Session W25: Pattern Formation, Chaos, Nonlinear Dynamics II

Concentration-measure theory of waves: new perspectives of the fundamentals of nonequilibrium statistical physics and mesoscopic physics

The foundations of statistical physics are currently under renovations. A number of scenarios for the arising of statistical physics in an individual system has been proposed. However, most studies have focused on equilibrium statistical physics. In this talk, I shall report our recent progress on the fundamentals of nonequilibrium statistical physics. I will show a theory for structures and fluctuations of waves in individual disordered media, using the mathematical tool of the concentration of measure. Applying this theory, we can see how the elementary nonequilibrium process of diffusion can arise from a single scattering state of wave equation, without the canonical hypothesis of ensembles. Furthermore, fluctuations associated with the change in incoming waves exhibit a new kind of universalities, which does not exist in conventional mesoscopic fluctuations associated with the change in disorder realizations, and originate from the coherence between the natural channels of waves – the transmission eigenchannels.
  

Coherent Perfect Absorption in Chaotic Microwave Graphs

Coherent perfect absorption (CPA) has been of great interest to the physics community, in part because it represents the time-reversed version of a laser. The CPA process works by taking waves of particular amplitude and phase (coherent illumination) from multiple input channels and causing them to interfere and to be completely absorbed by losses in the system. Its implementation promises the realization of a novel family of extremely efficient absorbers, tunable and highly-selective notch filters, and high-efficiency energy conversion systems. We experimentally demonstrate the concept of coherent perfect absorption in a microwave graph constructed from coaxial cables connected by Tee-junctions. By adding a simple variable lossy attenuator into the system, we can effectively identify the CPA frequencies as the complex zeros of the scattering matrix which crosses the real axis and achieve perfect absorption in this chaotic setup. Most importantly, our experimental set-up allows us to demonstrate that the concept of CPA can be extended beyond the case where time-reversal (TR) symmetry holds, by introducing a circulator into the microwave graph.   

Tailored Noise Correlations and Generalized Levy Wave Dynamics in Multimode Systems

Multi-mode systems, like cavities, fibers etc, often suffer from the presence of environmental noise which causes mode mixing and subsequent interferences between various modes. In many occasions the study of the exact wave dynamic is a formidable task due to the many degrees of freedom that have to be taken into account in the equations of motion that describe such systems. Instead, a statistical theory of wave propagation might be a best way to describe the transport in such framework. We have utilized a Random Matrix Theory modeling which allow us to study the spreading of an initial mode excitation in the mode-space due to the environmental noise. Using this method we have developed a systematic approach that enforces a generalized Levy-type wave dynamics with a power law that it is imposed from the noise correlations. Our theoretical predictions have been tested in realistic circumstances like in paraxial light propagation in multimode fibers with tailored quenched disorder or in quantum dots with time-modulated boundaries.   

: Efficient Hybrid Model of Field and Energy Flow in Interconnected Wave Chaotic Systems

Predicting energy flow through interconnected complex billiards is of keen interest to many fields. The Random Coupling Model (RCM) has been successfully applied to predicting the electromagnetic (EM) field statistical properties of various wave chaotic systems. Recent studies extend RCM to networks of coupled systems with multiple connecting channels [arXiv:1909.03827]. However the model becomes computationally costly as more billiards are added to the network. The Power Balance Model (PWB) can produce fast predictions for the averaged power density of waves in electrically-large systems. However the fluctuations of the wave field are lost in PWB, and many other mean-field approaches. Here we combine the best aspects of each model to create a hybrid treatment and study the EM fields in arrays of coupled complex systems. The proposed hybrid approach provides both mean and fluctuation information of the EM fields without the full computational demand of RCM. We compare the hybrid model predictions with experiments on linear cascades of overmoded cavities of various degrees of loss and find good agreement. The range of validity and applicability of the hybrid method is also discussed.   

