A fingering instability can develop at the interface between two fluids when the more mobile fluid is injected into the less-mobile one. For example, viscous fingering appears when a less viscous (i.e., more mobile) fluid displaces a more viscous (and hence less mobile) one in a porous medium. Fingering can also be due to a local change in mobility arising when a precipitation reaction locally decreases the permeability. We numerically analyze the properties of the related precipitation fingering patterns occurring when an $A+B\to C$ chemical reaction takes place, where $A$ and $B$ are reactants in solution and $C$ is a solid product. We show that, similarly to reactive viscous fingering patterns, the precipitation fingering structures differ depending on whether $A$ invades $B$ or vice versa. This asymmetry can be related to underlying asymmetric concentration profiles developing when diffusion coefficients or initial concentrations of the reactants differ. In contrast to reactive viscous fingering, however, precipitation fingering patterns appear at shorter time scales than viscous fingers because the solid product $C$ has a diffusivity tending to zero which destabilizes the displacement. Moreover, contrary to reactive viscous fingering, the system is more unstable with regard to precipitation fingering when the high-concentrated solution is injected into the low-concentrated one or when the faster diffusing reactant displaces the slower diffusing one.

The pretransitional behavior of dodecylcyanobiphenyl (12CB) (isotropic–smectic-$A$–solid mesomorphism) with $d=50\mathrm{nm}\mathrm{BaTi}{\mathrm{O}}_3$ nanoparticles (NPs) linked to the cubic phase was monitored via temperature studies of dielectric constant. Tests were carried out in the isotropic, liquid crystal mesomorphic, and solid phases. For each phase transition the same value of the critical exponent $\alpha \sim 0.5$ was obtained, including nanocolloids. All phase transitions show the weakly discontinuous nature. The temperature metric of the discontinuity $\mathrm{\Delta }T$ notably decreases when adding nanoparticles. The addition of nanoparticles first decreases the dielectric constant by approximately $50\%$ in comparison with pure 12CB, but already for a concentration $\sim x=0.4\%$ NP an increase over $50\%$ takes place. It is notable that for the latter concentration unique hallmarks of the pretransitional effect emerge also for the solid-mesophase transition. All these indicate the important impact of nanoparticles on multimolecular mesoscale fluctuations.

The ding-a-ling model is a kind of half lattice and half hard-point-gas (HPG) model. The original ding-a-ling model proposed by Casati et al. does not conserve total momentum and has been found to exhibit normal heat conduction behavior. Recently, a modified ding-a-ling model which conserves total momentum has been studied and normal heat conduction has also been claimed. In this work, we propose a full-lattice ding-a-ling model without hard point collisions where total momentum is also conserved. We investigate the heat conduction and energy diffusion of this full-lattice ding-a-ling model with three different nonlinear inter-particle potential forms. For symmetrical potential lattices, the thermal conductivities diverges with lattice length and their energy diffusions are superdiffusive signaturing anomalous heat conduction. For asymmetrical potential lattices, although the thermal conductivity seems to converge as the length increases, the energy diffusion is definitely deviating from normal diffusion behavior indicating anomalous heat conduction as well. No normal heat conduction behavior can be found for the full-lattice ding-a-ling model.

Among Markovian processes, the hallmark of Lévy flights is superdiffusion, or faster-than-Brownian dynamics. Here we show that Lévy laws, as well as Gaussian distributions, can also be the limit distributions of processes with long-range memory that exhibit very slow diffusion, logarithmic in time. These processes are path dependent and anomalous motion emerges from frequent relocations to already visited sites. We show how the central limit theorem is modified in this context, keeping the usual distinction between analytic and nonanalytic characteristic functions. A fluctuation-dissipation relation is also derived. Our results may have important applications in the study of animal and human displacements.

