Diffusion coefficients are intrinsically random in subdiffusion attributable to power-law trapping. Using deterministic biased and unbiased diffusion models, we investigate the Einstein relation for single trajectories in subdiffusion. The difference in the generalized Lyapunov exponent between biased and unbiased deterministic diffusions is related to the velocity under a bias. By Hopf's ergodic theorem, the ratios between the velocities and the Lyapunov exponents for single trajectories converge to a universal constant, which is proportional to the strength of the bias. Based on a certain transport coefficient obtained from a single trajectory, we provide a relation for the transport coefficients divided by the Lyapunov exponent and generalize the Einstein relation for single trajectories.

We study an evolutionary snowdrift game with the unconditional imitation updating rule on regular lattices. Detailed numerical simulations establish the structure of plateaus and discontinuous jumps of the equilibrium cooperation frequency $f_c$ as a function of the cost-to-benefit ratio $r$. By analyzing the stability of local configurations, it is found that the transitions occur at values of $r$ at which there are changes in the ranking of the payoffs to the different local configurations of agents using different strategies. Nonmonotonic behavior of $f_c\left(r\right)$ at the intermediate range of $r$ is analyzed in terms of the formation of blocks of agents using the cooperative strategy that are stabilized by agents inside the block due to the updating rule. For random initial condition with 50%–50% agents of different strategies randomly dispersed, cooperation persists in the whole range of $r$ and the level of cooperation is higher than that in the well-mixed case in a wide range of $r$. These results are in sharp contrast to those based on the replicator updating rule. The sensitivity to initial states with different fractions of cooperative agents is also discussed. The results serve to illustrate that both the spatial structure and the updating rule are important in determining the level of cooperation in a competing population. When extreme initial states are used where there are very few agents of a strategy in a background of the opposite strategy, the result would depend on the stability of the clusters formed by the initially minority agents.

We present dynamic measurements of a lattice gas during phase separation, which we use as an analogy for self-assembly of equilibrium ordered structures. We use two approaches to quantify the degree of reversibility of this process: First, we count events in which bonds are made and broken; second, we use correlation-response measurements and fluctuation-dissipation ratios to probe reversibility during different time intervals. We show how correlation and response functions can be related directly to microscopic (ir)reversibility and we discuss the time dependence and observable dependence of these measurements, including the role of fast and slow degrees of freedom during assembly.

We report measurements of the pressure profile in the outlet plane of a discharging silo. We observe that, whatever the preparation of the granular system, a dynamic Janssen effect is at play: the apparent mass of the grains (i.e., the part of their mass sustained by the base) is significantly smaller than their actual mass because of the redirection of the weight to the lateral wall of the container. The pressure profiles reveal a significant decrease in pressure in the vicinity of the outlet as the system discharges, whereas the flow rate remains constant. The measurements are thus a direct experimental proof that the flow rates of granular material through an aperture are not controlled by the local stress conditions.

Disordered soft materials, such as fibrous networks in biological contexts, exhibit a nonlinear elastic response. We study such nonlinear behavior with a minimal model for networks on lattice geometries with simple Hookian elements with disordered spring constant. By developing a mean-field approach to calculate the differential elastic bulk modulus for the macroscopic network response of such networks under large isotropic deformations, we provide insight into the origins of the strain stiffening and softening behavior of these systems. We find that the nonlinear mechanics depends only weakly on the lattice geometry and is governed by the average network connectivity. In particular, the nonlinear response is controlled by the isostatic connectivity, which depends strongly on the applied strain. Our predictions for the strain dependence of the isostatic point as well as the strain-dependent differential bulk modulus agree well with numerical results in both two and three dimensions. In addition, by using a mapping between the disordered network and a regular network with random forces, we calculate the nonaffine fluctuations of the deformation field and compare them to the numerical results. Finally, we discuss the limitations and implications of the developed theory.

Animals are able to cope with the noise, uncertainties, and complexity of the real world. Often even elementary living beings, equipped with very limited sensory organs, are able to reach regions favorable to their existence, using simple stochastic policies. In this paper we discuss a minimalistic stochastic behavioral rule, inspired from bacteria chemotaxis, which is able to increase the value of a specified evaluation function in a similar manner. In particular, we prove that, under opportune assumptions, the direction that is taken with maximum probability by an agent that follows this rule corresponds to the optimal direction. The rule does not require a specific agent dynamics, needs no memory for storing observed states, and works in generic $n$-dimensional spaces. It thus reveals itself interesting for the control of simple sensing robots as well.

