The long-time behavior of the first moment for the output signal of a fractional oscillator with fluctuating frequency subjected to an external periodic force is considered. Colored fluctuations of the oscillator eigenfrequency are modeled as a dichotomous noise. The viscoelastic type friction kernel with memory is assumed as a power-law function of time. Using the Shapiro-Loginov formula, exact expressions for the response to an external periodic field and for the complex susceptibility are presented. On the basis of the exact formulas it is demonstrated that interplay of colored noise and memory can generate a variety of cooperation effects, such as multiresonances versus the driving frequency and the friction coefficient as well as stochastic resonance versus noise parameters. The necessary and sufficient conditions for the cooperation effects are also discussed. Particularly, two different critical memory exponents have been found, which mark dynamical transitions in the behavior of the system.

While the density profile of a granular shock front can be obtained by the conventional treatment of supersonic fluids, its temperature profile is very different from that in ordinary shocks. We study the density and temperature profiles of a traveling granular shock generated by piling up metal spheres in a closed bottom quasi-two-dimensional channel. We successfully account for the temperature profile in the granular shock using a simple kinetic theory in terms of energy transfer from the mean flow direction to the transverse direction. Contrary to ordinary fluids and previous granular shock experiments, the granular shock width is found to increase with the inflow rate.

The stochastic opening and closing of voltage-gated ion channels produce noise in neurons. The effect of this noise on the neuronal performance has been modeled using either an approximate or Langevin model based on stochastic differential equations or an exact model based on a Markov process model of channel gating. Yet whether the Langevin model accurately reproduces the channel noise produced by the Markov model remains unclear. Here we present a comparison between Langevin and Markov models of channel noise in neurons using single compartment Hodgkin-Huxley models containing either ${\text{Na}}^{+}$ and ${\text{K}}^{+}$, or only ${\text{K}}^{+}$ voltage-gated ion channels. The performance of the Langevin and Markov models was quantified over a range of stimulus statistics, membrane areas, and channel numbers. We find that in comparison to the Markov model, the Langevin model underestimates the noise contributed by voltage-gated ion channels, overestimating information rates for both spiking and nonspiking membranes. Even with increasing numbers of channels, the difference between the two models persists. This suggests that the Langevin model may not be suitable for accurately simulating channel noise in neurons, even in simulations with large numbers of ion channels.

Directional control of microtubule shuttles via microfabricated tracks is key to the development of controlled nanoscale mass transport by kinesin motor molecules. Here we develop and test a model to quantitatively predict the stochastic behavior of microtubule guiding when they mechanically collide with the sidewalls of lithographically patterned tracks. By taking into account appropriate probability distributions of microscopic states of the microtubule system, the model allows us to theoretically analyze the roles of collision conditions and kinesin surface densities in determining how the motion of microtubule shuttles is controlled. In addition, we experimentally observe the statistics of microtubule collision events and compare our theoretical prediction with experimental data to validate our model. The model will direct the design of future hybrid nanotechnology devices that integrate nanoscale transport systems powered by kinesin-driven molecular shuttles.

The determination of mean first-passage time (MFPT) for random walks in networks is a theoretical challenge, and is a topic of considerable recent interest within the physics community. In this paper, according to the known connections between MFPT, effective resistance, and the eigenvalues of graph Laplacian, we first study analytically the MFPT between all node pairs of a class of growing treelike networks, which we term deterministic uniform recursive trees (DURTs), since one of its particular cases is a deterministic version of the famous uniform recursive tree. The interesting quantity is determined exactly through the recursive relation of the Laplacian spectra obtained from the special construction of DURTs. The analytical result shows that the MFPT between all couples of nodes in DURTs varies as $N\text{ln}N$ for large networks with node number $N$. Second, we study trapping on a particular network of DURTs, focusing on a special case with the immobile trap positioned at a node having largest degree. We determine exactly the average trapping time (ATT) that is defined as the average of FPT from all nodes to the trap. In contrast to the scaling of the MFPT, the leading behavior of ATT is a linear function of $N$. Interestingly, we show that the behavior for ATT of the trapping problem is related to the trapping location, which is in comparison with the phenomenon of trapping on fractal T-graph although both networks exhibit tree structure. Finally, we believe that the methods could open the way to exactly calculate the MFPT and ATT in a wide range of deterministic media.

We study the decay of survival probability at quantum phase transitions with infinitely degenerate ground levels at critical points. For relatively long times, the semiclassical theory predicts power-law decay of the survival probability in systems with $d=1$ and exponential decay in systems with sufficiently large $d$, where $d$ is the degrees of freedom of the classical counterpart of the system. The predictions are checked numerically in four models.