Breakdown of the Metastable State in the β Fermi-Pasta-Ulam-Tsingou Lattice

We investigate numerically the formation and disappearance of Fermi-Pasta-Ulam-Tsingou (FPUT) recurrences in the β FPUT lattice, with special consideration to the lifespan of the metastable state. First, we consider different initial energies (E) and system sizes (N), to determine which initial parameters cause FPUT recurrences to form. We find that for large N, recurrences cease to form above a critical Eβ, a product of the energy and nonlinear parameter β. We explore the relationship between their disappearance and the breakdown of the metastable state to explain this critical Eβ, and also to estimate the time scale to the metastable state breakdown. In so doing, we define and measure a quantity called “shareable energy” and use this to show that systems with a negative β have a much lower level of energy sharing than those with positive β.   

The β Fermi-Pasta-Ulam-Tsingou Recurrence Problem

One of the most remarkable and longest-studied problems in nonlinear dynamics is Fermi-Pasta-Ulam-Tsingou (FPUT) recurrences. We perform a thorough investigation of the first FPUT recurrence in the β-FPUT chain for both β > 0 and β < 0. We show numerically that the rescaled FPUT recurrence time T_r =t_r(N + 1)^-3 depends, for large N, only on the parameter S ≡ Eβ(N + 1). Our numerics also reveal that for large |S|, T_r is proportional to |S|^-1/2 for both β > 0 and β < 0 but with different multiplicative constants. We numerically study the continuum limit and find the recurrence time closely follows the |S|^-1/2 scaling and can be interpreted in terms of mKdV solitons, and the difference in the multiplicative factors between positive and negative β arises from soliton-kink interactions. We complement our numerical results with analytical considerations in the nearly linear regime (small |S|) and the highly nonlinear regime (large |S|). For the former, we extend previous results using a shifted-frequency perturbation theory and find a closed form for T_r, which depends only on S. In the latter regime, we show that the soliton theory correctly predicts T_r ~ |S|^-1/2 in the continuum limit.   

Statistical model and universality class for interacting puffs in transitional turbulence

To understand the universality class of the laminar-turbulent transition in pipe flow, we measure how turbulent regions known as puffs proliferate and interact. Because of the huge length and time scales required to reach the critical regime, we model the dynamics from the effective forces we measure in experiments. Away from the transition, we find a crystal-like pattern with spatio-temporal intermittency. Renormalization group analysis and numerical simulations of both discrete and continuous models of puff interactions show that interactions are irrelevant at the 1+1-dimensional directed percolation fixed point but several finite-size artifacts mask the critical scaling regime in practice.   

A reduced model for a single cardiac cell

The leading cause of sudden cardiac death is due to ventricular fibrillation (VF). VF is a heart rhythm disease that occurs from irregular dynamics behavior-discordant alternas. Therefore, it is vital to investigate the dynamics of a cardiac cell. In the past, typical mathematical models for dynamics of a cardiac cell involve intricate interplay between the membrane potential, ion channels, calcium cycling, etc, which would reproduce realistic responses such as cardiac alternas. Although above-mentioned models can reproduce genuine dynamics of a cardiac cell, they are generally complicated to analyze due to their high dimensional phase space. Hence, we propose a reduced model derived from an existing cardiac ionic model, and show that this three-variable dynamical system exhibits similar bifurcation diagram as that of the ionic model. The dynamical response and bifurcation behavior of a cardiac cell are investigated with the proposed reduced model.   

Diffusive behavior in walking droplets

Fluid droplets walking on a vibrating fluid bath have been observed to display deterministic diffusion. We present an experimental and theoretical investigation of such droplets. In our experiments a droplet is placed into an annular region on a vibrating fluid bath. The droplet motion becomes increasingly diffusive as the bath vibration is intensified above the Faraday wave threshold. This is also captured in our hydrodynamic – kinematic models, which shows close agreement between theory and experiments. Since the model can be studied at a much higher spatio-temporal resolutions than experiments, we use the model to numerically investigate bifurcations and chaotic dynamics suggested by experiments. Finally, we briefly discuss the possibility of model reduction to mitigate computational costs.   

Sloppy model analysis of dynamical systems near bifurcations

In dynamical systems, bifurcations refer to topological changes in phase space trajectories in response to relatively small changes in parameter values. They are useful for identifying tipping points between qualitatively different types of behaviors such a phase transitions or from regulated to cancerous cellular pathways. Bifurcations are classified by the nature of the topological change in phase space with classes being typified by a "normal form", indicating that only a few parameters are responsible for driving a system through a bifurcation. Sloppy models provide a framework for identifying relevant parameters in a data-driven way. We apply sloppy model analysis to several dynamical systems near their bifurcations. We show that after an appropriate coarse-graining procedure, sloppy model analysis is able to correctly identify the bifurcation parameters. This suggests that sloppy model analysis can be used to identify the relevant control knobs in multi-parameter models of complex dynamical systems in a data-driven way.   

