Session U25: Pattern Formation, Chaos, Nonlinear Dynamics I

Creating novel patterns with localized control in non-linear reaction-diffusion systems

The Gray-Scott model has been subject of numerous investigations. Due to the nonlinear nature of the coupled reaction-diffusion equations, the system exhibits interesting behavior for certain parameter sets. In many previous studies of this system investigators have used a limited range of parameter values dictated by neglecting diffusion effects. Through systematic parameter adjustment we are able to observe and characterize novel system patterns that were previously overlooked. Localized control routines are applied to the system creating long range periodic order from a chaotic parameter space region. We present a comprehensive view of the effects of control routines applied to this system.   

Motion and Chain formation of Conductive Spheres on a Horizontal Insulator Surface Immersed in Dielectric Liquids

This work presents an analysis of the dynamics of conductive, possibly charged spheres resting on the horizontal surface of an ideal insulator, both immersed in dielectric liquids. The system is subject to a uniform background field with components normal and parallel to the surface. The inclusion of dielectric liquids complicates the problem by adding the induced charge electrophoresis effect. In this work, for individual sphere, the dependence of the total force that drives its motion on the properties of the dielectric liquid is shown. For a pair of spheres, the conditions under which they can be pulled together to form a chain are derived. Experimental results are presented to qualitatively verify the theoretical analysis. This is a complex electromechanical system that has been studied for branched pattern formation. However, no detailed analysis of the sphere motion and chain formation has been conducted. This work will bridge this gap, as well as inform some sensor designs for engineering applications.   

Classical many-body chaos across Kosterlitz-Thouless and Ising transitions in two dimensions

Chaos, the sensitivity to the initial condition, lies at the foundation of statistical mechanics. Chaotic systems are characterized by a growth rate, the Lyapunov exponent λ_L , and a velocity for ballistic spread, the butterfly velocity v_B , of local perturbation. Here we study the temperature dependence of the chaotic behavior across thermal phase transitions in a well-known classical spin system, the XXZ model on a square lattice. We tune the finite-temperature phase transition from the Kosterlitz-Thouless (KT) to Ising universality class by changing the anisotropy and find the temperature (T ) dependence of λ_L, v_B and the diffusion coefficient D across these transitions. For both the KT and Ising cases, we find a crossover in λ_L(T ) across the transitions. On the contrary, a naive extraction of v_B(T ) assuming a ballistic spread of perturbation leads to a non-monotonic temperature dependence of v_B across the transitions. In the KT case, we show that even though the systems show subdiffusive behaviors at intermediate times below transitions due to spin-waves, the spread of perturbation is actually superballistic due to algebric decay of spatial correlations in KT phase.   

New parameters for an alternative characterization of the nematic transition for rods deposition on 2D lattices

The problem of excluded volume deposition of rigid rods of length k unit cells over square lattices is revisited. Two new features are introduced: a) two new short-distance complementary order parameters (called Π and Σ) are defined, calculated and discussed to deal with the phases present as coverage increases; b) the interpretation is now done beginning at the high-coverage locally ordered phase (present regardless of the k value) which allows to interpret the low coverage nematic phase as an ergodicity breakdown present for k ≥ 7. In addition data analysis offers also some novelty as both mutability (dynamical information theory method) and Shannon entropy (static distribution analysis) will be invoked to further characterize the phases. Moreover, mutability and Shannon entropy are compared between themselves reporting their advantages and disadvantages for dealing with this problem. The advantages of parameter Π over any other parameter is also established.   

Deterministic Phase Transitions and Self-Organization in Logistic Cellular Automata

We present a simple extension in which a single parameter tunes the dynamics of Cellular Automata by consequently expanding their discrete state space into a Cantor Set. Such an implementation serves as a potent platform for further investigation of several emergent phenomena, including deterministic phase transitions, pattern formation, autocatalysis and self-organization. We first apply this approach to Conway’s Game of Life and observe sudden changes in asymptotic dynamics of the system accompanied by emergence of complex propagators. Incorporation of the new state space with system features is used to explain the transitions and formulate the tuning parameter range where the propagators adaptively survive by investigating their autocatalytic local interactions. Similar behavior is present when the same recipe is applied to Rule 90, an outer totalistic elementary one-dimensional CA. The latter case shows that deterministic transitions between classes of CA can be achieved by tuning a single parameter continuously.   