Incoherent neutron scattering by water confined in carbon nanohorns was measured with the backscattering spectrometer SPHERES and analyzed in exemplary breadth and depth. Quasielastic spectra admit $\delta$-plus-Kohlrausch fits over a wide $q$ and $T$ range. From the $q$ and $T$ dependence of fitted amplitudes and relaxation times, however, it becomes clear that the fits do not represent a uniform physical process, but that there is a crossover from localized motion at low $T$ to diffusive $\alpha$ relaxation at high $T$. The crossover temperature of about 210 to 230 K increases with decreasing wave number, which is incompatible with a thermodynamic strong-fragile transition. Extrapolated diffusion coefficients $D\left(T\right)$ indicate that water motion is at room temperature about 2.5 times slower than in the bulk; in the supercooled state this factor becomes smaller. At even higher temperatures, where the $\alpha$ spectrum is essentially flat, a few percentages of the total scattering go into a Lorentzian with a width of about $1.6\mu \mathrm{eV}$, probably due to functional groups on the surface of the nanohorns.

In this paper, we present an estimation of the conductivity of composites constituted of identical spheres embedded in a host material. A family of polarization integral equations for the localization problem is constructed and the operator is then minimized to yield an optimal integral equation. As a result, the corresponding Neumann series converges with the fastest rate and can be used to estimate the effective conductivity. By combining this series and integral approximation, one can derive explicit expressions for the overall property using expansions in Fourier domain. For random hard-sphere systems, relations to structure factors and triplet structure factors have been made and Kirkwood superposition approximation is used to evaluate the effective conductivity, taking into account third-order correlations. This presents an original means to account for the statistical information up to third-order correlation when determining the effective properties of composite materials.

It is known that introducing a stochastic resetting in a random-walk process can lead to the emergence of a stationary state. Here we study this point from a general perspective through the derivation and analysis of mesoscopic (continuous-time random walk) equations for both jump and velocity models with stochastic resetting. In the case of jump models it is shown that stationary states emerge for any shape of the waiting-time and jump length distributions. The existence of such state entails the saturation of the mean square displacement to an universal value that depends on the second moment of the jump distribution and the resetting probability. The transient dynamics towards the stationary state depends on how the waiting time probability density function decays with time. If the moments of the jump distribution are finite then the tail of the stationary distributions is universally exponential, but for Lévy flights these tails decay as a power law whose exponent coincides with that from the jump distribution. For velocity models we observe that the stationary state emerges only if the distribution of flight durations has finite moments of lower order; otherwise, as occurs for Lévy walks, the stationary state does not exist, and the mean square displacement grows ballistically or superdiffusively, depending on the specific shape of the distribution of movement durations.

We examine immunostaining experimental data for the formation of stripe 2 of even-skipped (eve) transcripts on D. melanogaster embryos. An estimate of the factor converting immunofluorescence intensity units into molecular numbers is given. The analysis of the eve dynamics at the region of stripe 2 suggests that the promoter site of the gene has two distinct regimes: an earlier phase when it is predominantly activated until a critical time when it becomes mainly repressed. That suggests proposing a stochastic binary model for gene transcription on D. melanogaster embryos. Our model has two random variables: the transcripts number and the state of the source of mRNAs given as active or repressed. We are able to reproduce available experimental data for the average number of transcripts. An analysis of the random fluctuations on the number of eves and their consequences on the spatial precision of stripe 2 is presented. We show that the position of the anterior or posterior borders fluctuate around their average position by $\sim 1\%$ of the embryo length, which is similar to what is found experimentally. The fitting of data by such a simple model suggests that it can be useful to understand the functions of randomness during developmental processes.

A theory of acoustic vector solitons of self-induced transparency of the Rayleigh wave is constructed. A thin resonance transition layer on an elastic surface is considered using a model of a two-dimensional gas of impurity paramagnetic atoms or quantum dots. Explicit analytical expressions for the profile and parameters of the Rayleigh vector soliton with two different oscillation frequencies is obtained, as well as simulations of this nonlinear surface acoustic wave with realistic parameters, which can be used in acoustic experiments. It is shown that the properties of a surface vector soliton of the Rayleigh wave depend on the parameters of the resonance layer, the elastic medium, and the transverse structure of the surface acoustic wave.

