Long linear polymers in strongly disordered media are well described by self-avoiding walks (SAWs) on percolation clusters and a lot can be learned about the statistics of these polymers by studying the length distribution of SAWs on percolation clusters. This distribution encompasses 2 distinct averages, viz., the average over the conformations of the underlying cluster and the SAW conformations. For the latter average, there are two basic options, one being static and one being kinetic. It is well known for static averaging that if the disorder of the underlying medium is weak, this disorder is redundant in the sense the renormalization group; i.e., differences to the ordered case appear merely in nonuniversal quantities. Using dynamical field theory, we show that the same holds true for kinetic averaging. Our main focus, however, lies on strong disorder, i.e., the medium being close to the percolation point, where disorder is relevant. Employing a field theory for the nonlinear random resistor network in conjunction with a real-world interpretation of the corresponding Feynman diagrams, we calculate the scaling exponents for the shortest, the longest, and the mean or average SAW to 2-loop order. In addition, we calculate to 2-loop order the entire family of multifractal exponents that governs the moments of the the statistical weights of the elementary constituents (bonds or sites of the underlying fractal cluster) contributing to the SAWs. Our RG analysis reveals that kinetic averaging leads to renormalizability whereas static averaging does not, and hence, we argue that the latter does not lead to a well-defined scaling limit. We discuss the possible implications of this finding for experiments and numerical simulations which have produced widespread results for the exponent of the average SAW. To corroborate our results, we also study the well-known Meir-Harris model for SAWs on percolation clusters. We demonstrate that the Meir-Harris model leads back up to 2-loop order to the renormalizable real-world formulation with kinetic averaging if the replica limit is consistently performed at the first possible instant in the course of the calculation.

The dynamic behavior of a hard-sphere colloidal suspension was studied by x-ray photon correlation spectroscopy and small-angle x-ray scattering over a wide range of particle volume fractions. The short-time mobility of the particles was found to be smaller than that of free particles even at relatively low concentrations, showing the importance of indirect hydrodynamic interactions. Hydrodynamic functions were derived from the data, and for moderate particle volume fractions ($\Phi \le$ 0.40) there is good agreement with earlier many-body theory calculations by Beenakker and Mazur [Physica A 120, 349 (1984)]. Important discrepancies appear at higher concentrations, above $\Phi \approx$ 0.40, where the hydrodynamic effects are overestimated by the Beenakker-Mazur theory, but predicted accurately by an accelerated Stokesian dynamics algorithm developed by Banchio and Brady [J. Chem. Phys. 118, 10323 (2003)]. For the relaxation rates, good agreement was also found between the experimental data and a scaling form predicted by the mode coupling theory. In the high concentration range, with the fluid suspensions approaching the glass transition, the long-time diffusion coefficient was compared with the short-time collective diffusion coefficient to verify a scaling relation previously proposed by Segrè and Pusey [Phys. Rev. Lett. 77, 771 (1996)]. We discuss our results in view of previous experimental attempts to validate this scaling law [L. Lurio et al., Phys. Rev. Lett. 84, 785 (2000)].

Models of smectic-$C$ liquid-crystal elastomers predict that they can display soft elasticity, in which the shape of the elastomer changes at no energy cost. The amplitude of the soft mode and the accompanying shears are dependent on the orientation of the layer normal and the director with respect to the stretch axis. We demonstrate that in some geometries the director is forced to rotate perpendicular to the stretch axis, causing lateral expansion of the sample—a negative Poisson's ratio. Current models do not include the effect of imperfections that must be present in the physical sample. We investigate the effect of a simple model of these imperfections on the soft modes in monodomain smectic-$C$ elastomers in a variety of geometries. When stretching parallel to the layer normal (with imposed strain) the elastomer has a negative stiffness once the director starts to rotate. We show that this is a result of the negative Poisson's ratio in this geometry through a simple scalar model.

