We consider the problem of distinguishing between different rates of percolation under noise. A statistical model of percolation is constructed allowing for the birth and death of edges as well as the presence of noise in the observations. This graph-valued stochastic process is composed of a latent and an observed nonstationary process, where the observed graph process is corrupted by type-I and type-II errors. This produces a hidden Markov graph model. We show that for certain choices of parameters controlling the noise, the classical (Erdős-Rényi) percolation is visually indistinguishable from a more rapid form of percolation. In this setting, we compare two different criteria for discriminating between these two percolation models, based on the interquartile range (IQR) of the first component's size, and on the maximal size of the second-largest component. We show through data simulations that this second criterion outperforms the IQR of the first component's size, in terms of discriminatory power. The maximal size of the second component therefore provides a useful statistic for distinguishing between different rates of percolation, under physically motivated conditions for the birth and death of edges, and under noise. The potential application of the proposed criteria for the detection of clinically relevant percolation in the context of applied neuroscience is also discussed.

The interaction of three different sized (diameter 10, 18, and 28 nm) anionic silica nanoparticles with two model proteins—cationic lysozyme [molecular weight (MW) 14.7 kDa)] and anionic bovine serum albumin (BSA) (MW 66.4 kDa) has been studied by UV-vis spectroscopy, dynamic light scattering (DLS), and small-angle neutron scattering (SANS). The adsorption behavior of proteins on the nanoparticles, measured by UV-vis spectroscopy, is found to be very different for lysozyme and BSA. Lysozyme adsorbs strongly on the nanoparticles and shows exponential behavior as a function of lysozyme concentration irrespective of the nanoparticle size. The total amount of adsorbed lysozyme, as governed by the surface-to-volume ratio, increases on lowering the size of the nanoparticles for a fixed volume fraction of the nanoparticles. On the other hand, BSA does not show any adsorption for all the different sizes of the nanoparticles. Despite having different interactions, both proteins induce similar phase behavior where the nanoparticle-protein system transforms from one phase (clear) to two phase (turbid) as a function of protein concentration. The phase behavior is modified towards the lower concentrations for both proteins with increasing the nanoparticle size. DLS suggests that the phase behavior arises as a result of the nanoparticles’ aggregation on the addition of proteins. The size-dependent modifications in the interaction potential, responsible for the phase behavior, have been determined by SANS data as modeled using the two-Yukawa potential accounting for the repulsive and attractive interactions in the systems. The protein-induced interaction between the nanoparticles is found to be short-range attraction for lysozyme and long-range attraction for BSA. The magnitude of attractive interaction irrespective of protein type is enhanced with increase in the size of the nanoparticles. The total (attractive+repulsive) potential leading to two-phase formation is found to be more attractive for larger sized nanoparticles. The nanoparticle aggregates are characterized by mass fractal.

Crack events developed during uniaxial compression of cortical bones cut from femurs of developing pigs of several ages (4, 12, and 20 weeks) generate avalanches. These avalanches have been investigated by acoustic emission analysis techniques. The avalanche energies are power-law distributed over more than four decades. Such behavior indicates the absence of characteristic scales and suggests avalanche criticality. The statistical distributions of energies and waiting times depend on the pig age and indicate that bones become stronger, but less ductile, with increasing age. Crack propagation is equally age-dependent. Older pigs show, on average, larger cracks with a time distribution similar to those of aftershocks in earthquakes, while younger pigs show only statistically independent failure events.

As a minimal mathematical model generating cascade analogous to that of the Navier-Stokes turbulence in the inertial range, we propose a one-dimensional partial-differential-equation model that conserves the integral of the squared vorticity analog (enstrophy) in the inviscid case. With a large-scale random forcing and small viscosity, we find numerically that the model exhibits the enstrophy cascade, the broad energy spectrum with a sizable correction to the dimensional-analysis prediction, peculiar intermittency, and self-similarity in the dynamical system structure.

This study presents the confinement influences of Aharonov-Bohm (AB) flux and electric and magnetic fields directed along the $z$ axis and encircled by quantum plasmas on the hydrogen atom. The all-inclusive effects result in a strongly attractive system while the localizations of quantum levels change and the eigenvalues decrease. We find that the combined effect of the fields is stronger than a solitary effect and consequently there is a substantial shift in the bound state energy of the system. We also find that to perpetuate a low-energy medium for the hydrogen atom in quantum plasmas, a strong electric field and weak magnetic field are required, whereas the AB flux field can be used as a regulator. The application of the perturbation technique utilized in this paper is not restricted to plasma physics; it can also be applied in molecular physics.

Consider a one-dimensional branching Brownian motion and rescale the coordinate and time so that the rates of branching and diffusion are both equal to 1. If $X_1\left(t\right)$ is the position of the rightmost particle of the branching Brownian motion at time $t$, the empirical velocity $c$ of this rightmost particle is defined as $c=X_1\left(t\right)/t$. Using the Fisher-Kolmogorov-Petrovsky-Piscounov equation, we evaluate the probability distribution $\mathcal{P}(c,t)$ of this empirical velocity $c$ in the long-time $t$ limit for $c>2$. It is already known that, for a single seed particle, $\mathcal{P}(c,t)\sim exp[-(c^2/4-1)t]$ up to a prefactor that can depend on $c$ and $t$. Here we show how to determine this prefactor. The result can be easily generalized to the case of multiple seed particles and to branching random walks associated with other traveling-wave equations.