We derive the single integrodifferential wave equation for the probability density function of the position of a classical one-dimensional Lévy walk with continuous sample paths. This equation involves a classical wave operator together with memory integrals describing the spatiotemporal coupling of the Lévy walk. It is valid at all times, not only in the long time limit, and it does not involve any large-scale approximations. It generalizes the well-known telegraph or Cattaneo equation for the persistent random walk with the exponential switching time distribution. Several non-Markovian cases are considered when the particle's velocity alternates at the gamma and power-law distributed random times. In the strong anomalous case we obtain the asymptotic solution to the integrodifferential wave equation. We implement the nonlinear reaction term of Kolmogorov-Petrovsky-Piskounov type into our equation and develop the theory of wave propagation in reaction-transport systems involving Lévy diffusion.

For discrete-state stochastic systems obeying Markovian dynamics, we establish the counterpart of the conditional reversibility theorem obtained by Gallavotti for deterministic systems [Ann. de l'Institut Henri Poincaré (A) 70, 429 (1999)]. Our result states that stochastic trajectories conditioned on opposite values of entropy production are related by time reversal, in the long-time limit. In other words, the probability of observing a particular sequence of events, given a long trajectory with a specified entropy production rate $\sigma$, is the same as the probability of observing the time-reversed sequence of events, given a trajectory conditioned on the opposite entropy production, $-\sigma$, where both trajectories are sampled from the same underlying Markov process. To obtain our result, we use an equivalence between conditioned (“microcanonical”) and biased (“canonical”) ensembles of nonequilibrium trajectories. We provide an example to illustrate our findings.

A multimode optical fiber with a truncated transverse cross section acts as a powerful versatile support to investigate the wave features of complex ray dynamics. In this paper, we concentrate on the case of a geometry inducing mixed dynamics. We highlight that regular modes associated with stable periodic orbits present an enhanced spatial intensity localization. We report the statistics of the inverse participation ratio whose features are analogous to those of Anderson localized modes. Our study is supported by both numerical and experimental results on the spatial localization and spectral regularity of the regular modes.

The reduced dynamics for dark and bright soliton chains in the one-dimensional nonlinear Schrödinger equation is used to study the behavior of collective compression waves corresponding to Toda lattice solitons. We coin the term hypersoliton to describe such solitary waves riding on a chain of solitons. It is observed that in the case of dark soliton chains, the formulated reduction dynamics provides an accurate an robust evolution of traveling hypersolitons. As an application to Bose-Einstein condensates trapped in a standard harmonic potential, we study the case of a finite dark soliton chain confined at the center of the trap. When the central chain is hit by a dark soliton, the energy is transferred through the chain as a hypersoliton that, in turn, ejects a dark soliton on the other end of the chain that, as it returns from its excursion up the trap, hits the central chain repeating the process. This periodic evolution is an analog of the classical Newton's cradle.

A simple but efficient spectral approach for analyzing the community structure of complex networks is introduced. It works the same way for all types of networks, by spectrally splitting the adjacency matrix into a “unipartite” and a “multipartite” component. These two matrices reveal the structure of the network from different perspectives and can be analyzed at different levels of detail. Their entries, or the entries of their lower-rank approximations, provide measures of the affinity or antagonism between the nodes that highlight the communities and the “gateway” links that connect them together. An algorithm is then proposed to achieve the automatic assignment of the nodes to communities based on the information provided by either matrix. This algorithm naturally generates overlapping communities but can also be tuned to eliminate the overlaps.

Solid-state nanopores are single-molecule sensors that measure changes in ionic current as charged polymers such as DNA pass through. Here, we present comprehensive experiments on the length, voltage, and salt dependence of the frequency of double-stranded DNA translocations through conical quartz nanopores with mean opening diameter 15 nm. We observe an entropic barrier-limited, length-dependent translocation frequency at 4M LiCl salt concentration and a drift-dominated, length-independent translocation frequency at 1M KCl salt concentration. These observations are described by a unifying convection-diffusion equation, which includes the contribution of an entropic barrier for polymer entry.

