Session X25: Pattern Formation, Chaos, Nonlinear Dynamics III

Operator scrambling, hypersensitivity, and quantum Lyapunov spectrum

This work presents a model study of the quantum inverted harmonic oscillator (IHO), an archetype example for quantum chaotic systems. It is shown that the IHO exhibits hypersensitivity to perturbations, i.e., the Loschmidt echo decays as a double exponential. A quantum Lyapunov spectrum has been introduced to detect the exponential operator growth in chaotic systems. The IHO is demonstrated to possess a quantum Lyapunov spectrum that is identical to the classical one.   

Non-equilibrium Renormalization Group Fixed-Points of the Quantum Potts Chain

We devise a renormalization group recursion relation of the Loschmidt amplitude of the quantum Potts chain. The non-equilibrium fixed-points of the renormalization procedure determine the dynamical phases of the system, giving rising to a dynamical quantum phase transition. We will also discuss how there fixed-points affect the dynamical phase transition of the disordered Potts chain and the Loschmidt amplitude of the excited states.   

Laminar Chaos in a Mackey-Glass feedback circuit with Variable Time-Delay

Laminar chaos is a newly discovered chaotic behavior theoretically predicted in 2018 [1], which can be characterized by steady-state phases separated by short and irregular burst-like transitions. It has been observed experimentally in an optoelectronic variable time-delay feedback system implemented with a field-programmable gate array (FPGA) [2]. The Mackey-Glass model, developed to simulate the physiological mechanism of red blood cells, is a common time-delay system that can yield a wide range of periodic and chaotic dynamics. We report here the observation of laminar chaos in an electronic Mackey-Glass feedback circuit, the first observation of its kind, implemented with an Arduino board to produce variable time delays. This unique system implementation allows us to easily create audio signals from laminar chaos that contrast with signals generated by a system with a constant time delay.

[1] D. Müller, A. Otto, and G. Radons, Phys. Rev. Lett. 120, 084102 (2018)
[2] J. Hart, R. Roy, D. Müller, A. Otto, and G. Radons,123, Phys. Rev. Lett. 123, 154101 (2019)   

Theory of gating in recurrent neural networks

Understanding the emergent dynamics of networks of neurons is a central challenge in theoretical neuroscience. Most of the work in understanding the dynamics of these networks has focused on models with `additive interactions', where the input to a neuron is a weighted sum of the output of the rest of the network. However, there is ample evidence from neurophysiology that neurons can have gating or multiplicative interactions, where e.g. one neuron can dynamically decide whether another neuron is influenced by the rest of the network. Such gating interactions lead to qualitatively different behavior of single neurons, and are likely to have even more dramatic effects on the collective behavior of a network. Furthermore, researchers in machine learning have found that gating interactions facilitate training of model neural networks. Thus, gating can have significant implications for information processing. We leverage tools from the field theory of disordered systems to develop a theory of gating in a canonical neural network model. Our theory allows us to elucidate the dynamical aspects of gating which are important for the network's information processing capabilities.   

Short-term forecasting of hyperchaotic time series by noisy echo state network

We have applied a noisy echo state network, wherein pseudorandom numbers subject to uniform distribution are input to the reservoir nodes, to the short-term forecasting of a hyperchaotic time series generated by a star network of nonidentical Lorenz subsystems. The chaotic dynamics have five positive Lyapunov exponents with a Lyapunov dimension exceeding 12. Although the predictive model incurs a large prediction error, it is capable of reproducing the geometric structure of the hyperchaotic attractor with sufficient fidelity. We discuss these results in terms of Ueda’s view of chaos, wherein chaotic dynamical behavior is recognized as a piecewise deterministic process with intervening stochastic processes such as numerical round-off errors and perturbations caused by experimental measurements.   

Critical branching processes in digital memcomputing machines

Memcomputing is a novel computing paradigm that employs time non-locality (memory) to solve combinatorial optimization problems. It can be realized in practice by means of non-linear dynamical systems whose point attractors represent the solutions of the original problem. It has been previously shown that during the solution search digital memcomputing machines go through a transient phase of avalanches (instantons) that promote dynamical long-range order. By employing mean-field arguments we predict that the distribution of the avalanche sizes follows a Borel distribution typical of critical branching processes with exponent τ=3/2. We corroborate this analysis by solving various random 3-SAT instances of the Boolean satisfiability problem. The numerical results indicate a power-law distribution with exponent τ=1.51±0.02, in very good agreement with the mean-field analysis. This indicates that memcomputing machines self-tune to a critical state in which avalanches are characterized by a branching process, and that this state persists across the majority of their evolution. [1]