In this work, a pseudocompressible approximation relevant for turbulent mixing flows encountered in shock tubes is derived. The asymptotic analysis used for this purpose puts forward the role played by four dimensionless numbers on the flow compressibility, namely, the turbulent, deformation, stratification, and buoyancy force Mach numbers. The existence of rapid distortion and diffusion-dissipation regimes is also accounted for in the analysis. Some consequences of the derived pseudocompressible approximation on statistical turbulence models are discussed. In particular, the evolutions of the density variance and flux are examined, as well as the turbulent transport of energy. The different aspects of this study are assessed by performing a direct numerical simulation of a shock tube flow configuration.

Deterministic route to turbulence creation in 2D wall boundary layer is shown here by solving full Navier-Stokes equation by dispersion relation preserving (DRP) numerical methods for flow over a flat plate excited by wall and free stream excitations. Present results show the transition caused by wall excitation is predominantly due to nonlinear growth of the spatiotemporal wave front, even in the presence of Tollmien-Schlichting (TS) waves. The existence and linear mechanism of creating the spatiotemporal wave front was established in Sengupta, Rao and Venkatasubbaiah [Phys. Rev. Lett. 96, 224504 (2006)] via the solution of Orr-Sommerfeld equation. Effects of spatiotemporal front(s) in the nonlinear phase of disturbance evolution have been documented by Sengupta and Bhaumik [Phys. Rev. Lett. 107, 154501 (2011)], where a flow is taken from the receptivity stage to the fully developed 2D turbulent state exhibiting a $k^{-3}$ energy spectrum by solving the Navier-Stokes equation without any artifice. The details of this mechanism are presented here for the first time, along with another problem of forced excitation of the boundary layer by convecting free stream vortices. Thus, the excitations considered here are for a zero pressure gradient (ZPG) boundary layer by (i) monochromatic time-harmonic wall excitation and (ii) free stream excitation by convecting train of vortices at a constant height. The latter case demonstrates neither monochromatic TS wave, nor the spatiotemporal wave front, yet both the cases eventually show the presence of $k^{-3}$ energy spectrum, which has been shown experimentally for atmospheric dynamics in Nastrom, Gage and Jasperson [Nature 310, 36 (1984)]. Transition by a nonlinear mechanism of the Navier-Stokes equation leading to $k^{-3}$ energy spectrum in the inertial subrange is the typical characteristic feature of all 2D turbulent flows. Reproduction of the spectrum noted in atmospheric data (showing dominance of the $k^{-3}$ spectrum over the $k^{-5/3}$ spectrum in Nastrom et al.) in laboratory scale indicates universality of this spectrum for all 2D turbulent flows. Creation of universal features of 2D turbulence by a deterministic route has been established here for the first time by solving the Navier-Stokes equation without any modeling, as has been reported earlier in the literature by other researchers.

The immiscible displacement of oil by water in a circular microchannel was investigated. A fused silica microchannel with an inner diameter of 250 μm and a length of 7 cm was initially filled with a viscous silicone oil. Only water then was injected into the channel. We describe our flow observations based on the two-dimensional images captured in the middle of the channel. The water finger displaced the oil and left an oil film on the channel wall. While the oil was being displaced at the core, the flow resistance decreased, which resulted in increases in water flow rate and inertia. Eventually, the water finger reached the channel exit and formed a core-annular flow pattern. The wavelength of the waves formed at the oil-water interface also increased with the increase in inertia. The initially symmetric interfacial waves became asymmetric with time. Also, the water core shifted from the center of the channel and left a thinner oil film on one side of the microchannel. Under all flow rates tested in this study, as long as the water was continuously injected, the water core was stable and no breakup into droplets was observed. We also discuss the flow stability based on nonlinear and linear stability analyses performed on the core-annular flow. Compared to the linear analysis, which ignores the inertia effects, the nonlinear analysis, which includes the inertia effects, predicts longer interfacial wavelengths by a factor of $\frac{1}{\sqrt{1-\frac{a_o}{2}({\mathrm{We}}_w+{\mathrm{We}}_o\frac{a_o^2}{1-a_o^2})}}$ where ${\mathrm{We}}_w$ and ${\mathrm{We}}_o$ are the Weber numbers of the water and the oil phases, respectively, and $a_o$ is the unperturbed water core radius made dimensionless by the channel radius.

