We study a model for the motion of a tracer particle inside an active gel, exposing the properties of the van Hove distribution of the particle displacements. Active events of a typical force magnitude can give rise to non-Gaussian distributions having exponential tails or side peaks. The side peaks are predicted to appear when the local bulk elasticity of the gel is large enough and few active sources are dominant. We explain the regimes of the different distributions and study the structure of the side peaks for active sources that are susceptible to the elastic stress that they cause inside the gel. We show how the van Hove distribution is altered by both the duty cycle of the active sources and their susceptibility, and suggest it as a sensitive probe to analyze microrheology data in active systems with restoring elastic forces.

We consider $n$ particles diffusing freely in a domain. The boundary contains absorbing escape regions, where the particles can escape, and traps, where the particles can be captured. Modeled after biological examples such as receptors in the synaptic cleft and ambush predators waiting for prey, these traps, or capture regions, must recharge between captures. We are interested in characterizing the time courses of the number of particles remaining in the domain, the number of cumulative captures, and the number of available capture regions. We find that under certain conditions, the number of cumulative captures increases linearly in time with a slope and duration determined explicitly by the recharge rate of the capture regions. This recharge rate also determines the mean and variance of the clearance time, defined as the time it takes for all particles to leave the domain. Further, we find that while a finite recharge rate will always result in a lower number of captured particles when compared to instantaneous recharging, it can either increase or decrease the amount of variability. Lastly, we extend the model to partially absorbing traps in order to investigate the dynamics of receptor activation within an idealized synaptic cleft. We find that the width of the domain controls the amount of time that these receptors are activated, while the number of receptors controls the amplitude of activation. Our mathematical results are derived from considering this system in several ways: as a full spatial diffusion process with recharging traps, as a continuous-time Markov process on a discrete state space, and as a system of ordinary differential equations in a mean-field approximation.

We study mode locking in a canonical model of gradient frequency neural networks under periodic forcing. The canonical model is a generic mathematical model for a network of nonlinear oscillators tuned to a range of distinct frequencies. It is mathematically more tractable than biological neuron models and allows close analysis of mode-locking behaviors. Here we analyze individual modes of synchronization for a periodically forced canonical model and present a complete set of driven behaviors for all parameter regimes available in the model. Using a closed-form approximation, we show that the Arnold tongue (i.e., locking region) for $k:m$ synchronization gets narrower as $k$ and $m$ increase. We find that numerical simulations of the canonical model closely follow the analysis of individual modes when forcing is weak, but they deviate at high forcing amplitudes for which oscillator dynamics are simultaneously influenced by multiple modes of synchronization.

Chirality is an essential evolutionary-conserved physical aspect of swimming microorganisms. However, the role of chirality on the hydrodynamics of such microswimmers is still being elucidated. Hydrodynamic theories have so far predicted that, under a torque-free condition satisfied in the system of microswimmers, a rotlet dipole generating a twisting flow is the leading-order singularity of the chiral flow field. Nevertheless, such a chiral flow field has never been experimentally detected. Here we explore a hydrodynamic field generated in a system of a chiral microswimmer, where a droplet of a cholesteric liquid crystal (CLC) exhibits helical and spinning motions in surfactant solutions due to a chiral nonequilibrium cross coupling between the rotation and the Marangoni flow. Combining measurement of the flow field around the spinning CLC droplets and a computational flow modeling, we revealed that the CLC droplets generate a flow field of a rotlet dipole. Remarkably, we found that the chiral component of the flow field decays with distance $r$ as $r^{-3}$, which is consistent with the theoretical prediction for the flow field produced by a point singularity of a rotlet dipole. Our findings will promote the understanding of roles of chirality on the hydrodynamics in active matter as well as liquid crystals.

