The presented work probes the fundamentals of Knudsen forces. Using the direct simulation Monte Carlo (DSMC) method, the flows induced by temperature inhomogeneity within a representative configuration and the Knudsen force acting on a heated microbeam are captured as functions of Knudsen number in the entire flow regime. Both flow strength and Knudsen force peak in the transition regime and negative Knudsen force absent in experimental data is observed. The mechanisms of the thermally induced flows and Knudsen forces are studied. It has been found that thermal edge flow is the main driven source for the formation of the Knudsen force on microbeams and domain configuration plays an important role in the process.

Anomalous diffusion in liquids and the solid-liquid phase transition (melting) are studied in two-dimensional Yukawa systems. The self-intermediate scattering function (self-ISF), calculated from simulation data, exhibits a temporal decay, or relaxation, with a characteristic relaxation time. This decay is found to be useful for distinguishing normal and anomalous diffusion in a liquid, and for identifying the solid-liquid phase transition. For liquids, a scaling of the relaxation time with length scale is found. For the solid-liquid phase transition, the shape of the self-ISF curve is found to be a sensitive indicator of phase. Friction has a significant effect on the timing of relaxation, but not the melting point.

We propose a modified power method for computing the subdominant eigenvalue ${\lambda }_2$ of a matrix or continuous operator. While useful both deterministically and stochastically, we focus on defining simple Monte Carlo methods for its application. The methods presented use random walkers of mixed signs to represent the subdominant eigenfunction. Accordingly, the methods must cancel these signs properly in order to sample this eigenfunction faithfully. We present a simple procedure to solve this sign problem and then test our Monte Carlo methods by computing ${\lambda }_2$ of various Markov chain transition matrices. As $\left|{\lambda }_2\right|$ of this matrix controls the rate at which Monte Carlo sampling relaxes to a stationary condition, its computation also enabled us to compare efficiencies of several Monte Carlo algorithms as applied to two quite different types of problems. We first computed ${\lambda }_2$ for several one- and two-dimensional Ising models, which have a discrete phase space, and compared the relative efficiencies of the Metropolis and heat-bath algorithms as functions of temperature and applied magnetic field. Next, we computed ${\lambda }_2$ for a model of an interacting gas trapped by a harmonic potential, which has a mutidimensional continuous phase space, and studied the efficiency of the Metropolis algorithm as a function of temperature and the maximum allowable step size $\Delta$. Based on the ${\lambda }_2$ criterion, we found for the Ising models that small lattices appear to give an adequate picture of comparative efficiency and that the heat-bath algorithm is more efficient than the Metropolis algorithm only at low temperatures where both algorithms are inefficient. For the harmonic trap problem, we found that the traditional rule of thumb of adjusting $\Delta$ so that the Metropolis acceptance rate is around 50% is often suboptimal. In general, as a function of temperature or $\Delta$, ${\lambda }_2$ for this model displayed trends defining optimal efficiency that the acceptance ratio does not. The cases studied also suggested that Monte Carlo simulations for a continuum model are likely more efficient than those for a discretized version of the model.

The transport of angular momentum in the outward direction is the fundamental requirement for accretion to proceed in an accretion disk. This objective can be achieved if the accretion flow is turbulent. Instabilities are one of the sources for the turbulence. We study a differentially rotating compressive flow in the presence of nonvanishing radial and azimuthal magnetic field and demonstrate the occurrence of a high growth rate instability. This instability operates in a region where magnetic energy density exceeds the rotational energy density.

We analyze the low temperature properties of a system of classical Heisenberg spins on a hexagonal lattice with Kitaev couplings. For a lattice of $2N$ sites with periodic boundary conditions, the ground states form an $\left(N+1\right)$ dimensional manifold. We show that the ensemble of ground states is equivalent to that of a solid-on-solid model with continuously variable heights and nearest neighbor interactions, at a finite temperature. For temperature $T$ tending to zero, all ground states have equal weight, and there is no order by disorder in this model. We argue that the bond-energy bond-energy correlations at distance $R$ decay as $\frac{1}{R^2}$ at zero temperature. This is verified by Monte Carlo simulations. We also discuss the relation to the quantum spin-$S$ Kitaev model for large $S$, and obtain lower and upper bounds on the ground-state energy of the quantum model.

We investigate the dynamics of a disk shaped intruder sliding on a granular monolayer. The monolayer is on an inclined transparent plane, tilted at an angle much smaller than the angle of avalanche. A high speed camera allows us to measure the dynamics of both, the intruder (filming from top) and the grains (filming from below). We find a frictional force with a dependence on the speed of the intruder. Moreover, calculating a Reynolds-like number, it is possible to highlight the influence of the density of the beads that form the monolayer on the dynamics of the disk. We also find that the fluidization produced by the intruder’s action reduces substantially the effective friction coefficient.