This paper addresses the challenge of extracting meaningful information from measured bioelectric signals generated by complex, large scale physiological systems such as the brain or the heart. We focus on a combination of the well-known Laplacian eigenmaps machine learning approach with dynamical systems ideas to analyze emergent dynamic behaviors. The method reconstructs the abstract dynamical system phase-space geometry of the embedded measurements and tracks changes in physiological conditions or activities through changes in that geometry. It is geared to extract information from the joint behavior of time traces obtained from large sensor arrays, such as those used in multiple-electrode ECG and EEG, and explore the geometrical structure of the low dimensional embedding of moving time windows of those joint snapshots. Our main contribution is a method for mapping vectors from the phase space to the data domain. We present cases to evaluate the methods, including a synthetic example using the chaotic Lorenz system, several sets of cardiac measurements from both canine and human hearts, and measurements from a human brain.

The nonlinear dynamics of a balanced parity-time-symmetric optical microring arrangement are analytically investigated. By considering gain and loss saturation effects, the pertinent conservation laws are explicitly obtained in the Stokes domain, thus establishing integrability. Our analysis indicates the existence of two regimes of oscillatory dynamics and frequency locking, both of which are analogous to those expected in linear parity-time-symmetric systems. Unlike other saturable parity-time-symmetric systems considered before, the model studied in this work first operates in the symmetric regime and then enters the broken parity-time phase.

In this paper, we develop an analytical approach to studying random patterns of activity in excitable cells. Our analytical approach uses a two-state stochastic model of excitable system based on the electrophysiological properties of refractoriness and restitution, which characterize cell recovery after excitation. By applying the notion of probability density flux, we derive the distributions of transition times between states and the distribution of interspike interval (ISI) durations for a constant applied stimulus. The derived ISI distribution is unimodal and, provided that the time spent in the excited state is constant, can be approximated by a Rayleigh peak followed by an exponential tail. We then explore the role of the model parameters in determining the shape of the derived distributions and the ISI coefficient of variation. Finally, we use our analytical results to study simulation results from the stochastic Morris-Lecar neuron and from a three-state extension of the proposed stochastic model, which is capable of reproducing multimodal ISI histograms.

A kinetic theory is developed for exonuclease-deficient DNA polymerases, based on the experimental observation that the rates depend not only on the newly incorporated nucleotide, but also on the previous one, leading to the growth of Markovian DNA sequences from a Bernoullian template. The dependencies on nucleotide concentrations and template sequence are explicitly taken into account. In this framework, the kinetic and thermodynamic properties of DNA replication, in particular, the mean growth velocity, the error probability, and the entropy production are calculated analytically in terms of the rate constants and the concentrations. Theory is compared with numerical simulations for the DNA polymerases of T7 viruses and human mitochondria.

Kinetic theory and thermodynamics are applied to DNA polymerases with exonuclease activity, taking into account the dependence of the rates on the previously incorporated nucleotide. The replication fidelity is shown to increase significantly thanks to this dependence at the basis of the mechanism of exonuclease proofreading. In particular, this dependence can provide up to a 100-fold lowering of the error probability under physiological conditions. Theory is compared with numerical simulations for the DNA polymerases of T7 viruses and human mitochondria.

Using measurements of Lagrangian acceleration in turbulent rotating convection and accompanying direct numerical simulations, we show that the transition between turbulent states reported earlier [e.g., S. Weiss et al., Phys. Rev. Lett. 105, 224501 (2010)] is a boundary-layer transition between the Prandtl-Blasius type (typical of nonrotating convection) and Ekman type.

We use the shadow wave function formalism as a convenient model to study the fermion sign problem affecting all projector quantum Monte Carlo methods in continuum space. We demonstrate that the efficiency of imaginary-time projection algorithms decays exponentially with increasing number of particles and/or imaginary-time propagation. Moreover, we derive an analytical expression that connects the localization of the system with the magnitude of the sign problem, illustrating this behavior through numerical results. Finally, we discuss the computational complexity of the fermion sign problem and methods for alleviating its severity.

The eigenstate thermalization hypothesis (ETH) attempts to bridge the gap between quantum mechanical and statistical mechanical descriptions of isolated quantum systems. Here, we define unbiased measures for how well the ETH works in various regimes, by mapping general interacting quantum systems on regular lattices onto a single particle living on a high-dimensional graph. By numerically analyzing deviations from ETH behavior in the nonintegrable Ising model, we propose a quantity that we call the $n$-weight to democratically characterize the average deviations for all operators residing on a given number of sites, irrespective of their spatial structure. It appears to have a simple scaling form, which we conjecture to hold true for all nonintegrable systems. A closely related quantity, which we term the $n$-distinguishability, tells us how well two states can be distinguished if only $n$-site operators are measured. Along the way, we discover that complicated operators on average are worse than simple ones at distinguishing between neighboring eigenstates, contrary to the naive intuition created by the usual statements of the ETH that few-body (many-body) operators acquire the same (different) expectation values in nearby eigenstates at finite energy density. Finally, we sketch heuristic arguments that the ETH originates from the limited ability of simple operators to distinguish between quantum states of a system, especially when the states are subject to constraints such as roughly fixed energy with respect to a local Hamiltonian.