[1] S.R.B. Bearden, et al, 2019, EPL 127, 30005   

Connecting Dynamics and Trainability in Recurrent Neural Networks

Recurrent neural networks (RNNs) are well-suited for complex sequential learning tasks, but are notoriously difficult to train due to the problem of exploding or vanishing gradients (EVG) of the cost function. Local switch-like multiplicative “gates” were introduced to address this issue by modulating inter-neuron interactions and selectively updating the state of the network, promoting longer time scales. As such, these "gated" RNNs seem to mitigate the EVG, making training tractable. However, the specific role of each gate type on dynamics and training remains unclear. We take a dynamical systems perspective to study these questions for two popular gated RNN architectures: the Gated Recurrent Unit (GRU) and Long Short-Term Memory (LSTM). Using random matrix theory, we elucidate how gating enriches the repertoire of dynamical behavior expressed by these networks. Our approach furthermore sheds light on how gating is able to overcome the EVG by shaping asymptotic stability. Finally, we connect the intrinsic dynamics upon random parameter initialization to the subsequent ease of training GRUs and LSTMs.   

Misalignment-induced frequency locking in lasers

We study arrays of heterogeneous single-mode semiconductor lasers coupled through an external cavity with facet misalignments. This system can be modeled mathematically as a set of coupled nonlinear delay-differential equations. The heterogeneity, in the form of frequency detuning, is represented as parametric disorder between the oscillators, and the facet misalignments is represented by a small disordering of the time-delays in the coupling terms. In this system, we show that the introduction of time-delay disorder induces perfect frequency locking in a system that otherwise is unable to frequency-lock due to heterogeneous natural frequencies, as well as chaotic behavior.
  

Self-beating

There are two kinds of nonlinear mechanism. Beat is typical for the kind involving two or more frequencies. The other kind involves only one frequency due to large wave amplitude or weak media. Considering only linear materials and processes, we discover a new mechanism of self-beating to produce a wave function that does not appear to be linear. During a pulsed light being transmitted through a plasmonic slit, a portion of the light pulse transmits as a sub-pulse, while the rest is reflected at the exit interface, propagates a round-trip, and then reaches the exit again. These linear processes repeat. The superposition of transmitted sub-pulses with a phase delay in-between produces a periodic light that beats its original light frequency. Together with the plasma effects of non-uniform dispersion and sub-pulse spreading, self-beating can explain intrigue phenomena observed. The analytical model explains the complicated simulation results.   

Chaotic Dynamics Enhance the Sensitivity of Inner Ear Hair Cells

Hair cells of the auditory and vestibular systems can detect sounds that induce sub-nanometer vibrations of the hair bundle, below the stochastic noise levels of the surrounding fluid. Hair cells of certain species are known to oscillate without external stimulation. These spontaneous oscillations are believed to be a manifestation of an underlying active mechanism and may play a role in signal detection. We previously demonstrated experimentally that the spontaneous oscillations exhibit chaotic dynamics. We propose that the instabilities giving rise to chaotic dynamics are responsible for the extreme sensitivity of hair cells. We will present experimental measurements of spontaneous and driven hair bundle oscillations. By varying the conditions of the surrounding fluid, we were able to modulate the degree of chaos observed in the hair cell dynamics. We found that the hair bundle is most sensitive to small-amplitude stimulus when it is poised in the weakly chaotic regime. Further, we found that the response time to a force step decreases as the level of chaos is increased. These results agree well with our numerical simulations of a chaotic Hopf oscillator and suggest that chaos may be responsible for the extreme sensitivity and temporal resolution of hair cells.   

Nonlinear Coherent Perfect Absorbers

Coherent perfect absorber (CPA) is a resonant mechanism in which a cavity with a minimal amount of losses can completely absorb incident radiation with an appropriately tailored waveform. The cavity acts as an interferometric trap that confines the incident light for long times and thus amplifies the net absorption. Although in current studies, it was assumed that the medium inside the cavity is linear, there are realistic scenarios where nonlinear mechanisms are engaged.

