Nonequilibrium molecular dynamics simulations are used to study the motion of a ${\mathrm{C}}_{60}$ molecule on a graphene sheet subjected to a temperature gradient. The ${\mathrm{C}}_{60}$ molecule is actuated and moves along the system while it just randomly dances along the perpendicular direction. Increasing the temperature gradient increases the directed velocity of ${\mathrm{C}}_{60}$. It is found that the free energy decreases as the ${\mathrm{C}}_{60}$ molecule moves toward the cold end. The driving mechanism based on the temperature gradient suggests the construction of nanoscale graphene-based motors.

Even if noise titration cannot be satisfactorily used to prove the presence of chaos, it can still be used to detect nonlinear component in dynamics. Nevertheless, since the technique have the use of nonlinear models for one-step-ahead predictions, it requires an acute choice of modeling parameters, i.e., the number of terms and the nonlinearity degree of the models. Based on illustrative examples, we propose conditions under which the method of noise titration can be reliably applied to characterize nonlinearity in the time series. It is thus possible to compare different time series and state which one is governed by the strongest nonlinearity. For instance, it is shown that, when there is a single nonlinear term in the equations describing the system, the variable on which it acts can be identified among the others.

We study the propagation of quasidiscrete microwave solitons in a nonlinear left-handed coplanar waveguide coupled with split-ring resonators. By considering the relevant transmission line analog, we derive a nonlinear lattice model which is studied analytically by means of a quasidiscrete approximation. We derive a nonlinear Schrödinger equation, and find that the system supports bright envelope soliton solutions in a relatively wide subinterval of the left-handed frequency band. We perform systematic numerical simulations, in the framework of the nonlinear lattice model, to study the propagation properties of the quasidiscrete microwave solitons. Our numerical findings are in good agreement with the analytical predictions, and suggest that the predicted structures are quite robust and may be observed in experiments.

We consider the numerical integration of the Gross-Pitaevskii equation with a potential trap given by a time-dependent harmonic potential or a small perturbation thereof. Splitting methods are frequently used with Fourier techniques since the system can be split into the kinetic and remaining part, and each part can be solved efficiently using fast Fourier transforms. Splitting the system into the quantum harmonic-oscillator problem and the remaining part allows us to get higher accuracies in many cases, but it requires us to change between Hermite basis functions and the coordinate space, and this is not efficient for time-dependent frequencies or strong nonlinearities. We show how to build methods that combine the advantages of using Fourier methods while solving the time-dependent harmonic oscillator exactly (or with a high accuracy by using a Magnus integrator and an appropriate decomposition).

We study the long-time behavior of the probability density associated with the decoupled continuous-time random walk which is characterized by a superheavy-tailed distribution of waiting times. It is shown that, if the random walk is unbiased (biased) and the jump distribution has a finite second moment, then the properly scaled probability density converges in the long-time limit to a symmetric two-sided (an asymmetric one-sided) exponential density. The convergence occurs in such a way that all the moments of the probability density grow slower than any power of time. As a consequence, the reference random walk can be viewed as a generic model of superslow diffusion. A few examples of superheavy-tailed distributions of waiting times that give rise to qualitatively different laws of superslow diffusion are considered.

We consider a quantum gas of noninteracting particles confined in the expanding cavity and investigate the nature of the nonadiabatic force which is generated from the gas and acts on the cavity wall. First, with use of the time-dependent canonical transformation, which transforms the expanding cavity to the nonexpanding one, we can define the force operator. Second, applying the perturbative theory, which works when the cavity wall begins to move at time origin, we find that the nonadiabatic force is quadratic in the wall velocity and thereby does not break the time-reversal symmetry, in contrast with general belief. Finally, using an assembly of the transitionless quantum states, we obtain the nonadiabatic force exactly. The exact result justifies the validity of both the definition of the force operator and the issue of the perturbative theory. The mysterious mechanism of nonadiabatic transition with the use of transitionless quantum states is also explained. The study is done for both cases of the hard- and soft-wall confinement with the time-dependent confining length.