A crucial question recently raised in climate dynamics concerns abrupt climate transitions: Are they due to a tipping point (TP) being exceeded, or is fast noisy dynamics the cause of their excitation? In this respect, a case study based on a low-order ocean model is developed to show that in an excitable dynamical system perturbed by noise (a possible climate condition) the TPs may have limited physical meaning, with the coherence resonance mechanism being predominant. The analysis is based on an operational definition of stochastic TP, which accounts for the effect of noise and reconciles formally the TP and coherence resonance views.

The formalism of Kundu et al. [J. Stat. Mech. P03007 (2011)], for computing the large deviations of heat flow in harmonic systems, is applied to the case of single Brownian particle in a harmonic trap and coupled to two heat baths at different temperatures. The large-$\tau$ form of the moment generating function $\langle e^{-\lambda Q}\rangle \approx g\left(\lambda \right)\exp \left[\tau \mu \left(\lambda \right)\right]$, of the total heat flow $Q$ from one of the baths to the particle in a given time interval $\tau$, is studied and exact explicit expressions are obtained for both $\mu \left(\lambda \right)$ and $g\left(\lambda \right)$. For a special case of the single particle problem that corresponds to the work done by an external stochastic force on a harmonic oscillator coupled to a thermal bath, the large-$\tau$ form of the moment generating function is analyzed to obtain the exact large deviation function as well as the complete asymptotic forms of the probability density function of the work.

Many modern nanostructured materials and doped polymers are morphologically too complex to be interpreted by classical percolation theory. Here, we develop the concept of a hierarchical percolating (percolation-within-percolation) system to describe such complex materials and illustrate how to generalize the conventional percolation to double-level percolation. Based on Monte Carlo simulations, we find that the double-level percolation threshold is close to, but definitely larger than, the product of the local percolation thresholds for the two enclosed single-level systems. The deviation may offer alternative insights into physics concerning infinite clusters and open up new research directions for percolation theory.

The temporal evolution of a dilute granular gas, both in a compressible flow (uniform longitudinal flow) and in an incompressible flow (uniform shear flow), is investigated by means of the direct simulation Monte Carlo method to solve the Boltzmann equation. Emphasis is laid on the identification of a first “kinetic” stage (where the physical properties are strongly dependent on the initial state) subsequently followed by an unsteady “hydrodynamic” stage (where the momentum fluxes are well-defined non-Newtonian functions of the rate of strain). The simulation data are seen to support this two-stage scenario. Furthermore, the rheological functions obtained from simulation are well described by an approximate analytical solution of a model kinetic equation.

A vertical force applied to each of two colloids, trapped at a liquid-air interface, induces their logarithmic pairwise attraction. I recently showed [Phys. Rev. E 79, 011407 (2009)] that in clusters of size $R$ much larger than the capillary length $\lambda$, the attraction changes to that of a power law and is much stronger due to a many-body effect, and I derived two equations that describe the equilibrium coarse-grained meniscus profile and colloid density in such clusters. In this paper, this theory is shown also to describe small clusters with $R\ll$ $\lambda$ provided the number $N$ of colloids therein is sufficiently large. An analytical solution for a small circular cluster with an arbitrary short-range power-law pairwise repulsion is found. The energy of a cluster is obtained as a function of its radius $R$ and colloid number $N$. As in large clusters, the attraction force and energy universally scale with the distance $L$ between colloids as $L^{-3}$ and $L^{-2}$, respectively, for any repulsion forces. The states of an equilibrium cluster, predicted by the theory, are shown to be stable with respect to small perturbations of the meniscus profile and colloid density. The minimum number of colloids in a circular cluster, which sustains the thermal motion, is estimated. For standard parameters, it can be very modest, e.g., in the range 20–200, which is in line with experimental findings on reversible clusterization on a liquid-air interface.

Metastable gels formed by weakly attractive colloidal particles display a distinctive two-stage time-dependent settling behavior under their own weight. Initially, a space-spanning network is formed that, for a characteristic time, which we define as the lag time ${\tau }_{\mathrm{d}}$, resists compaction. This solidlike behavior persists only for a limited time. Gels whose age $t_{\mathrm{w}}$ is greater than ${\tau }_{\mathrm{d}}$ yield and suddenly collapse. We use a combination of confocal microscopy, rheology, and time-lapse video imaging to investigate both the process of sudden collapse and its microscopic origin in a refractive-index matched emulsion-polymer system. We show that the height $h$ of the gel in the early stages of collapse is well described by the surprisingly simple expression, $h\left(\tau \right)=h_0-A{\tau }^{\frac{3}{2}}$, with $h_0$ the initial height and $\tau =t_{\mathrm{w}}-{\tau }_{\mathrm{d}}$ the time counted from the instant where the gel first yields. We propose that this unexpected result arises because the colloidal network progressively builds up internal stress as a consequence of localized rearrangement events, which leads ultimately to collapse as thermal equilibrium is reestablished.