New type of oscillation death in coupled counter-rotating identical nonlinear oscillators

We study oscillatory and oscillation suppressed phases in coupled counter-rotating nonlinear oscillators. We demonstrate the existence of limit cycle, amplitude death, and oscillation death, and also clarify the Hopf, pitchfork, and infinite period bifurcations between them. Especially, the oscillation death is a new type of oscillation suppressions of which the inhomogeneous steady states are neutrally stable. We discuss the robust neutral stability of the oscillation death in non-conservative systems via the anti-PT-symmetric phase transitions at exceptional points in terms of non-Hermitian systems.   

Effects of the fluid flows on the stability of enzymatic chemical oscillations

Chemical oscillations are ubiquitous in nature and have a variety of promising applications. Chemical oscillations are usually analyzed within the context of a reaction-diffusion model framework. In real systems, the fluid flux modifies the chemical diffusion transport. To examine this effect, we consider a flow of chemical solution confined between two parallel walls forming a channel. The solution contains two reactants, A and B, which undergo transformations catalyzed by enzymes immobilized on the channel walls. Mutual transformations of the reactants provide a positive-negative feedback loop, which enables oscillations. We study the effect of the flow velocity on the stability boundary separating oscillating regimes from stationary distributions of chemicals in the channel. We show that the flow promotes the chemical transport in the system and, thereby, increases the amplitude and frequency of the oscillations. We support the predictions of the stability theory by the relevant numerical simulations. The findings can improve functionalities of micro-scale chemical reactors.   

Spatiotemporal dynamics of flame front instability described by an extended Kuramoto-Sivashinsky equation

We numerically study the spatiotemporal dynamics in an extended Kuramoto-Sivashinsky equation describing a freely propagating flame in a Hele-Shaw cell. The spatiotemporal dynamics of flame front undergoes a significant transition from low-dimensional to high-dimensional deterministic chaos with increasing the Rayleigh number. This is clearly identified by the permutation spectrum and the multiscale complexity-entropy causality plane. The Shannon entropy incorporating the probability of the degree in the interface network is useful for capturing the significant changes in randomness in the spatiotemporal dynamics.   

Dendritic crystal growth of ammonium nitrate and ammonium chloride

Dendritic crystal growth is an important example of nonequilibrium pattern formation that involves both nonlinear and noise-driven effects. The resulting large-scale structures are sensitively dependent on relatively small effects, such as surface tension, and on small anisotropies in those quantities. In this work, we present results for ammonium chloride dendrites, and compare them with new results for ammonium nitrate dendrites grown from supersaturated aqueous solution. This new system has been studied previously by van Driel et al.[1] and shown to exhibit several different morphologies, including both steady state dendritic growth and a state with persistent tip-splitting behavior. Specifically, we present new measurements of the tip radius ρ, growth speed v, and sidebranch spacing λ, along with initial estimates of the product Dd₀, where D is the chemical diffusion constant and d₀ is the capillary length, as well as the stability constant σ*=2d₀D/vρ². We discuss important similarities and differences between the two materials.

[1] Van Driel, C.A., Van der Heijden, A.E.D.M, Van Rosmalen, G.M., "Growth of Ammonium-Nitrate Phase-I and Phase-II Dendrites." J. Cryst. Growth 128 (1993) 229–233. https://doi.org/10.1016/0022-0248(93)90324-P   

Emergent Bose-Einstein statistics in classical non-equilibrium systems with scale selection

Non-equilibrium systems are ubiquitous in nature, from driven quantum matter and biological life forms to atmospheric and interstellar gases. Identifying and characterizing universal aspects of their dynamics continues to pose major conceptual challenges due to the absence of conserved quantities like energy. Here, we investigate the statistics of a broad class of non-equilibrium systems in which an intrinsic length-scale selection mechanism effectively constrains the dynamics of the microscopic degrees of freedom. As specific examples, we compare experimental observations of chaotic Faraday surface waves on water with simulations of a generalized Navier-Stokes equation for active fluids and quantum particle simulations in a random potential. Strikingly, we find that in all three cases the Fourier amplitudes of the energy density follow Bose-Einstein statistics. Furthermore, we show that the time evolution of these systems can be approximated by a sequence of randomly sampled monochromatic Gaussian fields, suggesting a unified view of non-equilibrium and equilibrium systems with length scale selection.   