Experiments by Poulard and Cazabat [Langmuir 21, 6270 (2005)] on spreading droplets of nematic liquid crystal (NLC) reveal a surprisingly rich variety of behavior, including at least two different emerging length scales resulting from a contact line instability. In earlier work [Cummings, Lin, and Kondic, Phys. Fluids 23, 043102 (2011)] we modified a lubrication model for NLCs due to Ben Amar and Cummings [Phys. Fluids 13, 1160 (2001)] and showed that, in a qualitative sense, it can account for two-dimensional (2D) versions of the observed behavior. In the present work we propose a different approach that allows us to explore the effect of anchoring variations on the substrate, again in a 2D geometry. This in turn gives a simple way to model the presence of defects, which are nearly always present in such flows. The present model leads to additional terms in the governing equation. We explore the influence of these additional terms for some simple flow scenarios to gain insight into their influence.

A long-standing issue in the field of long-time-stable, water-based, single-bubble sonoluminescence has been the close similarity of the spectra to that of blackbody radiation, the question being whether the similarity is just a weird coincidence, with the bubbles being, on the whole, transparent to their own radiation. One mechanism that has been suggested is the generation of a shock or, at least, a compression wave in the gas of the bubble. A footprint of such a wave would be segregation of species. We have investigated spectra from bubbles seeded with various mixtures of helium or neon with xenon or argon using a transformation, specific to our experimental setup and spectrometer, that was shown to allow for a single-parameter characterization of the spectra in some simpler situations. The surprising result of this investigation is that although no trace of segregation is found, the radiation seems to be highly thermalized in all cases.

A two-dimensional particle-in-cell simulation model is proposed to study the time evolution and energy deposition for ion clusters injected into magnetized two-component plasmas. The injection of an isolated ion cluster is studied in the case of weak and strong magnetic fields. For strong magnetic fields, the ions tend to deposit their energy smoothly along the trajectory of the cluster, due to the confinement by the strong magnetic fields. However, in the case of weak magnetic fields, a large amount of energy is deposited by the ions near the initial cluster injection position, where the cluster density is expected to be largest. We attribute these to the influences of interference effects between the cluster ions, which have close relations to the distances between the ions. Furthermore, the influences of various magnetic fields, injection angles, and injection velocities on the time evolution and energy deposition of a beam pulse, which contains several similar ion clusters, are investigated in detail. The influences of different magnetic fields on the beam pulse show similar to that of a single ion cluster. For increasing injection angles, the beam velocity perpendicular to the magnetic field increases, leading to increasing oscillations in the beam trajectory and energy deposition profile. Besides, the amount of energy that transferred from the beam pulse to the plasma increases as the beam injection velocity approaches the electron thermal velocity.

We derive analytical rogue wave solutions of variable-coefficient higher-order nonlinear Schrödinger equations describing the femtosecond pulse propagation via a transformation connected with the constant-coefficient Hirota equation. Then we discuss the propagation behaviors of controllable rogue waves, including recurrence, annihilation, and sustainment in a periodic distributed fiber system and an exponential dispersion decreasing fiber. Finally, we investigate nonlinear tunneling effects for rogue waves.

We study the counterion condensation on a two-dimensional charged disk in the limit of infinite dilution, and compare the energy–temperature relation obtained from the canonical free energy and microcanonical entropy. The microcanonical entropy is piecewise linear in energy, and is shown to be concave for all energies. As a result, even though the interactions are long-ranged, the energy–temperature relation and hence the counterion condensation transition points are identical in both the ensembles.

A three-phase contact line in a three-phase fluid system is modeled by a mean-field density functional theory. We use a variational approach to find the Euler-Lagrange equations. Analytic solutions are obtained in the two-phase regions at large distances from the contact line. We employ a triangular grid and use a successive overrelaxation method to find numerical solutions in the entire domain for the special case of equal interfacial tensions for the two-phase interfaces. We use the Kerins-Boiteux formula to obtain a line tension associated with the contact line. This line tension turns out to be negative. We associate line adsorption with the change of line tension as the governing potentials change.