Molecular motors are found throughout the cells of the human body and have many different and important roles. These micromachines move along filament tracks and have the ability to convert chemical energy into mechanical work that powers cellular motility. Different types of motors are characterized by different duty ratios, which is the fraction of time that a motor is attached to its filament. In the case of myosin II (a nonprocessive molecular machine with a low duty ratio), cooperativity between several motors is essential to induce motion along its actin filament track. In this work we use statistical mechanical tools to calculate the duty ratio of cooperative molecular motors. The model suggests that the effective duty ratio of nonprocessive motors that work in cooperation is lower than the duty ratio of the individual motors. The origin of this effect is the elastic tension that develops in the filament which is relieved when motors detach from the track.

Recent studies have shown that adaptive networks driven by simple local rules can organize into “critical” global steady states, providing another framework for self-organized criticality (SOC). We focus on the important convergence to criticality and show that noise and time-scale optimality are reached at finite values. This is in sharp contrast to the previously believed optimal zero noise and infinite time-scale separation case. Furthermore, we discover a noise-induced phase transition for the breakdown of SOC. We also investigate each of these three effects separately by developing models that reveal three generically low-dimensional dynamical behaviors: time-scale resonance, a simplified version of stochastic resonance, which we call steady-state stochastic resonance, and noise-induced phase transitions.

The fact that individuals will most likely behave differently in different situations begets the introduction of conditional strategies. Inspired by this, we study the evolution of cooperation in the spatial public goods game, where, besides unconditional cooperators and defectors, also different types of conditional cooperators compete for space. Conditional cooperators will contribute to the public good only if other players within the group are likely to cooperate as well but will withhold their contribution otherwise. Depending on the number of other cooperators that are required to elicit cooperation of a conditional cooperator, the latter can be classified in as many types as there are players within each group. We find that the most cautious cooperators, who require all other players within a group to be conditional cooperators, are the undisputed victors of the evolutionary process, even at very low synergy factors. We show that the remarkable promotion of cooperation is due primarily to the spontaneous emergence of quarantining of defectors, who become surrounded by conditional cooperators and are forced into isolated convex “bubbles” from which they are unable to exploit the public good. This phenomenon can be observed only in structured populations, thus adding to the relevance of pattern formation for the successful evolution of cooperation.

We propose a generalized one-dimensional kinetic equation for multidimensional reversible electron transfer (ET) reaction with a nonequilibrium situation as the initial condition. The rate constant for the forward reversible ET reaction obtained here consists of the rate for the corresponding irreversible ET reaction, and an extra term due to reversibility of the ET process which includes the rates of diffusion dynamics in the reactant and product wells. In order to understand the effect of reversibility, we consider back ET reaction in a system consisting of an electron donor-acceptor pair in a solvent modeled through low frequency solvent collective coordinates (multidimensional) characterized by the orientational polarization and slowly relaxing one-dimensional vibrational mode. We propose here a new generalized polarization energy functional corresponding to the extension of the continuum version for the same, which has opened up the possibility of inclusion of molecular nature of the solvent into the solvent reorganization energy. We then derive an exact expression for the ET rate for this model system. The numerical results calculated by using the proposed one-dimensional approach are shown to be in good agreement with the available experimental results. Non-Marcus free energy gap dependence of the rates observed here for the reversible and irreversible ET reactions are very close to each other in the barrierless region, while for other situations, the rate for the former process is found to be less than the latter. The extra term, which makes the difference between the rate constants for irreversible and reversible ET reactions, is found to be contributed by the diffusion dynamics from both reactant and product wells but the dominating contribution is provided mainly by the product well.

Mean-field analysis is an important tool for understanding dynamics on complex networks. However, surprisingly little attention has been paid to the question of whether mean-field predictions are accurate, and this is particularly true for real-world networks with clustering and modular structure. In this paper, we compare mean-field predictions to numerical simulation results for dynamical processes running on 21 real-world networks and demonstrate that the accuracy of such theory depends not only on the mean degree of the networks but also on the mean first-neighbor degree. We show that mean-field theory can give (unexpectedly) accurate results for certain dynamics on disassortative real-world networks even when the mean degree is as low as 4.