An exact spectral analysis of the Markov propagator for the voter model is presented for the complete graph and extended to the complete bipartite graph and uncorrelated random networks. Using a well-defined Martingale approximation in diffusion-dominated regions of phase space, which is almost everywhere for the voter model, this method is applied to compute analytically several key quantities such as exact expressions for the $m$ time-step propagator of the voter model, all moments of consensus times, and the local times for each macrostate. This spectral method is motivated by a related method for solving the Ehrenfest urn problem and by formulating the voter model on the complete graph as an urn model. Comparisons of the analytical results from the spectral method and numerical results from Monte Carlo simulations are presented to validate the spectral method.

This work casts the semiclassical zeta function in a form suitable for practical calculations of energy levels for rather general systems. To accomplish this, the zeta function is approximated by applying an initial-value representation (IVR) treatment to the traces of the transfer matrix that appear when the function is expanded in cumulants. Because this approach does not require searches for periodic orbits or special trajectories obeying double-ended boundary conditions, it is easily applicable to multidimensional systems with smooth potentials. Calculations are presented for the energy levels of three two-dimensional systems, including one that is classically integrable, one having mixed phase space, and one that is almost fully chaotic. The results show that the present treatment is far more numerically efficient than a previously proposed IVR method for the zeta function [Barak and Kay, Phys. Rev. E 88, 062926 (2013)]. The approach described here successfully resolves nearly all energy levels in the range investigated for the first two systems as well as energy levels in spectral regions that are not too highly congested for the highly chaotic system.

We consider different methods, which do not rely on numerical simulations of the percolation process, to approximate percolation thresholds in networks. We perform a systematic analysis on synthetic graphs and a collection of 109 real networks to quantify their effectiveness and reliability as prediction tools. Our study reveals that the inverse of the largest eigenvalue of the nonbacktracking matrix of the graph often provides a tight lower bound for true percolation threshold. However, in more than $40\%$ of the cases, this indicator is less predictive than the naive expectation value based solely on the moments of the degree distribution. We find that the performance of all indicators becomes worse as the value of the true percolation threshold grows. Thus, none of them represents a good proxy for the robustness of extremely fragile networks.

Variants of fluctuation theorems recently discovered in the statistical mechanics of nonequilibrium processes may be used for the efficient determination of high-dimensional integrals as typically occurring in Bayesian data analysis. In particular for multimodal distributions, Monte Carlo procedures not relying on perfect equilibration are advantageous. We provide a comprehensive statistical error analysis for the determination of the prior-predictive value (the evidence) in a Bayes problem, building on a variant of the Jarzynski equation. Special care is devoted to the characterization of the bias intrinsic to the method and statistical errors arising from exponential averages. We also discuss the determination of averages over multimodal posterior distributions with the help of a consequence of the Crooks relation. All our findings are verified by extensive numerical simulations of two model systems with bimodal likelihoods.

We investigate the relaxation of long-tailed distributions under stochastic dynamics that do not support such tails. Linear relaxation is found to be a borderline case in which long tails are exponentially suppressed in time but not eliminated. Relaxation stronger than linear suppresses long tails immediately, but may lead to strong transient peaks in the probability distribution. We also find that a $\delta$-function initial distribution under stronger than linear decay displays not one but two different regimes of diffusive spreading.

We investigate the nonlinear response of macromolecular thin films subjected to high-amplitude ultrasonic shear oscillation using a sphere-plane contact geometry. At a film thickness comparable to the radius of gyration, we observe the rheological properties intermediate between bulk and boundary nonlinear regimes. As the driving amplitude is increased, these films progressively exhibit oscillatory linear, microslip, and full slip regimes, which can be explained by the modified Coulomb friction law. At highest oscillation amplitudes, the interfacial adhesive failure takes place, being accompanied by a dewettinglike pattern. Moreover, the steady state sliding is investigated in thicker films with imposed shear stresses beyond the yield point. We find that applying high-amplitude shear ultrasound affects not only the yielding threshold but also the sliding velocity at a given shear load. A possible mechanism for the latter effect is discussed.

The present paper deals with the dynamics of bus networks, which consist of several identical dc bus systems connected by resistors. It is analytically guaranteed that the stability of a stand-alone dc bus system is equivalent to that of the networks, independent of the number of bus systems and the network topology. In addition, we show that a decentralized delayed-feedback control can stabilize an unstable operating point embedded within the networks. Moreover, this stabilization does not depend on the number of bus systems or the network topology. A systematic procedure for designing the controller is presented. Finally, the validity of the analytical results is confirmed through numerical examples.