We consider Rayleigh-Taylor and Richtmyer-Meshkov instabilities at the interface between two fluids, one or both of which may be viscous. We derive exact analytic expressions for the amplitude $\eta \left(t\right)$ in the linear regime when only one of the fluids is viscous. The more general case is solved numerically using Laplace transforms. We compare the exact solutions of the initial-value problem with the approximate solutions of the eigenvalue problem used in a simple expression for $\eta \left(t\right)$ in terms of two growth rates, ${\gamma }_{+}$ and ${\gamma }_{-}$. We then propose a hybrid model as an improvement on the approximate model. The hybrid model is based on the same expression for $\eta \left(t\right)$ in terms of ${\gamma }_{\pm }$ but uses exact eigenvalues for ${\gamma }_{+}$, the larger growth rate, and a relationship between ${\gamma }_{-}$ and ${\gamma }_{+}$. We also discuss two concepts: isogrowth wave number pairs and asymptotic decay. The first relies on viscosity in one or both fluids to identify perturbations of two different wavelengths having the same ${\gamma }_{+}$. The second concept, which is more general, can be found in viscous as well as inviscid fluids and requires only a specific initial growth rate ${\dot{\eta }}_0^{\mathrm{critical}}$ to force $\eta \left(t\right)\to 0$ as $t\to \infty$. We present several examples illustrating these two concepts and comparing exact, approximate, and hybrid treatments.

Smoldering is the slow, low-temperature, flameless burning of porous fuels and the most persistent type of combustion phenomena. It is a complex physical process that is not yet completely understood, but it is known that it is driven by heat transfer, mass transfer, and fuel chemistry. A specific case of high interest and complexity is fingering behavior. Fingering is an instability that occurs when a thin fuel layer burns against an oxygen current. These instabilities appear when conduction rather than convection is the dominant mode of heat transfer to the fuel ahead and the availability of oxygen is limited during the combustion of a thin fuel, such as paper. The pattern of the fingers can be characterized through the distance between them and their width, and can be classified into three different regimes: isolated fingers, tip-splitting fingers, or no fingers forming and a smooth continuous front. In this paper, a multilayer cellular automaton based on three governing principles (heat, oxygen, and fuel) is shown to reproduce all the regimes and the details of finger structures observed in previous experiments. It is shown how when oxygen is not limited, a smooth smoldering front is formed. If the oxygen speed decreases beyond a critical value, fingers appear first as tip-splitting fingers and later as isolated fingers, increasing the distance between them and decreasing their thickness. The oxygen consumed during oxidation influences these critical values with a positive correlation. This cellular automaton provides an alternative approach to simulate smoldering combustion in large systems over long times. That the model is able to reproduce the complex pattern formation seen in a fingering experiment validates the model. In the future, we could apply the model in various other geometries to make predictions on the outcome of smoldering combustion processes.

Using tools from large deviation theory, we study fluctuations of the heat current in a model of $d$-dimensional incompressible fluid driven out of equilibrium by a temperature gradient. We find that the most probable temperature fields sustaining atypical values of the global current can be naturally classified in an infinite set of curves, allowing us to exhaustively analyze their topological properties and to define universal profiles onto which all optimal fields collapse. We also compute the statistics of empirical heat current, where we find remarkable logarithmic tails for large current fluctuations orthogonal to the thermal gradient. Finally, we determine explicitly a number of cumulants of the current distribution, finding interesting relations between them.

Nonlinear interactions in focusing media between traveling solitons and the dispersive shocks produced by an initial discontinuity are studied using the one-dimensional nonlinear Schrödinger equation. It is shown that, when solitons travel from a region with nonzero background toward a region with zero background, they always pass through the shock structure without generating dispersive radiation. However, their properties (such as amplitude, velocity, and shape) change in the process. A similar effect arises when solitons travel from a region with zero background toward a region with nonzero background, except that, depending on its initial velocity, in this case the soliton may remain trapped inside the shocklike structure indefinitely. In all cases, the new soliton properties can be determined analytically. The results are validated by comparison with numerical simulations.