For the simulation of fiber systems, there exist several stochastic models: systems of straight nonoverlapping fibers, systems of overlapping bending fibers, or fiber systems created by sedimentation. However, there is a lack of models providing dense, nonoverlapping fiber systems with a given random orientation distribution and a controllable level of bending. We introduce a new stochastic model in this paper that generalizes the force-biased packing approach to fibers represented as chains of balls. The starting configuration is modeled using random walks, where two parameters in the multivariate von Mises–Fisher orientation distribution control the bending. The points of the random walk are associated with a radius and the current orientation. The resulting chains of balls are interpreted as fibers. The final fiber configuration is obtained as an equilibrium between repulsion forces avoiding crossing fibers and recover forces ensuring the fiber structure. This approach provides high volume fractions up to $72.0075\%$.

We use a path-integral approach to calculate the distribution $P(w,t)$ of the fluctuations in the work $w$ at time $t$ of a polymer molecule (modeled as an elastic dumbbell in a viscous solvent) that is acted on by an elongational flow field having a flow rate $\mathrm{\gamma ̇}$. We find that$P(w,t)$ is non-Gaussian and that, at long times, the ratio $P(w,t)/P(-w,t)$ is equal to $\exp [w/(k_{B}T)]$, independent of $\mathrm{\gamma ̇}$. On the basis of this finding, we suggest that polymers in elongational flows satisfy a fluctuation theorem.

Traditional fiber bundles models (FBMs) have been an effective tool to understand brittle heterogeneous systems. However, fiber bundles in modern nano- and bioapplications demand a new generation of FBM capturing more complex deformation processes in addition to damage. In the context of loose bundle systems and with reference to time-independent plasticity and soft biomaterials, we formulate a generalized statistical model for ductile fracture and nonlinear elastic problems capable of handling more simultaneous deformation mechanisms by means of two order parameters (as opposed to one). As the first rational FBM for coupled damage problems, it may be the cornerstone for advanced statistical models of heterogeneous systems in nanoscience and materials design, especially to explore hierarchical and bio-inspired concepts in the arena of nanobiotechnology. Applicative examples are provided for illustrative purposes at last, discussing issues in inverse analysis (i.e., nonlinear elastic polymer fiber and ductile Cu submicron bars arrays) and direct design (i.e., strength prediction).

Input-output tables describe the flows of goods and services between the sectors of an economy. These tables can be interpreted as weighted directed networks. At the usual level of aggregation, they contain nodes with strong self-loops and are almost completely connected. We derive two measures of node centrality that are well suited for such networks. Both are based on random walks and have interpretations as the propagation of supply shocks through the economy. Random walk centrality reveals the vertices most immediately affected by a shock. Counting betweenness identifies the nodes where a shock lingers longest. The two measures differ in how they treat self-loops. We apply both to data from a wide set of countries and uncover salient characteristics of the structures of these national economies. We further validate our indices by clustering according to sectors’ centralities. This analysis reveals geographical proximity and similar developmental status.

The modeling of epidemic-like processes on random networks has received considerable attention in recent years. While these processes are inherently stochastic, most previous work has been focused on deterministic models that ignore important fluctuations that may persist even in the infinite network size limit. In a previous paper, for a class of epidemic and rumor processes, we derived approximate models for the full probability distribution of the final size of the epidemic, as opposed to only mean values. In this paper we examine via direct simulations the adequacy of the approximate model to describe stochastic epidemics and rumors on several random network topologies: homogeneous networks, Erdös-Rényi (ER) random graphs, Barabasi-Albert scale-free networks, and random geometric graphs. We find that the approximate model is reasonably accurate in predicting the probability of spread. However, the position of the threshold and the conditional mean of the final size for processes near the threshold are not well described by the approximate model even in the case of homogeneous networks. We attribute this failure to the presence of other structural properties beyond degree-degree correlations, and in particular clustering, which are present in any finite network but are not incorporated in the approximate model. In order to test this “hypothesis”  we perform additional simulations on a set of ER random graphs where degree-degree correlations and clustering are separately and independently introduced using recently proposed algorithms from the literature. Our results show that even strong degree-degree correlations have only weak effects on the position of the threshold and the conditional mean of the final size. On the other hand, the introduction of clustering greatly affects both the position of the threshold and the conditional mean. Similar analysis for the Barabasi-Albert scale-free network confirms the significance of clustering on the dynamics of rumor spread. For this network, though, with its highly skewed degree distribution, the addition of positive correlation had a much stronger effect on the final size distribution than was found for the simple random graph.