Linear instabilities are analyzed in a physiologically based mean-field corticothalamic model and a reduced-parameter model derived from it. In both models, the stable zone corresponding to normal arousal states is bounded by a series of surfaces demarcating the onsets of instabilities. The stable zone is found to depend on delay and rate parameters, whose values have a simple relationship to the number of instabilities and dominant frequencies on the stable zone's boundary. The dominant frequencies of linear activity inside the stable zone are found to lie in clearly delineated regions, each corresponding to an instability surface on its boundary and having approximately the same dominant frequency. These regions are ordered in parameter space according to their dominant frequencies, and an instability associated with the intrathalamic loop is shown to have the highest frequency that can become unstable. This reveals an important role for the thalamus in controlling the stability and bandwidth of dynamics in the corticothalamic system as a whole. The reduced model is found to agree well with the full model in a wide region of parameter space and, thus, is a useful guide to the full model's dynamics.

We analytically study the input-output properties of a neuron whose active dendritic tree, modeled as a Cayley tree of excitable elements, is subjected to Poisson stimulus. Both single-site and two-site mean-field approximations incorrectly predict a nonequilibrium phase transition which is not allowed in the model. We propose an excitable-wave mean-field approximation which shows good agreement with previously published simulation results [Gollo et al., PLoS Comput. Biol. 5, e1000402 (2009)] and accounts for finite-size effects. We also discuss the relevance of our results to experiments in neuroscience, emphasizing the role of active dendrites in the enhancement of dynamic range and in gain control modulation.

Modularity-based partitioning methods divide networks into modules by comparing their structure against random networks conditioned to have the same number of nodes, edges, and degree distribution. We propose a novel way to measure modularity and divide graphs, based on conditional probabilities of the edge strength of random networks. We provide closed-form solutions for the expected strength of an edge when it is conditioned on the degrees of the two neighboring nodes, or alternatively on the degrees of all nodes comprising the network. We analytically compute the expected network under the assumptions of Gaussian and Bernoulli distributions. When the Gaussian distribution assumption is violated, we prove that our expression is the best linear unbiased estimator. Finally, we investigate the performance of our conditional expected model in partitioning simulated and real-world networks.

The flow of two immiscible fluids completely filling an enclosed cylinder and driven by the rotation of the bottom end wall is studied numerically. The simulations are in parameter regimes where there is significant advection of angular momentum, i.e., the disk rotation rate is fast compared to the viscous diffusion time. We consider two classes of scenarios. The first consists of cases that are straightforward to reproduce in physical experiments where only the rotation rate and the viscosity ratio of the fluids are varied. Then we isolate different forces acting on the system such as inertia, surface tension, and gravity by studying variations in individual governing parameters. The viscosity ratio determines how quickly the upper fluid equilibriates dynamically to the flow in the lower fluid and plays a major role in determining how vortex lines are bent in the neighborhood of the interface between the two fluids. This in turn determines the structure of the interfacial layer between the two swirling fluids, which is responsible for the flow in the upper fluid. The simulations show that even when there is significant interfacial deformation, both the dynamics and the equilibrium flow are dominated by vortex bending rather than vortex stretching. The simulations show that for the range of immiscible fluids considered, surface tension effects are significant. Increased surface tension reduces the degree to which the interface is deformed and the limit of zero surface tension is not an appropriate approximation.