We consider a possibility to realize self-accelerating motion of interacting states with effective positive and negative masses in the form of pairs of solitons in two-component BEC loaded in an optical-lattice (OL) potential. A crucial role is played by the fact that gap solitons may feature a negative dynamical mass, keeping their mobility in the OL. First, the respective system of coupled Gross-Pitaevskii equations (GPE) is reduced to a system of equations for envelopes of the lattice wave functions. Two generic dynamical regimes are revealed by simulations of the reduced system, viz., shuttle oscillations of pairs of solitons with positive and negative masses, and splitting of the pair. The coaccelerating motion of the interacting solitons, which keeps constant separation between them, occurs at the boundary between the shuttle motion and splitting. The position of the coacceleration regime in the system's parameter space can be adjusted with the help of an additional gravity potential, which induces its own acceleration, that may offset the relative acceleration of the two solitons, while gravity masses of both solitons remain positive. The numerical findings are accurately reproduced by a variational approximation. Collisions between shuttling or coaccelerating soliton pairs do not alter the character of the dynamical regime. Finally, regimes of the shuttle motion, coacceleration, and splitting are corroborated by simulations of the original GPE system, with the explicitly present OL potential.

We introduce a $k$-leaf removal algorithm as a generalization of the so-called leaf removal algorithm. In this pruning algorithm, vertices of degree smaller than $k$, together with their first nearest neighbors and all incident edges, are progressively removed from a random network. As the result of this pruning the network is reduced to a subgraph which we call the Generalized $k$-core ($Gk$-core). Performing this pruning for the sequence of natural numbers $k$, we decompose the network into a hierarchy of progressively nested $Gk$-cores. We present an analytical framework for description of $Gk$-core percolation for undirected uncorrelated networks with arbitrary degree distributions (configuration model). To confirm our results, we also derive rate equations for the $k$-leaf removal algorithm which enable us to obtain the structural characteristics of the $Gk$-cores in another way. Also we apply our algorithm to a number of real-world networks and perform the $Gk$-core decomposition for them.

Raman spectra in the low-frequency spectral range—between 5 and $90\mathrm{c}{\mathrm{m}}^{-1}$—were studied for multilamellar bilayers prepared with cholesterol (Chol) and phospholipids of three different types: doubly unsaturated lipids 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC), monounsaturated lipids 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC), and fully saturated lipids 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC). The narrow peak seen below 250 K and positioned between 9 and $18\mathrm{c}{\mathrm{m}}^{-1}$—depending on the system and temperature—was attributed to the vibrational eigenmode of a lipid monolayer. For the DOPC-Chol bilayer, the peak position and the peak width were found to monotonically increase and decrease, respectively, with the Chol concentration. For POPC-Chol and DMPC-Chol bilayers, these parameters revealed nonmonotonic concentration dependences, with an apparent minimum at the intermediate Chol content. The peak intensity was ascribed to interleaflet coupling. As in the literature, a coexistence of liquid-ordered and solid-ordered domains was suggested for the DMPC-Chol and POPC-Chol bilayers; the Chol concentration dependences of Raman peak parameters were discussed in line with this suggestion, under the assumption that the different composition of coexisting domains conserves upon cooling. We demonstrated that the obtained Raman data disagree with the suggested domain coexistence if the domain sizes are substantially larger than the lipid layer thickness.

In many natural and artificial devices diffusive transport takes place in confined geometries with corrugated boundaries. Such boundaries cause both entropic and hydrodynamic effects, which have been studied only for the case of spherical particles. Here we experimentally investigate the diffusion of particles of elongated shape confined in a corrugated quasi-two-dimensional channel. The elongated shape causes complex excluded-volume interactions between particles and channel walls which reduce the accessible configuration space and lead to novel entropic free-energy effects. The extra rotational degree of freedom also gives rise to a complex diffusivity matrix that depends on both the particle location and its orientation. We further show how to extend the standard Fick-Jacobs theory to incorporate combined hydrodynamic and entropic effects, so as, for instance, to accurately predict experimentally measured mean first passage times along the channel. Our approach can be used as a generic method to describe translational diffusion of anisotropic particles in corrugated channels.

