fragments distribution is proportional to it. We will also discuss the relation between our findings and the fragmentation of elastic rings studied by Sir N. Mott.   

Thermodynamic limit, quasi-stationary states and the range of pair interactions

``Quasi-stationary'' states are approximately time-independent out of equilibrium states which have been observed in a variety of systems of particles interacting by long-range interactions. We investigate here the conditions of their occurrence for a generic pair interaction $V(r \rightarrow \infty) \sim 1/r^\gamma$ with $\gamma > 0$, in $d>1$ dimensions. We generalize analytic calculations known for gravity in $d=3$ to determine the scaling parametric dependences of their relaxation rates due to two body collisions, and report extensive numerical simulations testing their validity. Our results lead to the conclusion that, for $\gamma < d-1$, the existence of quasi-stationary states is ensured by the large distance behavior of the interaction alone, while for $\gamma > d-1$ it is conditioned on the short distance properties of the interaction, requiring the presence of a sufficiently large soft-core in the interaction potential.   

Numerical study of Potts models with aperiodic modulations: influence on first-order transitions

We perform a numerical study of Potts models on a rectangular lattice with aperiodic interactions along one spatial direction. The number of states $q$ is such that the transition is a first-order one for the uniform model. The Wolff algorithm is employed, for many lattice sizes, allowing for a finite-size scaling analyses to be carried out. Three different self-dual aperiodic sequences are employed, such that the exact critical temperature is known: this leads to precise results for the exponents. We analyze models with $q=6$ and $15$ and show that the Harris-Luck criterion, originally introduced in the study of continuous transitions, is obeyed also for first-order ones. The new universality class that emerges for relevant aperiodic modulations depends on the number of states of the Potts model, as obtained elsewhere for random disorder, and on the aperiodic sequence. We determine the occurrence of log-periodic behavior, as expected for models with aperiodic modulated interactions.   

Boundaries Matter for Confined Colloidal Glasses

We confine dense colloidal suspensions within emulsion droplets to examine how confinement and properties of the confining medium affect the colloidal glass transition. Samples are imaged via fast confocal microscopy. By observing a wide range of droplet sizes and varying the viscosity of the external continuous phase, we separate finite size and boundary effects on particle motions within the droplet. Suspensions are composed of binary PMMA spheres in organic solvents while the external phases are simple mixtures of water and glycerol. In analogy with molecular super-cooled liquids and thin-film polymers, we find that confinement effects in colloidal systems are not merely functions of the finite size of the system, but are strongly dependent on the viscosity of the confining medium and interactions between particles and the interface of the two phases.   

Dynamics near shear-jamming for a dense granular system

This talk will present several systematic experimental studies of a two-dimensional, frictional dense granular system subjected to simple shear deformation. The first experiment consists of linear shear for densities smaller than the isotropic jamming point, and examines both the evolution of the average stress and the evolution of force network. These measures reveal three distinguishable regimes of the granular system with increasing shear strain: unjammed, fragile, and shear-jammed regimes. The second experiment uses small amplitude cyclic shear to probe the dynamical response of the states from the first experiment. For fragile or jammed regimes, cyclic shear drives the system through transient states that evolve towards relatively stable forces networks and system-averaged stress. The timescale of the transient increases rapidly as the system moves deeper into the fragile, or shear-jammed regimes. These experiments also involve particle tracking (displacements and rotations) to search for and characterize non-affine motion and spatial heterogeneity. There is a clear increase in particle diffusion with increasing density and shear strain amplitude, even when the system is still unjammed and experiences only minimal stress. When the system is fragile or jammed, the heterogeneity of particle displacements reveals subtle correlations with the force network.   

Elasticity of adherent active cells on a compliant substrate

We present a continuum mechanical model of rigidity sensing by livings cells adhering to a compliant substrate. The cell or cell colony is modeled as an elastic active gel, adapting recently developed continuum theories of active viscoelastic fluids. The coupling to the substrate enters as a boundary condition that relates the cell's deformation field to local stress gradients. In the presence of activity, the substrate induces spatially inhomogeneous contractile stresses and deformations, with a power law dependence of the total traction forces on cell or colony size. This is in agreement with recent experiments on keratinocyte colonies adhered to fibronectin coated surfaces. In the presence of acto-myosin activity, the substrate also enhances the cell polarization, breaking the cell's front-rear symmetry. Maximal polarization is observed when the substrate stiffness matches that of the cell, in agreement with experiments on stem cells.   

Differential geometry of the space of Ising models

We use information geometry to understand the emergence of simple effective theories, using an Ising model perturbed with terms coupling non-nearest-neighbor spins as an example. The Fisher information is a natural metric of distinguishability for a parameterized space of probability distributions, applicable to models in statistical physics. Near critical points both the metric components (four-point susceptibilities) and the scalar curvature diverge with corresponding critical exponents. However, connections to Renormalization Group (RG) ideas have remained elusive. Here, rather than looking at RG flows of parameters, we consider the reparameterization-invariant flow of the manifold itself. To do this we numerically calculate the metric in the original parameters, taking care to use only information available after coarse-graining. We show that under coarse-graining the metric contracts very anisotropically, leading to a ``sloppy'' spectrum with the metric's Eigenvalues spanning many orders of magnitude. Our results give a qualitative explanation for the success of simple models: most directions in parameter space become fundamentally indistinguishable after coarse-graining.   

Seeing and Sculpting Nematic Liquid Crystal Textures with the Thom construction

Nematic liquid crystals are the foundation for modern display technology and also exhibit topological defects that can readily be seen under a microscope. Recently, experimentalists have been able to create and control several new families of interesting defect textures, including reconfigurably knotted defect lines around colloids (Ljubljana) and the ``toron,'' a pair of hedgehogs bound together with a ring of double-twist between them (CU Boulder). We apply the Thom construction from algebraic topology to visualize 3 dimensional molecular orientation fields as certain colored surfaces in the sample. These surfaces turn out to be a generalization to 3 dimensions of the dark brushes seen in Schlieren textures of two-dimensional samples of nematics. Manipulations of these surfaces correspond to deformations of the nematic orientation fields, giving a hands-on way to classify liquid crystal textures which is also easily computable from data and robust to noise.