Here we develop the theoretical framework for the design of CPA waveforms under nonlinear conditions. As an example, we use a photonic circuit consisting of an array of coupled resonators with a nonlinear target embedded in the array. A general solution for the design of the incoming waveform is proposed, and a plethora of CPA frequencies that depend on the intensity of the incident radiation is found–a feature that was not present in linear CPAs. Furthermore, we have demonstrated that under appropriate conditions one can realize a CPA degeneracy, which resembles an exceptional point singularity.   

Recent advences on non-normalizable Boltzmann-Gibbs statistics and infinite-ergodic theory

The equilibrium state of a thermal system, in the presence of a strongly confining potential, is given by the famous Boltzmann-Gibbs distribution. This, along with the ergodic hypothesis, are hallmarks of statistical physics. If the potential is weakly confining, the Boltzmann factor is non-normalizable and the particle packet is constantly expanding. This gives rise to many questions. Among them: can we still infer the shape of the potential landscape, by observing the spatial distribution of the diffusing particles? How do we obtain ensemble and time-averaged observables in this case? And what is the entropy-energy relation in this system?

We show that the non-normalizable Boltzmann state is obtained from an entropy maximization principle, in the spirit of the canonical ensemble from standard thermodynamics, as well as from the ground state of the effective Hamiltonian of the system. The ergodic properties of important physical observables, like energy and occupation times in the system, are given by the Aaronson-Darlin-Kac theorem, and a generalized virial theorem is derived. Thus, our work extends the standard thermodynamic theory to a new class of potentials, and provides further strong evidence to the physical significance of infinite-ergodic theory.   

Experimental Observation of a Non-Normalizable Boltzmann State

The probability density function of a particle in equilibrium suspended in a fluid follows a Boltzmann distribution, P(x)=exp(-V(x)/k_BT)/Z, where V is the potential energy, k_BT is thermal energy, and Z is the normalizing partition function. However, there are cases where the Boltzmann distribution is not normalizable and the system cannot reach thermodynamic equilibrium. It has recently been shown theoretically and via simulations that, in many instances, a particle coupled to a heat bath approaches in the long time a non-normalizable Boltzmann state, such that P_t(x)=exp(-V(x)/k_BT-x²/4Dt)/Z_t, where D is the diffusion coefficient and Z_t~t^-α [1]. These systems violate the ergodic hypothesis, i.e. the time average of an observable does not converge to the ensemble average in the long time limit. Here, we test these predictions using functionalized polystyrene beads near a charged surface in liquid. The electrostatic potential V(x)~exp(-x/λ_D), with Debye length λ_D, yields a non-normalizable Boltzmann distribution. We track charged particles in 3D near surfaces bearing different surface charge densities. Our experiments reveal a non-normalizable Boltzmann state and ergodicity breaking in agreement with theoretical predictions.
[1] E. Aghion et al., Phys. Rev. Lett. 122, 010601 (2019)   

Experimental Search for Supradegeneracy

Recently a new statistical mechanical phenomenon has been
formally identified that turns some thermodynamic intuitions on their
heads: supradegeneracy [1]. Normally, population inversion (as in
lasers) requires nonequilibrium conditions, but supradegenerate
systems are predicted to exhibit inversion at equilibrium. If
realized, this phenomenon might help lead to improved photovoltaics and
to new types of thermophotovoltaics and batteries [2]. Although
supradegenerate systems do not appear in Nature, it seems likely that
manmade ones can be realized. This talk reviews the current laboratory
search for supradegeneracy in condensed matter systems, specifically, in
semiconductors and concentration-graded ionomer membranes.
[1] D.P. Sheehan and L.S. Schulman, Physica A 524, 100 (2019).
[2] D.P. Sheehan, Sustain. Energy. Tech. Assess., in press (2019).   

Lacunarity exponents in chaotic systems

Many physical processes result in very uneven, apparently random, distributions of matter, characterised by fluctuations of the local density varying over orders of magnitude. Examples include the distribution of stars within galaxies, distribution of debris floating on fluids, distribution of human population, and the distribution of small inertial particles in a turbulent flow. Many of these systems can be described by chaotic dynamical systems and the existence of a power-law describing the high-density regions is consistent with the notion that chaotic systems can have fractal invariant measures. However, the distribution of density in the sparse regions can also have a power-law distribution, with an exponent that we refer to as lacunarity exponent, and which is not related to the fractal properties of the system. Here, we discuss a robust mechanism that explains the wide occurrence of these power laws and gives analytical expressions for the lacunarity exponent in some cases of interest, including simple chaotic models and the problem of particles advected by fluid flow.