The behavior of a lattice Monte Carlo (LMC) algorithm (if it is designed correctly) must approach that of the continuum system that it is designed to simulate as the time step and the mesh step tend to zero. However, we show for an algorithm for unbiased particle diffusion that if one of these two parameters remains fixed, the accuracy of the algorithm is optimal for a certain finite value of the other parameter. In one dimension, the optimal algorithm with moves to the two nearest neighbor sites reproduces the correct second and fourth moments (and minimizes the error for the higher moments at large times) of the particle distribution and preserves the first two moments of the first-passage time distributions. In two and three dimensions, the same level of accuracy requires simultaneous moves along two axes (“diagonal” moves). Such moves attempting to cross an impenetrable boundary should be projected along the boundary, rather than simply rejected. We also treat the case of absorbing boundaries. We discuss the relation between optimally accurate LMC algorithms and a particular case of lattice Boltzmann (LB) algorithms for simulating diffusion and compare the computational efficiency of optimal LMC and optimal LB algorithms.

A unified optimization criterion for Carnot engines and refrigerators is proposed. It consists of maximizing the product of the heat absorbed by the working system times the efficiency per unit time of the device, either the engine or the refrigerator. This criterion can be applied to both low symmetric dissipation Carnot engines and refrigerators. For engines the criterion coincides with the maximum power criterion and then the Curzon-Ahlborn efficiency ${\eta }_{\mathrm{CA}}=1-\sqrt{T_c/T_h}$ is recovered, where $T_h$ and $T_c$ are the temperatures of the hot and cold reservoirs, respectively [Esposito, Kawai, Lindenberg, and Van den Broeck, Phys. Rev. Lett. 105, 150603 (2010)]. For refrigerators the criterion provides the counterpart of Curzon-Ahlborn efficiency for refrigerators ${\varepsilon }_{\mathrm{CA}}=[1/\left(\sqrt{1-(T_c/T_h)}\right]-1$, first derived by Yan and Chen for the particular case of an endoreversible Carnot-type refrigerator with linear (Newtonian) finite heat transfer laws [Yan and Chen, J. Phys. D: Appl. Phys. 23, 136 (1990)].

The problem of random walk is considered in one dimension in the simultaneous presence of a quenched random force field and long-range connections, the probability of which decays with the distance algebraically as $p_l\simeq \beta l^{-s}$. The dynamics are studied mainly by a numerical strong disorder renormalization group method. According to the results, for $s>2$ the long-range connections are irrelevant, and the mean-square displacement increases as $\langle x^2\left(t\right)\rangle \sim {\left(\ln t\right)}^{2/\psi }$ with the barrier exponent $\psi =1/2$, which is known in one-dimensional random environments. For $s<2$, instead, the quenched disorder is found to be irrelevant, and the dynamical exponent is $z=1$ like in a homogeneous environment. At the critical point, $s=2$, the interplay between quenched disorder and long-range connections results in activated scaling, however, with a nontrivial barrier exponent $\psi \left(\beta \right)$, which decays continuously with $\beta$ but is independent of the form of the quenched disorder. Upper and lower bounds on $\psi \left(\beta \right)$ are established, and numerical estimates are given for various values of $\beta$. Besides random walks, accurate numerical estimates of the graph dimension and the resistance exponent are given for various values of $\beta$ at $s=2$.

We construct complex networks from symbolic time series of particle properties within a dense quasistatically deforming granular assembly subjected to biaxial compression. The structure of the resulting networks embodies the evolving structural rearrangements in the granular material, in both contact forces and contact topologies. These rearrangements are usefully summarized through standard network statistics as well as building block motifs and community structures. Dense granular media respond to applied compression and shear by a process of self-organization to form two cooperatively evolving structures comprising the major load-bearing columnlike force chains, and the lateral trusslike three-cycle triangle topologies. We construct networks summarizing their individual evolution based on relationships between symbolic time series indicating a particle's chronological force chain and three-cycle membership histories. We test which particle membership histories are similar with respect to each other through the information theory-based measure of Hamming distance. The complex networks summarize the essential structural rearrangements, while the community structures within the networks partition the material into distinct zones of deformation, including interlacing subregions of failure inside the shear band. The taxonomy of granular rheology at the mesoscopic scale distills the inelastic structural rearrangements throughout loading history down to its core elements, and should lay bare an objective and physics-based formalism for thermodynamic internal variables and associated evolution laws.