We study firing behaviors and the transitions among them in small-world noisy neuronal networks with electrical synapses and information transmission delay. Each neuron is modeled by a two-dimensional Rulkov map neuron. The distance between neurons, which is a main source of the time delay, is taken into consideration. Through spatiotemporal patterns and interspike intervals as well as the interburst intervals, the collective behaviors are revealed. It is found that the networks switch from resting state into intermittent firing state under Gaussian noise excitation. Initially, noise-induced firing behaviors are disturbed by small time delays. Periodic firing behaviors with irregular zigzag patterns emerge with an increase of the delay and become progressively regular after a critical value is exceeded. More interestingly, in accordance with regular patterns, the spiking frequency doubles compared with the former stage for the spiking neuronal network. A growth of frequency persists for a larger delay and a transition to antiphase synchronization is observed. Furthermore, it is proved that these transitions are generic also for the bursting neuronal network and the FitzHugh-Nagumo neuronal network. We show these transitions due to the increase of time delay are robust to the noise strength, coupling strength, network size, and rewiring probability.

We study relations among the side-chain asymmetry, structure, and order-disorder transition (ODT) in hairy-rod-type poly(9,9-dihexylfluorene) (PF6) with two identical side chains and atactic poly(9-octyl-9-methyl-fluorene) (PF1-8) with two different side chains per repeat. PF6 and PF1-8 organize into alternating side-chain and backbone layers that transform into an isotropic phase at $T^{\mathrm{ODT}}\left(\mathrm{PF}6\right)$ and $T_{\mathrm{bi}}^{\mathrm{ODT}}\left(\mathrm{PF}1-8\right)$. We interpret polymers in terms of monodisperse and bidisperse brushes and predict scenarios $T^{\mathrm{ODT}}<T_{\mathrm{bi}}^{\mathrm{ODT}}$ and $T^{\mathrm{ODT}}\sim T_{\mathrm{bi}}^{\mathrm{ODT}}$ for high and low grafting densities (the side-chain length above or below the average grafting distance). Calorimetry and x-ray scattering indicate the condition $T^{\mathrm{ODT}}\left(\mathrm{PF}6\right)\sim T_{\mathrm{bi}}^{\mathrm{ODT}}\left(\mathrm{PF}1-8\right)$ following the low grafting prediction. PF6 side chains coming from the alternating backbone layers appear as two separate layers with thickness $H\left(\mathrm{PF}6\right)$, whereas PF1-8 side chains appear as an indistinguishable bilayer with a half thickness $H_{\mathrm{bilayer}}\left(\mathrm{PF}1-8\right)/2\approx H\left(\mathrm{PF}6\right)$. The low grafting density region is structurally possible but not certain for PF6 and confirmed for PF1-8.

We investigate energy dissipation and the distribution of particle velocities at the jamming transition for overdamped shear-driven frictionless disks in two dimensions at zero temperature. We find that the dissipation is caused by the fastest particles and that the fraction of particles responsible for the dissipation decreases towards zero as jamming is approached. These particles belong to an algebraic tail of the velocity distribution that approaches $\sim v^{-3}$ as jamming is approached. We further find that different measures of the velocity diverge differently, which means that concepts such as typical velocity may no longer be used, a finding that should have implications for analytical approaches to shear-driven jamming.

We developed a “brute-force” simulation method and conducted numerical “experiments” on homogeneous nucleation in an isotropic system at large driving forces (not small supersaturations) using the stochastic Ginzburg-Landau approach. Interactions in the system are described by the asymmetric (no external field), athermal (temperature-independent driving force), tangential (simple phase diagram) Hamiltonian, which has two independent “drivers” of the phase transition: supersaturation and thermal noise. We obtained the probability distribution function of the lifetime of the metastable state and analyzed its mean value as a function of the supersaturation, noise strength, and volume. We also proved the nucleation theorem in the mean-field approximation. The results allowed us to find the thermodynamic properties of the barrier state and conclude that at large driving forces the fluctuating volumes are not independent.

We focus on solitary waves generated in arrays of lightly contacting spherical elastic granules by shock forces of steep rise and slow decay durations and establish a priori: (i) whether the peak value of the resulting solitary wave would be greater than, equal to, or less than the peak value of the input shock force; (ii) the magnitude of the peak value of the solitary waves; (iii) the magnitude of the linear momentum in each solitary wave; (iv) the magnitude of the linear momentum added to the remaining granules if the first granule is ejected; and (v) a quantitative estimate of the effect of the granules’ radius, density, and stiffness on force amplification or mitigation. We have supported the analytical results by direct numerical simulations.