A replica-symmetry-breaking phase transition is predicted in a host of disordered media. The criticality of the transition has, however, long been questioned below its upper critical dimension, six, due to the absence of a critical fixed point in the renormalization-group flows at one-loop order. A recent two-loop analysis revealed a possible strong-coupling fixed point, but given the uncontrolled nature of perturbative analysis in the strong-coupling regime, debate persists. Here we examine the nature of the transition as a function of spatial dimension and show that the strong-coupling fixed point can go through a Hopf bifurcation, resulting in a critical limit cycle and a concomitant discrete scale invariance. We further investigate a different renormalization scheme and argue that the basin of attraction of the strong-coupling fixed point (or limit cycle) may stay finite for all dimensions.

We analyze the heat exchange distribution of open quantum systems undergoing a thermal relaxation process with a time-dependent effective temperature. We show that such processes arise, for example, if the dynamics maximizes the entropy production. Using a two-point measurement scheme, we find an expression for the heat moment generating function that depends solely on the system's partition function and on the thermalization function (i.e., the law of cooling) describing the effective temperature. Applications include the relaxation of free bosonic and fermionic modes, for which closed-form expressions for the time-dependent heat distribution function are derived. Multiple free modes with arbitrary dispersion relations are also briefly discussed. In the semiclassical limit our formula agrees with previous results of the literature for the heat distribution of an optically trapped nanoscopic particle far from equilibrium.

Motivated by several biological models such as the SIS model from epidemiology and the Tuckwell-Miura model describing cortical spreading depression, we investigate the types of wave solutions that can exist for reaction-diffusion systems of two equations in which the reaction terms are degenerate in the sense that they are linearly dependent. In particular, we show that there are surprising differences between the types of waves that occur in a single reaction-diffusion equation and the types of waves that occur in a degenerate system of two equations. Importantly, and in contrast to previously published results, we demonstrate that nonstationary pulse solutions can exist for a degenerate system of two equations but cannot exist for a single reaction-diffusion equation. We show that this has important consequences for the minimal model that can generate the types of waves observed in cortical spreading depression. On the other hand, stationary fronts can exist for both single reaction-diffusion equations and degenerate systems. However, for degenerate systems, such solutions cannot be accessed when perturbing a uniform rest state with a localized perturbation unless the diffusion coefficients of the two species are equal. We also give an explicit condition on the source term in a degenerate reaction-diffusion system that guarantees the existence of nonstationary and stationary pulse and front solutions. We use this approach to provide several examples of reaction terms that have analytical pulse and front solutions. We also show that the case in which one species cannot diffuse is singular in the sense that the degenerate reaction-diffusion system can admit infinite families of stationary piecewise constant solutions. We further show how such solutions can be accessed by perturbing a constant rest state with a localized continuous disturbance.

K-core and bootstrap percolation are widely studied models that have been used to represent and understand diverse deactivation and activation processes in natural and social systems. Since these models are considerably similar, it has been suggested in recent years that they could be complementary. In this manuscript we provide a rigorous analysis that shows that for any degree and threshold distributions heterogeneous bootstrap percolation can be mapped into heterogeneous k-core percolation and vice versa, if the functionality thresholds in both processes satisfy a complementary relation. Another interesting problem in bootstrap and k-core percolation is the fraction of nodes belonging to their giant connected components $P_{\infty b}$ and $P_{\infty c}$, respectively. We solve this problem analytically for arbitrary randomly connected graphs and arbitrary threshold distributions, and we show that $P_{\infty b}$ and $P_{\infty c}$ are not complementary. Our theoretical results coincide with computer simulations in the limit of very large graphs. In bootstrap percolation, we show that when using the branching theory to compute the size of the giant component, we must consider two different types of links, which are related to distinct spanning branches of active nodes.

