Synchronization of coupled noisy oscillators: Coarse-graining from continuous to discrete phases

The theoretical description of synchronization phenomena often relies on coupled units of continuous time noisy Markov chains with a small number of states in each unit. It is frequently assumed, either explicitly or implicitly, that coupled discrete-state noisy Markov units can be used to model mathematically more complex coupled noisy continuous phase oscillators. In this work we explore conditions that justify this assumption by coarse-graining continuous phase units. In particular, we determine the minimum number of states necessary to justify this correspondence for Kuramoto-like oscillators

Velocity Distribution in a Viscous Granular Gas

We investigate the velocity relaxation of a viscous one-dimensional granular gas, that is, one in which neither energy nor momentum is conserved in a collision. Of interest is the distribution of velocities in the gas as it cools, and the time dependence of the relaxation behavior. A Boltzmann equation of instantaneous binary collisions leads to a two-peaked distribution with each peak relaxing to zero velocity as 1/t while each peak also narrows as 1/t. Numerical simulations of grains on a line also lead to a double-peaked distribution that narrows as 1/t. A Maxwell approximation leads to a single-peaked distribution about zero velocity with power-law wings. This distribution narrows exponentially. In either case, the relaxing distribution is not of Maxwell-Boltzmann form

Synchronization of globally coupled two-state stochastic oscillators with a state dependent refractory period

We present a model of identical coupled two-state stochastic units each of which in isolation is governed by a fixed refractory period. The nonlinear coupling between units directly affects the refractory period, which now depends on the global state of the system and can therefore itself become time dependent. At weak coupling the array settles into a quiescent stationary state. Increasing coupling strength leads to a saddle node bifurcation, beyond which the quiescent state coexists with a stable limit cycle of nonlinear coherent oscillations. We explicitly determine the critical coupling constant for this transition

A continuous-time persistent random walk model for flocking

Random walkers characterized by random positions and random velocities lead to normal diffusion. A random walk was originally proposed by Einstein to model Brownian motion and to demonstrate the existence of atoms and molecules. Such a walker represents an inanimate particle driven by environmental fluctuations. On the other hand, there are many examples of so-called "persistent random walkers", including self-propelled particles that are able to move with almost constant speed while randomly changing their direction of motion. Examples include living entities (ranging from flagellated unicellular organisms to complex animals such as birds and fish), as well as synthetic materials. Here we discuss such persistent non-interacting random walkers as a model for active particles. We also present a model that includes interactions among particles, leading to a transition to flocking, that is, to a net flux where the majority of the particles move in the same direction. Moreover, the model exhibits secondary transitions that lead to clustering and more complex spatially structured states of flocking. We analyze all these transitions in terms of bifurcations using a number of mean field strategies (all to all interaction and advection-reaction equations for the spatially structured states), and compare these results with direct numerical simulations of ensembles of these interacting active particles

Finite group actions on reductive groups and buildings and tamely-ramified descent in Bruhat-Tits theory

The purpose of the paper is to give a new approach to tamely-ramified descent in Bruhat-Tits theory. This descent was first studied by Guy Rousseau in his thesis.Comment: 28 pages. arXiv admin note: text overlap with arXiv:1611.0743

Proposal for Quantum Simulation via All-Optically Generated Tensor Network States

We devise an all-optical scheme for the generation of entangled multimode photonic states encoded in temporal modes of light. The scheme employs a nonlinear down-conversion process in an optical loop to generate one- and higher-dimensional tensor network states of light. We illustrate the principle with the generation of two different classes of entangled tensor network states and report on a variational algorithm to simulate the ground-state physics of many-body systems. We demonstrate that state-of-the-art optical devices are capable of determining the ground-state properties of the spin-1/2 Heisenberg model. Finally, implementations of the scheme are demonstrated to be robust against realistic losses and mode mismatch.Comment: 6 pages main text plus 6 pages Supplementary Material and many figures. Updated to published version. Comments welcom

Self-Similarity in Random Collision Processes

Kinetics of collision processes with linear mixing rules are investigated analytically. The velocity distribution becomes self-similar in the long time limit and the similarity functions have algebraic or stretched exponential tails. The characteristic exponents are roots of transcendental equations and vary continuously with the mixing parameters. In the presence of conservation laws, the velocity distributions become universal.Comment: 4 pages, 4 figure

Species-conserved mechanisms of cognitive flexibility in complex environments

Flexible decision making in complex environments is a hallmark of intelligent behavior but the underlying learning mechanisms and neural computations remain elusive. Through a combination of behavioral, computational and electrophysiological analysis of a novel multidimensional rule-learning paradigm, we show that both rats and humans sequentially probe different behavioral strategies to infer the task rule, rather than learning all possible mappings between environmental cues and actions as current theoretical formulations suppose. This species-conserved process reduces task dimensionality and explains both observed sudden behavioral transitions and positive transfer effects. Behavioral strategies are represented by rat prefrontal activity and strategy-related variables can be decoded from magnetoencephalography signals in human prefrontal cortex. These mechanistic findings provide a foundation for the translational investigation of impaired cognitive flexibility.One-Sentence SummaryBoth rats and humans use behavioral strategies to infer task rules during multidimensional rule-learning