We investigate the velocity dependence of kinetic friction with a model that makes minimal assumptions on the actual mechanism of friction so that it can be applied at many scales, provided the system involves multicontact friction. Using a recently developed master equation approach, we investigate the influence of two concurrent processes. First, at a nonzero temperature, thermal fluctuations allow an activated breaking of contacts that are still below the threshold. As a result, the friction force monotonically increases with velocity. Second, the aging of contacts leads to a decrease of the friction force with velocity. Aging effects include two aspects: the delay in contact formation and aging of a contact itself, i.e., the change of its characteristics with the duration of stationary contact. All these processes are considered simultaneously with the master equation approach, giving a complete dependence of the kinetic friction force on the driving velocity and system temperature, provided the interface parameters are known.

Methods for computing Hashin-Shtrikman bounds and related self-consistent estimates of elastic constants for polycrystals composed of crystals having orthorhombic symmetry have been known for about three decades. However, these methods are underutilized, perhaps because of some perceived difficulties with implementing the necessary computational procedures. Several simplifications of these techniques are introduced, thereby reducing the overall computational burden, as well as the complications inherent in mapping out the Hashin-Shtrikman bounding curves. The self-consistent estimates of the effective elastic constants are very robust, involving a quickly converging iteration procedure. Once these self-consistent values are known, they may then be used to speed up the computations of the Hashin-Shtrikman bounds themselves. It is shown furthermore that the resulting orthorhombic polycrystal code can be used as well to compute both bounds and self-consistent estimates for polycrystals of higher-symmetry tetragonal, hexagonal, and cubic (but not trigonal) materials. The self-consistent results found this way are shown to be the same as those obtained using the earlier methods, specifically those methods designed specially for each individual symmetry type. But the Hashin-Shtrikman bounds found using the orthorhombic code are either the same or (more typically) tighter than those found previously for these special cases (i.e., tetragonal, hexagonal, and cubic). The improvement in the Hashin-Shtrikman bounds is presumably due to the additional degrees of freedom introduced into the available search space.

We study the Markov property of experimental velocity data of different homogeneous isotropic turbulent flows. In particular, we examine the stochastic “cascade” process of nested velocity increments $\xi \left(r\right):=u(x+r)-u\left(x\right)$ as a function of scale $r$ for different nesting structures. It was found in previous work that, for a certain nesting structure, the stochastic process of $\xi \left(r\right)$ has the Markov property for step sizes larger than the so-called Einstein-Markov coherence length $l_{\mathrm{EM}}$, which is of the order of magnitude of the Taylor microscale $\lambda$ [Phys. Lett. A 359, 335 (2006)]. We now show that, if a reasonable definition of the effective step size of the process is applied, this result holds independently of the nesting structure. Furthermore, we analyze the stochastic process of the velocity $u$ as a function of the spatial position $x$. Although this process does not have the exact Markov property, a characteristic length scale $l_{u\left(x\right)}\approx l_{\mathrm{EM}}$ can be identified on the basis of a statistical test for the Markov property. Using a method based on the matrix of transition probabilities, we examine the significance of the non-Markovian character of the velocity $u\left(x\right)$ for the statistical properties of turbulence.