The motion of an inertial ellipsoid in a creeping linear shear flow of a Newtonian fluid is studied numerically. This constitutes a fundamental system that is used as a basis for simulations and analysis of flows with heavy nonspherical particles. The torque on the ellipsoid is given analytically by Jeffery [Proc. R. Soc. London, Ser. A 102, 161 (1922)]. This torque is coupled with the angular-momentum equation for the particle. The motion is then governed by the Stokes number $\text{St}={\rho }_e\dot{\gamma }{l}^2/\mu$, where ${\rho }_e$ is the density of the ellipsoid, $\dot{\gamma }$ is the rate of shear, $l$ is the length of the major axis of the ellipsoid, and $\mu$ is the dynamic viscosity of the fluid. For low St (the numerical value depends on the aspect ratio of the particle), the particle motion is similar to the Jeffery orbits obtained for inertia-free particles with the addition of an orbit drift so that the particle eventually lies in the flow-gradient plane. At higher St, more drastic effects are seen. For particles oriented in the flow-gradient plane, the rotation rate increases rather abruptly to half the shear rate in a narrow range of St. For particles with other orientations, the motion goes from a kayaking motion to rotation around an oblique axis. It is suggested that, depending on aspect and density ratios, particle inertia might be sufficient to explain and model orbit drift observed previously at low Reynolds numbers. It is discussed how and when the assumption of negligible fluid inertia and strong particle inertia can be justified from a fundamental perspective for particles of different aspect ratios.

We report the generation of complex toroidal nanoscale vortices in nanochannel flows, through the employment of patterned surface patches with contrasting wettability characteristics. Through extensive molecular-dynamics simulations, we further delineate that complicated interactions between the wettability gradient-driven vortex structures and surface roughness-influenced flow features may give rise to interesting patterns in the flow field. We also establish the orientation specificities of these nanoscale vortices, and demonstrate that the surface roughness effects may potentially act to alter the flow rotationalities realized out of modulated surface patterning effects to a considerable extent, which may have interesting consequences toward designing nanoscale mixing elements.

We extend our earlier model for Rayleigh-Taylor and Richtmyer-Meshkov instabilities to the more general class of hydrodynamic instabilities driven by a time-dependent acceleration $g\left(t\right)$. Explicit analytic solutions for linear as well as nonlinear amplitudes are obtained for several $g\left(t\right)$s by solving a Schrödinger-like equation $d^2\eta /d{t}^2-g\left(t\right)kA\eta =0$, where $A$ is the Atwood number and $k$ is the wave number of the perturbation amplitude $\eta \left(t\right)$. In our model a simple transformation $k\to k_L$ and $A\to A_L$ connects the linear to the nonlinear amplitudes: ${\eta }^{nonlinear}\left(k,A\right)\sim \left(1/k_L\right)\text{ln}{\eta }^{linear}\left(k_L,A_L\right)$. The model is found to be in very good agreement with direct numerical simulations. Bubble amplitudes for a variety of accelerations are seen to scale with $s$ defined by $s=\int \sqrt{g\left(t\right)}dt$, while spike amplitudes prefer scaling with displacement $\Delta x=\int \left[\int g\left(t\right)dt\right]dt$.

We generalize the Flory-Stockmayer theory of percolation to a model of associating (patchy) colloids, which consists of hard spherical particles, having on their surfaces $f$ short-ranged-attractive sites of $m$ different types. These sites can form bonds between particles and thus promote self-assembly. It is shown that the percolation threshold is given in terms of the eigenvalues of a $m\times m$ matrix, which describes the recursive relations for the number of bonded particles on the $i\text{th}$ level of a cluster with no loops; percolation occurs when the largest of these eigenvalues equals unity. Expressions for the probability that a particle is not bonded to the giant cluster, for the average cluster size and the average size of a cluster to which a randomly chosen particle belongs, are also derived. Explicit results for these quantities are computed for the case $f=3$ and $m=2$. We show how these structural properties are related to the thermodynamics of the associating system by regarding bond formation as a (equilibrium) chemical reaction. This solution of the percolation problem, combined with Wertheim’s thermodynamic first-order perturbation theory, allows the investigation of the interplay between phase behavior and cluster formation for general models of patchy colloids.

We measured the splay $\left(K_{11}\right)$ and bend $\left(K_{33}\right)$ elastic constants in the nematic phase of a bent-core liquid crystal. In the vicinity of the nematic-isotropic transition temperature $K_{33}$ is proportional to the square of the order parameter. In the nematic range $K_{11}$ increases monotonically with decreasing temperature, whereas $K_{33}$ is practically independent of temperature and is smaller than $K_{11}$. $K_{33}$ exhibits a pretransitional slow divergence toward the transition temperature to the smectic phase and becomes slightly larger than $K_{11}$. The small $K_{33}$ is explained on the basis of strong coupling of the bent shape of the molecules with the bend distortion.

