Quantum walks: The first detected passage time problem.

Even after decades of research, the problem of first passage time statistics for quantum dynamics remains a challenging topic of fundamental and practical importance. Using a projective measurement approach, with a sampling time τ, we obtain the statistics of first detection events for quantum dynamics on a lattice, with the detector located at the origin. A quantum renewal equation for a first detection wave function, in terms of which the first detection probability can be calculated, is derived. This formula gives the relation between first detection statistics and the solution of the corresponding Schrödinger equation in the absence of measurement. We illustrate our results with tight-binding quantum walk models. We examine a closed system, i.e., a ring, and reveal the intricate influence of the sampling time τ on the statistics of detection, discussing the quantum Zeno effect, half dark states, revivals, and optimal detection. The initial condition modifies the statistics of a quantum walk on a finite ring in surprising ways. In some cases, the average detection time is independent of the sampling time while in others the average exhibits multiple divergences as the sampling time is modified. For an unbounded one-dimensional quantum walk, the probability of first detection decays like (time)^{(-3)} with superimposed oscillations, with exceptional behavior when the sampling period τ times the tunneling rate γ is a multiple of π/2. The amplitude of the power-law decay is suppressed as τ→0 due to the Zeno effect. Our work, an extended version of our previously published paper, predicts rich physical behaviors compared with classical Brownian motion, for which the first passage probability density decays monotonically like (time)^{-3/2}, as elucidated by Schrödinger in 1915.

Superaging correlation function and ergodicity breaking for Brownian motion in logarithmic potentials.

We consider an overdamped Brownian particle moving in a confining asymptotically logarithmic potential, which supports a normalized Boltzmann equilibrium density. We derive analytical expressions for the two-time correlation function and the fluctuations of the time-averaged position of the particle for large but finite times. We characterize the occurrence of aging and nonergodic behavior as a function of the depth of the potential, and we support our predictions with extensive Langevin simulations. While the Boltzmann measure is used to obtain stationary correlation functions, we show how the non-normalizable infinite covariant density is related to the superaging behavior.

Gas-phase detonation propagation in mixture composition gradients.

The propagation of detonations through several fuel-air mixtures with spatially varying fuel concentrations is examined numerically. The detonations propagate through two-dimensional channels, inside of which the gradient of mixture composition is oriented normal to the direction of propagation. The simulations are performed using a two-component, single-step reaction model calibrated so that one-dimensional detonation properties of model low- and high-activation-energy mixtures are similar to those observed in a typical hydrocarbon-air mixture. In the low-activation-energy mixture, the reaction zone structure is complex, consisting of curved fuel-lean and fuel-rich detonations near the line of stoichiometry that transition to decoupled shocks and turbulent deflagrations near the channel walls where the mixture is extremely fuel-lean or fuel-rich. Reactants that are not consumed by the leading detonation combine downstream and burn in a diffusion flame. Detonation cells produced by the unstable reaction front vary in size across the channel, growing larger away from the line of stoichiometry. As the size of the channel decreases relative to the size of a detonation cell, the effect of the mixture composition gradient is lessened and cells of similar sizes form. In the high-activation-energy mixture, detonations propagate more slowly as the magnitude of the mixture composition gradient is increased and can be quenched in a large enough gradient.

Fluctuations of time averages for Langevin dynamics in a binding force field.

We derive a simple formula for the fluctuations of the time average x(t) around the thermal mean (eq) for overdamped brownian motion in a binding potential U(x). Using a backward Fokker-Planck equation, introduced by Szabo, Schulten, and Schulten in the context of reaction kinetics, we show that for ergodic processes these finite measurement time fluctuations are determined by the Boltzmann measure. For the widely applicable logarithmic potential, ergodicity is broken. We quantify the large nonergodic fluctuations and show how they are related to a superaging correlation function.

Phase-field model of mode III dynamic fracture.

We introduce a phenomenological continuum model for the mode III dynamic fracture that is based on the phase-field methodology used extensively to model interfacial pattern formation. We couple a scalar field, which distinguishes between "broken" and "unbroken" states of the system, to the displacement field in a way that consistently includes both macroscopic elasticity and a simple rotationally invariant short-scale description of breaking. We report two-dimensional simulations that yield steady-state crack motion in a strip geometry above the Griffith threshold.

Microscopic selection of fluid fingering patterns.