Dense packing of hydrophobic residues in the cores of globular proteins determines their stability. Recently, we have shown that protein cores possess packing fraction $\varphi \approx 0.56$, which is the same as dense, random packing of amino-acid-shaped particles. In this article, we compare the structural properties of protein cores and jammed packings of amino-acid-shaped particles in much greater depth by measuring their local and connected void regions. We find that the distributions of surface Voronoi cell volumes and local porosities obey similar statistics in both systems. We also measure the probability that accessible, connected void regions percolate as a function of the size of a spherical probe particle and show that both systems possess the same critical probe size. We measure the critical exponent $\tau$ that characterizes the size distribution of connected void clusters at the onset of percolation. We find that the cluster size statistics are similar for void percolation in packings of amino-acid-shaped particles and randomly placed spheres, but different from that for void percolation in jammed sphere packings. We propose that the connected void regions are a defining structural feature of proteins and can be used to differentiate experimentally observed proteins from decoy structures that are generated using computational protein design software. This work emphasizes that jammed packings of amino-acid-shaped particles can serve as structural and mechanical analogs of protein cores, and could therefore be useful in modeling the response of protein cores to cavity-expanding and -reducing mutations.

First passage in a stochastic process may be influenced by the presence of an external confining potential, as well as “stochastic resetting” in which the process is repeatedly reset back to its initial position. Here, we study the interplay between these two strategies, for a diffusing particle in a one-dimensional trapping potential $V\left(x\right)$, being randomly reset at a constant rate $r$. Stochastic resetting has been of great interest as it is known to provide an “optimal rate” ($r_{*}$) at which the mean first passage time is a minimum. On the other hand, an attractive potential also assists in the first capture process. Interestingly, we find that for a sufficiently strong external potential, the advantageous optimal resetting rate vanishes (i.e., $r_{*}\to 0$). We derive a condition for this optimal resetting rate vanishing transition, which is continuous. We study this problem for various functional forms of $V\left(x\right)$, some analytically, and the rest numerically. We find that the optimal rate $r_{*}$ vanishes with a deviation from the critical strength of the potential as a power law with an exponent $\beta$ which appears to be universal.

The second law of thermodynamics requires the overall thermal current to flow from hot to cold. However, it does not forbid a local thermal current from flowing from cold to hot. By coupling a harmonic system of three masses connected by a few springs to two Langevin reservoirs at different temperatures, a local atypical thermal current is found to flow from cold to hot in the steady state while the overall thermal current is still from hot to cold. The direction of the local thermal current can be tuned by the mass, spring constant, and system-reservoir coupling. The local thermal current can vanish if the parameters are tuned to proper values. We also consider nonlinear effect from the system-substrate coupling and find that the local atypical thermal current survives in the presence of the nonlinear potential. Moreover, the local atypical thermal current is robust against asymmetry of the system-reservoir coupling, inhomogeneity of the nonlinear potential, and additions of more masses and springs. In molecular or nanomechanical systems where the setup may find its realization, the direction of the local thermal current may be controlled by mechanical or electromagnetic means, which may lead to applications in information storage.

We study the problem of finding the smallest set of nodes in a network whose removal results in an empty $k$-core, where the $k$-core is the subnetwork obtained after the iterative removal of all nodes of degree smaller than $k$. This problem is also known in the literature as finding the minimal contagious set. The main contribution of our work is an analysis of the performance of the recently introduced corehd algorithm [Zdeborová, Zhang, and Zhou, Sci. Rep. 6, 37954 (2016)] on random graphs taken from the configuration model via a set of deterministic differential equations. Our analyses provide upper bounds on the size of the minimal contagious set that improve over previously known bounds. Our second contribution is a heuristic called the weak-neighbor algorithm that outperforms all currently known local methods in the regimes considered.