Mean-field models of the brain approximate spiking dynamics by assuming that each neuron responds to its neighbors via a naive spatial average that neglects local fluctuations and correlations in firing activity. In this paper we address this issue by introducing a rigorous formalism to enable spatial coarse-graining of spiking dynamics, scaling from the microscopic level of a single type 1 (integrator) neuron to a macroscopic assembly of spiking neurons that are interconnected by chemical synapses and nearest-neighbor gap junctions. Spiking behavior at the single-neuron scale $\ell \approx 10\mu \mathrm{m}$ is described by Wilson's two-variable conductance-based equations [H. R. Wilson, J. Theor. Biol. 200, 375 (1999)], driven by fields of incoming neural activity from neighboring neurons. We map these equations to a coarser spatial resolution of grid length $B\ell$, with $B\gg 1$ being the blocking ratio linking micro and macro scales. Our method systematically eliminates high-frequency (short-wavelength) spatial modes $\vec{q}$ in favor of low-frequency spatial modes $\vec{Q}$ using an adiabatic elimination procedure that has been shown to be equivalent to the path-integral coarse graining applied to renormalization group theory of critical phenomena. This bottom-up neural regridding allows us to track the percolation of synaptic and ion-channel noise from the single neuron up to the scale of macroscopic population-average variables. Anticipated applications of neural regridding include extraction of the current-to-firing-rate transfer function, investigation of fluctuation criticality near phase-transition tipping points, determination of spatial scaling laws for avalanche events, and prediction of the spatial extent of self-organized macrocolumnar structures. As a first-order exemplar of the method, we recover nonlinear corrections for a coarse-grained Wilson spiking neuron embedded in a network of identical diffusively coupled neurons whose chemical synapses have been disabled. Intriguingly, we find that reblocking transforms the original type 1 Wilson integrator into a type 2 resonator whose spike-rate transfer function exhibits abrupt spiking onset with near-vertical takeoff and chaotic dynamics just above threshold.

Electroclinic measurements are reported for two chiral liquid crystals above their bulk chiral isotropic–nematic phase transition temperatures. It is found that an applied electric field $E$ induces a rotation θ $[\propto {\rm E}]$ of the director in the very thin paranematic layers that are induced by the cell's two planar-aligning substrates. The magnitude of the electroclinic coefficient $d\theta /\mathit{dE}$ close to the transition temperature is comparable to that of a bulk chiral nematic, as well as to that of a parasmectic region above a bulk isotropic-to-chiral smectic-$A$ phase. However, $d\theta /\mathit{dE}$ in the paranematic layer varies much more slowly with temperature than in the parasmectic phase, and its relaxation time is slower by more than three orders of magnitude than that of the bulk chiral nematic electroclinic effect.

Equilibrium and nonequilibrium molecular dynamics simulations were conducted in order to evaluate the hypothesis that the hydrodynamic slip length is a surface property. The system under investigation was water confined between two graphite layers to form nanochannels of different sizes (3–8 nm). The water-carbon interaction potential was calibrated by matching wettability experiments of graphitic-carbon surfaces free of airborne hydrocarbon contamination. Three equilibrium theories were used to calculate the hydrodynamic slip length. It was found that one of the recently reported equilibrium theories for the calculation of the slip length featured confinement effects, while the others resulted in calculations significantly hindered by the large margin of error observed between independent simulations. The hydrodynamic slip length was found to be channel-size independent using equilibrium calculations, i.e., suggesting a consistency with the definition of a surface property, for 5-nm channels and larger. The analysis of the individual trajectories of liquid particles revealed that the reason for observing confinement effects in 3-nm nanochannels is the high mobility of the bulk particles. Nonequilibrium calculations were not consistently affected by size but by noisiness in the smallest systems.