We study the issue of the selection of viscous fingering patterns in the limit of small surface tension. Through detailed simulations of anisotropic fingering, we demonstrate conclusively that no selection independent of the small-scale cutoff (macroscopic selection) occurs in this system. Rather, the small-scale cutoff completely controls the pattern, even on short time scales, in accordance with the theory of microscopic solvability. We demonstrate that ordered patterns are dynamically selected only for not too small surface tensions. For extremely small surface tensions, the system exhibits chaotic behavior and no regular pattern is realized.

Nonlinear lattice model of viscoelastic mode III fracture.

We study the effect of general nonlinear force laws in viscoelastic lattice models of fracture, focusing on the existence and stability of steady-state mode III cracks. We show that the hysteretic behavior at small driving is very sensitive to the smoothness of the force law. At large driving, we find a Hopf bifurcation to a straight crack whose velocity is periodic in time. The frequency of the unstable bifurcating mode depends on the smoothness of the potential, but is very close to an exact period-doubling instability. Slightly above the onset of the instability, the system settles into a exactly period-doubled state, presumably connected to the aforementioned bifurcation structure. We explicitly solve for this new state and map out its velocity-driving relation.

Two state behavior in a solvable model of beta-hairpin folding.

Understanding the mechanism of protein secondary structure formation is an essential part of the protein-folding puzzle. Here we describe a simple model for the formation of a beta hairpin, motivated by the fact that folding of a beta hairpin captures much of the basic physics of protein folding. The modeled hairpin is composed of two interacting Gaussian chains with one pairwise (two-body) and two many-body interactions. We show that these many-body interactions, arising from side chain packing effects, are responsible for producing an "all-or-none" folding transition. We also estimate the (single exponential) folding/unfolding rate via calculating the thermodynamic weight of the "critical" droplet/bubble.

How does a beta -hairpin fold/unfold? competition between topology and heterogeneity in a solvable model.

We study the competition between topological effects and sequence inhomogeneities in determining the thermodynamics and the un/folding kinetics of a beta-hairpin. Our work utilizes a new exactly solvable model that allows for arbitrary configurations of native contacts. In general, the competition between heterogeneity and topology results in a crossover of the dominant transition state. Interestingly, near this crossover, the single reaction coordinate picture can be seriously misleading. Our results also suggest that inferring the folding pathway from unfolding simulations is not always justified.

Arrested cracks in nonlinear lattice models of brittle fracture.

We generalize lattice models of brittle fracture to arbitrary nonlinear force laws and study the existence of arrested semi-infinite cracks. Unlike what is seen in the discontinuous case studied to date, the range in driving displacement for which these arrested cracks exist is very small. Also, our results indicate that small changes in the vicinity of the crack tip can have an extremely large effect on arrested cracks. Finally, we briefly discuss the possible relevance of our findings to recent experiments.

Steady-state cracks in viscoelastic lattice models.

We study the steady-state motion of mode III cracks propagating on a lattice exhibiting viscoelastic dynamics. The introduction of a Kelvin viscosity eta allows for a direct comparison between lattice results and continuum treatments. Utilizing both numerical and analytical (Wiener-Hopf) techniques, we explore this comparison as a function of the driving displacement Delta and the number of transverse rows N. At any N, the continuum theory misses the lattice-trapping phenomenon; this is well known, but the introduction of eta introduces some new twists. More importantly, for large N even at large Delta, the standard two-dimensional elastodynamics approach completely misses the eta-dependent velocity selection, as this selection disappears completely in the leading order naive continuum limit of the lattice problem.

The legal and scientific basis for FDA's assertion of jurisdiction over cigarettes and smokeless tobacco.

On August 28, 1996, the US Food and Drug Administration (FDA) asserted jurisdiction over cigarettes and smokeless tobacco under the Federal Food, Drug, and Cosmetic Act. Under this Act, a product is a "drug" or "device" subject to FDA jurisdiction if it is "intended to affect the structure or any function of the body." The FDA determined that nicotine in cigarettes and smokeless tobacco does "affect the structure or any function of the body" because nicotine causes addiction and other pharmacological effects. The FDA then determined that these pharmacological effects are "intended" because (1) a scientific consensus has emerged that nicotine is addictive; (2) recent studies have shown that most consumers use cigarettes and smokeless tobacco for pharmacological purposes, including satisfying their addiction to nicotine; and (3) newly disclosed evidence from the tobacco manufacturers has revealed that the manufacturers know that nicotine causes pharmacological effects, including addiction, and design their products to provide pharmacologically active doses of nicotine. The FDA thus concluded that cigarettes and smokeless tobacco are subject to FDA jurisdiction because they contain a "drug," nicotine, and a "device" for delivering this drug to the body.