A dense mixture of granules and liquid often shows a severe shear thickening and is called a dilatant fluid. We construct a fluid dynamics model for the dilatant fluid by introducing a phenomenological state variable for a local state of dispersed particles. With simple assumptions for an equation of the state variable, we demonstrate that the model can describe basic features of the dilatant fluid such as the stress-shear rate curve that represents discontinuous severe shear thickening, hysteresis upon changing shear rate, and instantaneous hardening upon external impact. An analysis of the model reveals that the shear thickening fluid shows an instability in a shear flow for some regime and exhibits the shear thickening oscillation (i.e., the oscillatory shear flow alternating between the thickened and the relaxed states). The results of numerical simulations are presented for one- and two-dimensional systems.

We report measurements of the temperature variations of physical parameters in ambient-temperature nematic liquid crystal mixtures of bent-core (BC) and rodlike molecules (5CB): birefringence $\Delta n$; static dielectric constants ${\epsilon }_{\left|\right|}$ and ${\epsilon }_{\perp }$; splay $K_{11}$ and bend $K_{33}$ elastic constants; rotational viscosity ${\gamma }_1$; and diffusion coefficients $D_{\left|\right|}$ and $D_{\perp }$ of a microsphere. Both $\Delta n$ and ${\epsilon }_{\left|\right|}$ decreases rapidly with increasing BC concentration, whereas ${\epsilon }_{\perp }$ remains almost constant. At a shifted temperature (e.g., $T-T_{NI}=-10{}^{\circ }$C), $K_{11}$ increases by $\sim$50$\%$ and $K_{33}$ decreases by $\sim$80$\%$ compared to pure 5CB when the BC concentration is increased to $\sim$43 mol $\%$ in the mixture. Viscosities parallel and perpendicular to the director, ${\eta }_{\left|\right|}$, ${\eta }_{\perp }$, which are nearly equal to the Miesowicz viscosities ${\eta }_2$ and ${\eta }_3$, respectively, were obtained by $D_{\left|\right|}$ and $D_{\perp }$ using the Stokes-Einstein relation. Both the viscosities at room temperature increase by 60 and 50 times, respectively, whereas ${\gamma }_1$ increases by 180 times (at $\sim$43 mol $\%$) compared to the corresponding values of pure 5CB. The stiffening of $K_{11}$ and exorbitantly large enhancement in all the viscosities at a higher mol $\%$ of BC indicate that the viscoelastic properties are highly impacted by the presence of smectic clusters of BC molecules that results from the restricted free rotation of the molecules along the bow axis in the nematic phase. A possible attachment model of smectic type clusters of BC molecules surrounding the microparticle is presented.

Recent single-molecule experiments have observed that formation of a plectonemically supercoiled region in a stretched, twisted DNA proceeds via abrupt formation of a small plectonemic “bubble.” A detailed mesoscopic model is presented for the formation of plectonemic domains, including their positional entropy, and the influence of small chiral loops or “curls” along the extended DNA. Curls begin to appear just before plectoneme formation, and are more numerous at low salt concentrations ($<20$ mM univalent ions) and at low forces ($<0.5$ pN). However, plectonemic domains quickly become far more stable slightly beyond the transition to supercoiling at moderate forces and physiological salt conditions. At the supercoiling transition, for shorter DNAs (2 kb) only one supercoiled domain appears, but for longer DNAs at lower forces ($<0.5$ pN) positional entropy favors formation of more than one plectonemic domain; a similar effect occurs for low salt. Although they are not the prevalent mode of supercoiling, curls are a natural transition state for binding of DNA-loop-trapping enzymes; we show how addition of loop-trapping enzymes can modify the supercoiling transition. The behavior of DNA torque is also discussed, including the effect of the measurement apparatus torque stiffness, which can play a role in determining how large the torque “overshoot” is at the buckling transition.