A motile bacterium swims by generating flow in its surrounding liquid. Anisotropy of the suspending liquid significantly modifies the swimming dynamics and corresponding flow signatures of an individual bacterium and impacts collective behavior. We study the interactions between swimming bacteria in an anisotropic environment exemplified by lyotropic chromonic liquid crystal. Our analysis reveals a significant localization of the bacteria-induced flow along a line coaxial with the bacterial body, which is due to strong viscosity anisotropy of the liquid crystal. Despite the fact that the average viscosity of the liquid crystal is two to three orders of magnitude higher than the viscosity of pure water, the speed of bacteria in the liquid crystal is of the same order of magnitude as in water. We show that bacteria can transport a cargo (a fluorescent particle) along a predetermined trajectory defined by the direction of molecular orientation of the liquid crystal. We demonstrate that while the hydrodynamic interaction between flagella of two close-by bacteria is negligible, the observed convergence of the swimming speeds as well as flagella waves' phase velocities may occur due to viscoelastic interaction between the bacterial bodies.

We study the dynamics of nonaligning, noninteracting self-propelled particles confined to a box in two dimensions. In the strong confinement limit, when the persistence length of the active particles is much larger than the size of the box, particles stay on the boundary and align with the local boundary normal. It is then possible to derive the steady-state density on the boundary for arbitrary box shapes. In nonconvex boxes, the nonuniqueness of the boundary normal results in hysteretic dynamics and the density is nonlocal, i.e., it depends on the global geometry of the box. These findings establish a general connection between the geometry of a confining box and the behavior of an ideal active gas it confines, thus providing a powerful tool to understand and design such confinements.

Completely positive quantum operations are frequently discussed in the contexts of statistical mechanics and quantum information. They are customarily given by maps forming positive operator-values measures. To intuitively understand the physical meanings of such abstract operations, the method of phase-space representations is examined. This method enables one to grasp the operations in terms of the classical statistical notions. As an example of physical importance, here, the phase-space representation of the completely positive quantum operation arising from the single-mode subdynamics of the two-mode squeezed vacuum state, which maps from the vacuum state at vanishing temperature to mixed states with perfect decoherence including the thermal state, is studied. It is found in the $P$ representation that remarkably this operation is invertible, implying that coherence lost by the quantum operation can be recovered.

We implement Brownian dynamics to investigate the static properties of colloidal particles confined anisotropically and interacting via a potential which can be tailored in a repulsive-attractive-respulsive fashion as the interparticle distance increases. A diverse number of structural phases are self-assembled, which were classified according to two aspects, that is, their macroscopic and microscopic patterns. Concerning the microscopic phases we found the quasicrystalline, triangular, square, and mixed orderings, where this latter is a combination of square and triangular cells in a $3\times 2$ proportion, i.e., the so-called $(3^3,4^2)$ Archimedian lattice. On the macroscopic level the system could self-organize in a compact or perforated single cluster surrounded or not by fringes. All the structural phases are summarized in detailed phases diagrams, which clearly show that the different phases are extended as the confinement potential becomes more anisotropic.

We report a direct measurement of the friction coefficient ${\xi }_c$ of two fluctuating contact lines formed on a fiber surface when a long glass fiber intersects the two water-air interfaces of a thin soap film. The glass fiber of diameter $d$ in the range of 0.4–4 $\mu \text{m}$ and length 100–300 $\mu \text{m}$ is glued onto the front end of a rectangular cantilever used for atomic force microscopy. As a sensitive mechanical resonator, the hanging fiber probe can accurately measure a minute change of its viscous damping caused by the soap film. By measuring the broadening of the resonant peak of the hanging fiber probe with varying viscosity $\eta$ of the soap film and different surface treatments of the glass fiber, we confirm that the contact line dissipation obeys a universal scaling law, ${\xi }_c=\alpha \pi d\eta$, where the coefficient $\alpha =1.1\pm 0.3$ is insensitive to the change of liquid-solid contact angle. The experimental result is in good agreement with the numerical result based on the phase field model under the generalized Navier boundary conditions.

It has been observed that dynamo action occurs in the von-Kármán-Sodium (VKS) experiment only when the rotating disks and the blades are made of soft iron. The purpose of this paper is to numerically investigate the role of soft iron in the VKS dynamo scenario. This is done by using a mean-field model based on an axisymmetric mean flow, a localized permeability distribution, and a localized $\alpha$ effect modeling the action of the small velocity scales between the blades. The action of the rotating blades is modeled by an axisymmetric effective permeability field. Key properties of the flow giving to the numerical magnetic field a geometric structure similar to that observed experimentally are identified. Depending on the permeability of the disks and the effective permeability of the blades, the dynamo that is obtained is either oscillatory or stationary. Our numerical results confirm the leading role played by the ferromagnetic impellers. A scenario for the VKS dynamo is proposed.

