Active Brownian Particles. From Individual to Collective Stochastic Dynamics

We review theoretical models of individual motility as well as collective dynamics and pattern formation of active particles. We focus on simple models of active dynamics with a particular emphasis on nonlinear and stochastic dynamics of such self-propelled entities in the framework of statistical mechanics. Examples of such active units in complex physico-chemical and biological systems are chemically powered nano-rods, localized patterns in reaction-diffusion system, motile cells or macroscopic animals. Based on the description of individual motion of point-like active particles by stochastic differential equations, we discuss different velocity-dependent friction functions, the impact of various types of fluctuations and calculate characteristic observables such as stationary velocity distributions or diffusion coefficients. Finally, we consider not only the free and confined individual active dynamics but also different types of interaction between active particles. The resulting collective dynamical behavior of large assemblies and aggregates of active units is discussed and an overview over some recent results on spatiotemporal pattern formation in such systems is given.Comment: 161 pages, Review, Eur Phys J Special-Topics, accepte

Anticipating Critical Transitions with Nonlinearity, Periodicity and Heterogeneity

Many natural and engineering systems may switch abruptly from one stable state to another due to a small perturbation to the system's state or a small change in the underlining conditions. In ecosystems, for example, extinctions of species or desertification can occur rapidly. Therefore, critical transitions can be dangerous to a number of systems, and it could be very beneficial if monitoring or early warning methods were available while the system is still in the healthy regime. The approach of critical transitions in many natural and engineering systems is accompanied by a phenomenon called critical slowing down. Theoretical and experimental studies have suggested that responses to small perturbations become increasingly slow when these systems are near critical transitions. Statistics such as variance, autocorrelation calculated from time series data have been proposed as early warning signals to anticipate the system's approach to a transition point. The problem of anticipating critical transitions becomes more complicated when other factors come into play. Factors such as nonlinearity, periodicity and heterogeneity can alter the behavior of the system, and thus affect the applicability of generic early warning signals. This thesis examines the effect of these factors on the critical transition of a system, and develops new data-driven approaches accordingly. To deal with and exploit the existence of nonlinearity in the system, recoveries from large instead of small perturbations are used to calculate the recovery rates of the system versus amplitudes. Under the circumstances of periodicity, recovery rates are calculated discretely via the Poincare section. Using experimental and computational data, we show that a combination of using recoveries from large perturbations and calculating recovery rates using the Poincare section can be highly effective in terms of anticipating critical transitions for systems with parametric resonance. Moreover, this thesis develops new early warning signals for spatially extended systems based on the eigenvalues of the covariance matrix. We mathematically show that the dominance of the largest eigenvalue of the covariance matrix can be used as an early warning signal by establishing the relationship between the eigenvalues of the covariance matrix and the eigenvalues of the force matrix. This new set of early warning signals are especially useful when the system has strong spatial heterogeneity. Lastly, this thesis investigates the influence of the choice of hyper-parameters, such as moving window size, sample rate, detrending methods, on the robustness of several early warning signals. General rules regarding data preparation and hypothesis testing are proposed.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/145907/1/shychen_1.pd

Regenerative memory in time-delayed neuromorphic photonic resonators

We investigate a photonic regenerative memory based upon a neuromorphic oscillator with a delayed self-feedback (autaptic) connection. We disclose the existence of a unique temporal response characteristic of localized structures enabling an ideal support for bits in an optical buffer memory for storage and reshaping of data information. We link our experimental implementation, based upon a nanoscale nonlinear resonant tunneling diode driving a laser, to the paradigm of neuronal activity, the FitzHugh-Nagumo model with delayed feedback. This proof-of-concept photonic regenerative memory might constitute a building block for a new class of neuron-inspired photonic memories that can handle high bit-rate optical signals

Keldysh Field Theory for Driven Open Quantum Systems

Recent experimental developments in diverse areas - ranging from cold atomic gases over light-driven semiconductors to microcavity arrays - move systems into the focus, which are located on the interface of quantum optics, many-body physics and statistical mechanics. They share in common that coherent and driven-dissipative quantum dynamics occur on an equal footing, creating genuine non-equilibrium scenarios without immediate counterpart in condensed matter. This concerns both their non-thermal flux equilibrium states, as well as their many-body time evolution. It is a challenge to theory to identify novel instances of universal emergent macroscopic phenomena, which are tied unambiguously and in an observable way to the microscopic drive conditions. In this review, we discuss some recent results in this direction. Moreover, we provide a systematic introduction to the open system Keldysh functional integral approach, which is the proper technical tool to accomplish a merger of quantum optics and many-body physics, and leverages the power of modern quantum field theory to driven open quantum systems.Comment: 73 pages, 13 figure

The Bar--Halo Interaction--I. From Fundamental Dynamics to Revised N-body Requirements

Only through resonances can non-axisymmetric features such as spiral arms and bars exert torques over large scales and change the overall structure of a near-equilibrium galaxy. We describe the resonant interaction mechanism in detail and derive explicit criteria for the particle number required to simulate these dynamical processes accurately using N-body simulations and illustrate them with numerical experiments. To do this, we perform direct numerical solution of perturbation theory and make detailed comparisons with N-body simulations. The criteria include: sufficient particle coverage in phase space near the resonance and enough particles to minimize gravitational potential fluctuations that will change the dynamics of the resonant encounter. Some of our more surprising findings are as follows. First, the Inner-Lindblad-like resonance (ILR), responsible for coupling the bar to the central halo cusp, requires almost 10^9 equal mass particles within the virial radius for a Milky-Way-like bar in an NFW profile. Second, orbits that linger near the resonance receive more angular momentum than orbits that move through the resonance quickly. Small-scale fluctuations present in state-of-the-art particle-particle simulations can knock orbits out of resonance, preventing them from lingering and, thereby, decrease the torque. The required particle numbers are sufficiently high for scenarios of interest that apparent convergence in particle number is misleading: the convergence is in the noise-dominated regime. State-of-the-art simulations are not adequate to follow all aspects of secular evolution driven by the bar-halo interaction. We present a procedure to test the requirements for individual N-body codes for the actual problem of interest. [abridged]Comment: 30 pages, 19 figures, submitted to Monthly Notices. For paper with figures at full resolution: http://www.astro.umass.edu/~weinberg/weinberg_katz_1.ps.g