We present a simple model of a free-energy transducer made of allosterically coupled two ratchet subsystems. Each of the subsystems transports particles from one particle reservoir to another. The coupling of the subsystems imposes correlated transitions of the potential profiles of the two subsystems. As a result, a downhill flux in one subsystem with higher chemical-potential difference drives an uphill flux in the other subsystem with lower chemical-potential difference. The direction of the driven flux inverts depending on the direction of the driving flux. The ratio between the fluxes conveyed by the two subsystems is variable and nonstoichiometric. By selecting appropriate parameters, the maximum ratio of the driven flux to driving flux and maximum free-energy transducing efficiency reaches some 90 and 40%, respectively. At a stalled state, the driven flux vanishes while the driving flux remains finite. The allosteric model enables explicit analysis of the timing between binding-unbinding of particles and transitions of potential profile. The behavior of the model is similar to but different from that of the alternate access model, which is a biochemical model for active transport proteins. Our model works also as a regulatory system. We suggest that the correlated transitions of the subsystems (subunits or domains) through allosteric interaction are the origin of the diverse functions of the protein machineries.

We consider the continuous-time random walk of a particle in a two-dimensional self-affine quenched random potential of Hurst exponent $H>0$. The corresponding master equation is studied via the strong disorder renormalization procedure introduced in Monthus and Garel [J. Phys. A: Math. Theor. 41, 255002 (2008)]. We present numerical results on the statistics of the equilibrium time $t_{\text{eq}}$ over the disordered samples of a given size $L\times L$ for $10\le L\le 80$. We find an “infinite disorder fixed point,” where the equilibrium barrier ${\Gamma }_{\text{eq}}\equiv \text{ln}{t}_{\text{eq}}$ scales as ${\Gamma }_{\text{eq}}=L^{H}u$ where $u$ is a random variable of order $O\left(1\right)$. This corresponds to a logarithmically slow diffusion $\left|\vec{r}\left(t\right)-\vec{r}\left(0\right)\right|\sim {\left(\text{ln}t\right)}^{1/H}$ for the position $\vec{r}\left(t\right)$ of the particle.

Neuronal networks interact via spike trains. How the spike trains are transformed by neuronal networks is critical for understanding the underlying mechanism of information processing in the nervous system. Both the rate and synchrony of the spikes can affect the transmission, while the relationship between them has not been fully understood. Here we investigate the mapping between input and output spike trains of a neuronal network in terms of firing rate and synchrony. With large enough input rate, the working mode of the neurons is gradually changed from temporal integrators into coincidence detectors when the synchrony degree of input spike trains increases. Since the membrane potentials of the neurons can be depolarized to near the firing threshold by uncorrelated input spikes, small input synchrony can cause great output synchrony. On the other hand, the synchrony in the output may be reduced when the input rate is too small. The case of the feedforward network can be regarded as iterative process of such an input-output relationship. The activity in deep layers of the feedforward network is in an all-or-none manner depending on the input rate and synchrony.

Traffic is essential for many dynamic processes on networks. The efficient routing strategy [G. Yan, T. Zhou, B. Hu, Z. Q. Fu, and B. H. Wang, Phys. Rev. E 73, 046108 (2006)] can reach a very high capacity of more than ten times of that with shortest path strategy. In this paper, we propose a global dynamic routing strategy for network systems based on the information of the queue length of nodes. Under this routing strategy, the traffic capacity is further improved. With time delay of updating node queue lengths and the corresponding paths, the system capacity remains constant, while the travel time for packets increases.

This paper presents a general analysis and a concrete example of the catastrophic case of a discontinuity-induced bifurcation in so-called Filippov nonsmooth dynamical systems. Such systems are characterized by discontinuous jumps in the right-hand sides of differential equations across a phase space boundary and are often used as physical models of stick-slip motion and relay control. Sliding bifurcations of periodic orbits have recently been shown to underlie the onset of complex dynamics including chaos. In contrast to previously analyzed cases, in this work a periodic orbit is assumed to graze the boundary of a repelling sliding region, resulting in its abrupt destruction without any precursive change in its stability or period. Necessary conditions for the occurrence of such catastrophic grazing-sliding bifurcations are derived. The analysis is illustrated in a piecewise-smooth model of a stripline resonator, where it can account for the abrupt onset of self-modulating current fluctuations. The resonator device is based around a ring of NbN containing a microbridge bottleneck, whose switching between normal and super conducting states can be modeled as discontinuous, and whose fast temperature versus slow current fluctuations are modeled by a slow-fast timescale separation in the dynamics. By approximating the slow component as Filippov sliding, explicit conditions are derived for catastrophic grazing-sliding bifurcations, which can be traced out as parameters vary. The results are shown to agree well with simulations of the slow-fast model and to offer a simple explanation of one of the key features of this experimental device.