It is often believed that the Rayleigh-Lagrange formalism for classical dissipative systems is unable to encompass forces described by nonlinear functions of the velocities. Here we show that this is indeed a misconception.

We introduce variable focused local search algorithms for satisfiabiliity problems. Usual approaches focus uniformly on unsatisfied clauses. The methods described here work by focusing on random variables in unsatisfied clauses. Variants are considered where variables are selected uniformly and randomly or by introducing a bias towards picking variables participating in several unsatistified clauses. These are studied in the case of the random 3-SAT problem, together with an alternative energy definition, the number of variables in unsatisfied constraints. The variable-based focused Metropolis search (V-FMS) is found to be quite close in performance to the standard clause-based FMS at optimal noise. At infinite noise, instead, the threshold for the linearity of solution times with instance size is improved by picking preferably variables in several UNSAT clauses. Consequences for algorithmic design are discussed.

While $ABC$ triblock copolymers are known to form a plethora of dual-mode (i.e., order-on-order) nanostructures, bicomponent $ABA$ triblock copolymers normally self-assemble into single morphologies at thermodynamic incompatibility levels up to the strong-segregation regime. In this study, we employ on-lattice Monte Carlo simulations to examine the phase behavior of molecularly asymmetric $A_{1}B{A}_2$ copolymers possessing chemically identical endblocks differing significantly in length. In the limit of superstrong segregation, interstitial micelles composed of the minority $A_2$ endblock are observed to arrange into two-dimensional hexagonal arrays along the midplane of $B$-rich lamellae in compositionally symmetric $(50:50 A:B)$ copolymers. Simulations performed here establish the coupled molecular-asymmetry and incompatibility conditions under which such micelles form, as well as the temperature dependence of their aggregation number. Beyond an optimal length of the $A_2$ endblock, the propensity for interstitial micelles to develop decreases, and the likelihood for colocation of both endblocks in the $A_1$-rich lamellae increases. Interestingly, the strong-segregation theory of Semenov developed to explain the formation of free micelles by diblock copolymers accurately predicts the onset of interstitial micelles confined at nanoscale dimensions between parallel lamellae.

Feller studied a branching process and found that the distribution for this process approximately obeys a diffusion equation [W. Feller, in Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability (University of California Press, Berkeley and Los Angeles, 1951), pp. 227–246]. This diffusion equation and its generalizations play an important role in many scientific problems, including, physics, biology, finance, and probability theory. We work under the assumption that the fundamental solution represents a probability density and should account for all of the probability in the problem. Thus, under the circumstances where the random process can be irreversibly absorbed at the boundary, this should lead to the presence of a Dirac delta function in the fundamental solution at the boundary. However, such a feature is not present in the standard approach (Laplace transformation). Here we require that the total integrated probability is conserved. This yields a fundamental solution which, when appropriate, contains a term proportional to a Dirac delta function at the boundary. We determine the fundamental solution directly from the diffusion equation via spectral decomposition. We obtain exact expressions for the eigenfunctions, and when the fundamental solution contains a Dirac delta function at the boundary, every eigenfunction of the forward diffusion operator contains a delta function. We show how these combine to produce a weight of the delta function at the boundary which ensures the total integrated probability is conserved. The solution we present covers cases where parameters are time dependent, thereby greatly extending its applicability.

We present a model for a random walk with memory, phenomenologically inspired in a biological system. The walker has the capacity to remember the time of the last visit to each site and the step taken from there. This memory affects the behavior of the walker each time it reaches an already visited site modulating the probability of repeating previous moves. This probability increases with the time elapsed from the last visit. A biological analog of the walker is a frugivore, with the lattice sites representing plants. The memory effect can be associated with the time needed by plants to recover its fruit load. We propose two different strategies, conservative and explorative, as well as intermediate cases, leading to nonintuitive interesting results, such as the emergence of cycles.

Flows through pipes, such as crude oil through pipelines, are the most common and important method of transportation of fluids. To enhance the flow output along the pipeline requires reducing viscosity and suppressing turbulence simultaneously and effectively. Unfortunately, no method is currently available to accomplish both goals simultaneously. Here we show that electrorheology provides an efficient solution. When a strong electric field is applied along the flow direction in a small section of pipeline, the field polarizes and aggregates the particles suspended inside the base liquid into short chains along the flow direction. Such aggregation breaks the rotational symmetry and makes the fluid viscosity anisotropic. In the directions perpendicular to the flow, the viscosity is substantially increased, effectively suppressing the turbulence. Along the flow direction, the viscosity is significantly reduced; thus the flow along the pipeline is enhanced. Recent field tests with a crude oil pipeline fully confirm the theoretical results.