Coarse-Graining and Renormalization without Locality

The renormalization group (RG) describes how to systematically coarse-grain physical models and calculate their scaling behavior near a critical point. This approach has been immensely successful for systems where interactions are local, with a high degree of symmetry that effectively reduces the number of parameters that can contribute to the behavior in the ultraviolet limit. Here, we introduce a scheme inspired by the recent work on coarse-graining neural dynamics [1], which is capable of detecting infra-red behavior directly from experimental data without explicit reference to locality or symmetry. Specifically, our approach selects maximally correlated pairs of system elements, taking the correlations themselves as a proxy for local interaction, and compresses their activity using the Information Bottleneck method, while preserving the information that the compressed variables contain about the next-closest scale. Repeatedly applying such transformations recovers the renormalization group flow for the coupling strengths and variation of nearest-neighbor correlations with length scale for data taken from a 2D Ising system on the square lattice, showing the viability of this data-driven approach.

[1] L. Meshulam, et al. Phys. Rev. Lett. 123, 178103 (2019).   

Statistics of coherent subsets in the Kuramoto model for uniform distributions

Experiments in synchronization and simulations of the Kuramoto model reveal that small subsets of oscillators tend to synchronize into coherent clumps at moderate coupling strengths below the critical coupling strength. Such clumps are not present in the thermodynamic limit, and so are intrinsically a finite size effect. We develop a theory that predicts the critical coupling for the formation of individual coherent subsets. Applying these new conditions to a finite population sampled from a uniform distribution, we obtain known results, such as a discontinuous transition to synchrony. We also obtain a lower estimate for the critical coupling of this transition that is consistent with simulation and previous results for uniformly sampled oscillators. This opens a new route for computing finite size effects, which we will present.   

Emergence of Complexity in Self-Limited Assemblies of Nanoparticles

Inorganic nanoparticles (NPs) have the ability to self-organize into variety of structures with sophisticated and dynamic geometries. Many of them are self-limited due to repulsive electrostatic interactions, which manifests in monodispersity of mesoscale superstructures formed from hundreds of polydispersed NPs. Remarkably, the organization of self-assembled NP systems can rival in complexity to those found in biology which reflects the biomimetic behavior of nanoscale inorganic matter. In this talk, I will address the mechanism and variety of self-limited assemblies of NPs and mechanisms of emergent complexity especially for chiral NPs. Self-limited assemblies of the latter
will illustrate the importance of subtle anisotropic effects stemming from collective behavior of NPs and non-additivity of their interactions. While these phenomena present obvious fundamental significance for instance as a path to spontaneous compartmentalization and protocells, the practicality of self-organization of nanoparticles will be discussed in relation to charge polarization-based optoelectronics, quantum optics, and chiral catalysis.
  

Topological Control of Synchronization Patterns: Trading Symmetry for Stability

Symmetries are ubiquitous in network systems and have profound impacts on the observable dynamics. At the most fundamental level, many synchronization patterns are induced by underlying network symmetry, and a high degree of symmetry is believed to enhance the stability of identical synchronization. Yet, here we show that the synchronizability of almost any symmetry cluster in a network of identical nodes can be enhanced precisely by breaking its structural symmetry. This counterintuitive effect holds for generic node dynamics and arbitrary network structure and is, moreover, robust against noise and imperfections typical of real systems, which we demonstrate by implementing a state-of-the-art optoelectronic experiment. These results lead to new possibilities for the topological control of synchronization patterns, which we substantiate by presenting an algorithm that optimizes the structure of individual clusters under various constraints   

Phase Synchronization in 2D Kuramoto Model

We study a system of identical Kuramoto oscillators in the presence of Gaussian white noise. The oscillators are arranged on a two-dimensional periodic lattice and interact with their nearest neighbors only [1-2]. In the thermodynamic limit, the stationary states at very low temperatures (limit T→0) are well described by the presence of topological defects (vortices and anti-vortices) in the phase-field of the oscillators. We apply duality transformation on a Hamiltonian [3] to explore the underlying vortex structure.
The system exhibits a phase transition from stochastic synchronization to de-synchronization as temperature varies. We investigate the nature of the phase transition and calculate the critical temperature numerically. The relaxation dynamics is also studied. In the low-temperature regime, the system relaxes to equilibrium algebraically whereas, at high temperature, the decay is exponential. Moreover, the system, at very low temperature belongs to the EW universality class yielding the dynamic exponent z=2 [4].

[1] Hong H, Park H and Choi M Y 2005 Phys. Rev. E 72 036217
[2] Lee T E, Tam H, Refael G, Rogers L and Cross M C 2010 Phys. Rev. E 82 036202
[3] Witthaut D and Timme M 2014 Phys. Rev. E 90 032917
[4] Forrest B M and Tang L-H 1990 J. Stat. Phys. 60 1-2