We show that rotating shallow water dynamics possesses an approximate (adiabatic-type) positive quadratic invariant, which exists not only at midlatitudes (where its analog in the quasigeostrophic equation has been previously investigated), but near the equator as well (where the quasigeostrophic equation is inapplicable). When deriving the extra invariant we find two kinds of small denominators: (i) those due to the triad resonances (as in the case of the quasigeostrophic equation) and (ii) those due to the equatorial limit, when the Rossby radius of deformation becomes infinite. We show that both kinds of small denominators can be canceled. The presence of the extra invariant can lead to the generation of zonal jets. We find that this tendency should be especially pronounced near the equator. A similar invariant occurs in magnetically confined fusion plasmas and can lead to the emergence of zonal flows.

The inertial migration of a two-dimensional elastic capsule in a Poiseuille flow was studied over the Reynolds number range 1 $⩽$ Re $⩽$ 100. The lateral migration velocity, slip velocity, and the deformation and inclination angle of the capsule were investigated by varying the lateral position, Reynolds number, capsule-to-channel size ratio ($\lambda$), membrane stretching coefficient ($\varphi$), and membrane bending coefficient ($\gamma$). During the initial transient motion, the lateral migration velocity increased with increasing Re and $\lambda$, but decreased with increases in $\varphi ,\gamma$, and the lateral distance from the wall. On the other hand, the deformation of the capsule increased and the inclination angle became smaller as Re, $\varphi ,\hspace{0.5em}\gamma$, and the distance from the wall decreased. The initial behavior of the capsule was influenced by variation in the initial lateral position ($y_0$), but the equilibrium position of the capsule was not affected by such variation. The balance between the wall effect and the shear gradient effect determined the equilibrium position. As Re increased, the equilibrium position initially shifted closer to the wall and then moved toward the channel center. A peak in the equilibrium position was observed near Re $=$ 30 for $\lambda$ $=$ 0.1, and the peak shifted to higher Re as $\lambda$ increased. Depending on the lateral migration velocity, the equilibrium position moved toward the centerline for larger $\lambda$, but moved toward the wall for larger $\varphi \hspace{0.5em}\mathrm{and}\hspace{0.5em}\gamma$.

We examine the exponentiation of the Raman forward scattering instability in modulated plasma channels computationally and analytically. An evolution equation for the complex phases of the Raman scattered waves treating the spatial localization and discrete nature of the channel modes is derived. Simulations with WAKE [P. Mora and T. M. Antonsen Jr., Phys. Plasmas 4, 217 (1997)] verify the theory in the linear growth regime and provide insight into the nonlinear stage of the instability when cascading and pump depletion play a role. We find that the exponentiation in modulated channels depends on two factors: the increase in coupling due to the increased plasma wavenumber in the high-density regions of the channel and a decreased coupling due to the reduced longitudinal spatial coherence. For the parameters considered, simulations show that the finite extent of the pump pulse is more significant in determining the exponentiation than phase mixing due to the transverse variation of the channel. Both the theory and simulations confirm that modulated channels allow for the stable guiding of longer pulses than nonmodulated channels.

This work extends the results of a recently developed theory of a rather complete thermodynamic formalism for discrete-state, continuous-time Markov processes with and without detailed balance. We investigate whether and in what way the thermodynamic structure is invariant in a multiscale stochastic system, that is, whether the relations between thermodynamic functions of state and process variables remain unchanged when the system is viewed at different time scales and resolutions. Our results show that the dynamics on a fast time scale contribute an entropic term to the internal energy function $u_S(x)$ for the slow dynamics. Based on the conditional free energy $u_S(x)$, we can then treat the slow dynamics as if the fast dynamics is nonexistent. Furthermore, we show that the free energy, which characterizes the spontaneous organization in a system without detailed balance, is invariant with or without the fast dynamics: The fast dynamics is assumed to reach stationarity instantaneously on the slow time scale; it has no effect on the system’s free energy. The same cannot be said for the entropy and the internal energy, both of which contain the same contribution from the fast dynamics. We also investigate the consequences of time-scale separation in connection to the concepts of quasi-stationarity and steady adiabaticity introduced in the phenomenological steady-state thermodynamics.