A mathematical approach to the effects of gender bias and cross gender interactions on careers in STEM.

Creating a mathematical model to predict participation in STEM fields would allow communities such as the APS understand better how to influence participation in physics. We propose a model that simulates the progression of women in male-dominated fields starting from the equations for a model predator-prey type system. We include interactions between the dominant and non-dominant populations and investigate how they effect the percent of the non-dominant population. We believe that such a model effectively capters some of the dynamics erported by researchers directly collecting data about the experiences of women in these fields.   

Hysteresis Experiments in Coupled Oscillators near Daido's Aging Transition

One way to experimentally characterize collective oscillations is to measure the magnitude of the steady-state response. In order to examine more dynamic aspects of the behavior, we have measured the hysteresis of the collective response as we vary the collective bifurcation value. Specifically, in our collection of electronic oscillators, we produce a hysteresis by sequentially lowering then raising the bifurcation setting of each oscillator such that the collective bifurcation parameter goes below and above is critical value. The shape of the hysteresis loop exhibits a strong dependence on the rate at which we drive the system through the transition. We find that the area of the hysteresis loop exhibits a square-root dependence on the driving frequency. This is in contrast to feromagnetic and feroelectric systems, which exhibit power-law frequency dependence with much weaker exponents.   

Conformal field theory and the web of quantum chaos diagnostics

We study three prominent diagnostics of chaos and scrambling in the context of two-dimensional conformal field theory: the spectral form factor, out-of-time-ordered correlators, and unitary operator entanglement. With the observation that all three quantities may be obtained by different
analytic continuations of the torus partition function, we address the connections and distinctions between the information that each quantity provides us. In this process, we study the emergence of irrationality from “large-N” limits of rational conformal field theories (RCFTs) as well as the explicit breakdown of rationality for theories with central charges greater than the number of their conserved currents. Our analysis begins to elucidate the intermediate dynamical behavior of theories that bridge the gap between integrable RCFTs and maximally chaotic holographic CFTs.   

Cryptographic analysis of chaotic systems

In computer science, hash functions are elementary operators that convert arbitrary-length inputs into finite-length outputs. We describe a direct analogy between these functions and the trajectories of particles advected by fluid flows, and we show that, when the governing flow is chaotic, hydrodynamic hash functions exhibit statistical properties typically associated with hash functions used for digital cryptography. These include non-invertibility, sensitivity to initial input, and avoidance of collisions—a phenomenon in which two similar inputs produce identical outputs. We show that this analogy originates from the tendency of certain chaotic flows to braid together particle trajectories across space in time in an irreducible manner, and we describe how techniques used to probe the properties of digital hash functions can be used to characterize the properties of flows when only limited observational data is available. Our findings have potential applications in microfluidic proof-of-work systems, as well as for characterizing large-scale transport in ocean flows and living systems.   

Universality in Kinetic Models of Circadian Rhythms in Arabidopsis thaliana

Adapting to the 24-hour periodic environment on the Earth, plants have evolved sets of chemical reactions that regulate their circadian rhythms. Over the past fifteen years, researchers studying these circadian reactions in the common laboratory plant Arabidopsis thaliana have developed eleven, increasingly elaborate, chemical kinetic models based on genetic feedback loops. Each model consists of a system of coupled nonlinear ordinary differential equations. We find these models are all situated near a Hopf bifurcation in parameter space. This suggests that there may be some biological significance corresponding to this mathematical property.
To illustrate the special nature of these systems, we numerically compute the solutions to the kinetic models for Arabidopsis thaliana. Separately, we perform a weakly nonlinear analysis on each model near bifurcation to predict the amplitude and frequency of the oscillating concentration of chemical species from the Stuart-Landau amplitude equation. By scaling the numerical frequencies and amplitudes by our theoretical predictions, we show that the solutions to all these models collapse into a universal parameter-free form. We further comment on some implications of our results for improving future modeling efforts.