In this work, we extend previous analyses of ac electro-osmosis to account for the combined action of two experimentally relevant effects: (i) Faradaic currents from electrochemical reactions at the electrodes and (ii) differences in ion mobilities of the electrolyte. In previous works, the ac electro-osmotic motion has been analyzed theoretically under the assumption that only forces in the diffuse (Debye) layer are relevant. Here, we first show that if the ion mobilities of a 1-1 aqueous solution are different, the charged zone expands from the Debye layer to include the diffusion layer. We later include the Faradaic currents and, as an attempt to explore both factors simultaneously, we perform a thin-layer, low-frequency, linear analysis of the system. Finally, the model is applied to the case of an electrolyte actuated by a traveling-wave signal. A steady liquid motion in opposite direction to the applied signal is predicted for some ranges of the parameters. This could serve as a partial explanation for the observed flow reversal in some experiments.

We consider the dynamics of dilute monodisperse bubbly liquid confined by two plane solid walls and subject to small-amplitude high-frequency oscillations normal to the walls. The initial state corresponds to the uniform distribution of bubbles and motionless liquid. The period of external driving is assumed much smaller than typical relaxation times for a single bubble but larger than the period of volume eigenoscillations. The time-averaged description accounting for the two-way coupling between the liquid and the bubbles is applied. We show that the model predicts accumulation of bubbles in thin sheets parallel to the walls. These singular structures, which are formally characterized by infinitely thin width and infinitely high concentration, are referred to as bubbly screens. The formation of a bubbly screen is described analytically in terms of a self-similar solution, which is in agreement with numerical simulations. We study the evolution of bubbly screens and detect a one-dimensional stationary state, which is shown to be unconditionally unstable.

This paper studies the spatial structure of decaying chemical fields generated by a chaotic-advection flow and maintained by a spatially smooth chemical source. Previous work showed that in a regime where diffusion can be neglected (large Péclet number), the structures are filamental or smooth depending on the relative strength of the chemical dynamics and the stirring induced by the flow. The scaling exponent, ${\gamma }_q$, of the $q\text{th}$-order structure function depends, at leading order, linearly on the ratio of the rate of decay of the chemical processes, $\alpha$, and the average rate of divergence of neighboring fluid parcel trajectories (Lyapunov exponent), $\overline{h}$. Under a homogeneous stretching approximation, ${\gamma }_q/q=\text{max}\left\{\alpha /\overline{h},1\right\}$ which implies that a well-defined filamental-smooth transition occurs at $\alpha =\overline{h}$. This approximation has been improved by using the distribution of finite-time Lyapunov exponents to characterize the inhomogeneous stretching of the flow. However, previous work focused more on the behavior of the exponents as $q$ varies and less on the effects of $\alpha$ and hence the implications for the filamental-smooth transition. Here we set out the precise relation between the stretching rate statistics and the scaling exponents and emphasize that the latter are determined by the distribution of the finite-size (rather than finite-time) Lyapunov exponents. We clarify the relation between the two distributions. We show that the corrected exponents, ${\tilde{\gamma }}_q$, depend nonlinearly on $\alpha$ with ${\tilde{\gamma }}_q<{\gamma }_q$ for ${\tilde{\gamma }}_q<q$. The magnitude of the correction to the homogeneous stretching approximation, ${\tilde{\gamma }}_q-{\gamma }_q$, grows as $\alpha$ increases, reaching a maximum when the leading-order transition is reached $\left(\alpha =\overline{h}\right)$. The implication of these results is that there is no well-defined bulk filamental-smooth transition. Instead it is the case that the chemical field is unambiguously smooth for $\alpha >h_{\text{max}}$, where $h_{\text{max}}$ denotes the maximum finite-time Lyapunov exponent and unambiguously filamental for $\alpha <\overline{h}$, with an intermediate character for $\alpha$ between these two values. Theoretical predictions are confirmed by numerical results obtained for a linearly decaying chemistry coupled to a renewing type of flow together with careful calculations of the Crámer function.

We propose a simple thermal diode passively acting as a rectifier of heat current. The key design of the diode is the size asymmetry of the areas in contact with two distinct heat baths. The heat-conducting medium is liquid, inside of which gaslike regions are induced depending on the applied conditions. Simulating nanoscale systems of this diode, the rectification of heat current is demonstrated. If the packing density of the medium and the working regime of temperature are properly chosen, the heat current is effectively cut off when the heat bath with narrow contact is hotter, but it flows normally under opposite temperature conditions. In the former case, the gaslike region is induced in the system and it acts as a thermal insulator because it covers the entire narrow area of contact with the bath.