Disordered networks are found in many natural and artificial materials, from gels or cytoskeletal structures to metallic foams or bones. Here, the energy distribution in this type of networks is modeled, taking into account the orientation of the struts. A correlation between the orientation and the energy per unit volume is found and described as a function of the connectivity in the network and the relative bending stiffness of the struts. If one or both parameters have relatively large values, the struts aligned in the loading direction present the highest values of energy. On the contrary, if these have relatively small values, the highest values of energy can be reached in the struts oriented transversally. This result allows explaining in a simple way remodeling processes in biological materials, for example, the remodeling of trabecular bone and the reorganization in the cytoskeleton. Additionally, the correlation between the orientation, the affinity, and the bending-stretching ratio in the network is discussed.

We analyze coupled FitzHugh-Nagumo oscillators on various network topologies, in particular random diluted and scale-free topologies, under the influence of noise. Similarly to globally coupled excitable units, noise acts as control parameter: changing monotonically its strength, the collective dynamical behavior varies from stable equilibrium solutions to coherent firing of a large fraction, and for even stronger noise to incoherent firing leading to chaotic behavior of the excitable elements. For strong noise the system is less sensitive to the network topology. The specific topology enters via the degree of nodes and determines the average number of spikes. Apart from bifurcation regions, it is the ratio of noise intensity to size that determines the dynamical behavior of average values. Specific behavior such as limit cycles may then be realized for strong noise and large systems or for low noise and small systems. Within bifurcation regions, the actual values of noise intensity and system-size matter independently. Here we analyze in more detail phase portraits of small systems. For a given noise intensity and network topology we have studied the regularity of signals as a function of time. We observe the phenomenon of system-size resonance for a whole interval of noise intensities as long as the degree distribution is homogeneous, so that no fine tuning of the noise is needed. Therefore it is plausible that natural systems make actually use of noise when noise is unavoidably present.

High-quality measurements of seismic activities around the world provide a wealth of data and information that are relevant to understanding of when earthquakes may occur. If viewed as complex stochastic time series, such data may be analyzed by methods that provide deeper insights into their nature, hence leading to better understanding of the data and their possible implications for earthquakes. In this paper, we provide further evidence for our recent proposal [P. Mansour et al., Phys. Rev. Lett. 102, 014101 (2009)] for the existence of a transition in the shape of the probability density function (PDF) of the successive detrended increments of the stochastic fluctuations of Earth’s vertical velocity $V_z$, collected by broadband stations before moderate and large earthquakes. To demonstrate the transition, we carried out extensive analysis of the data for $V_z$ for 12 earthquakes in several regions around the world, including the recent catasrophic one in Haiti. The analysis supports the hypothesis that before and near the time of an earthquake, the shape of the PDF undergoes significant and discernable changes, which can be characterized quantitatively. The typical time over which the PDF undergoes the transition is about 5–10 h prior to a moderate or large earthquake.

In this work we perform a detailed study of the scaling properties of Lyapunov vectors (LVs) for two different one-dimensional Hamiltonian lattices: the Fermi-Pasta-Ulam and ${\Phi }^4$ models. In this case, characteristic (also called covariant) LVs exhibit qualitative similarities with those of dissipative lattices but the scaling exponents are different and seemingly nonuniversal. In contrast, backward LVs (obtained via Gram-Schmidt orthonormalizations) present approximately the same scaling exponent in all cases, suggesting it is an artificial exponent produced by the imposed orthogonality of these vectors. We are able to compute characteristic LVs in large systems thanks to a “bit reversible” algorithm, which completely obviates computer memory limitations.

The vortices emerging in rotating turbulent Rayleigh-Bénard convection in water at Rayleigh number $\text{Ra}=6.0\times {10}^8$ are investigated using stereoscopic particle image velocimetry and by direct numerical simulation. The so-called $Q$ criterion is used to detect the vortices from velocity fields. This criterion allows distinguishing vorticity- and strain-dominated regions in the flow by decomposing the velocity gradient tensor into symmetric and antisymmetric parts. Vortex densities, mean vortex radii and mean vortex circulations are calculated at two horizontal cross-sections of the cylindrical flow domain and at several rotation rates, described by the Taylor number which takes values between $3.0\times {10}^8$ and $7.7\times {10}^{10}$. Separate statistics are calculated for cyclonic and anticyclonic vortices. Vortex densities and mean vortex radii are mostly independent of the Taylor number except very close to the bottom and top plates where more vortices are detected when the Taylor number is raised (rotation increases). The vortex population close to the plate consists mostly of cyclones while further into the bulk of the domain a similar amount of cyclones and anticyclones is found. The cyclonic vortices contain more circulation than the anticyclones. The same vortex analysis of the simulation results at additional vertical positions revealed that the vortices are formed in a boundary layer on the plate with a thickness of approximately two Ekman lengths.

A Green’s function for a generalized two-dimensional (2D) fluid in an unbounded domain (the so-called $\alpha$ turbulence system) is discussed. The generalized 2D fluid is characterized by a relationship between an advected quantity $q$ and the stream function $\psi$: namely, $q=-{\left(-\Delta \right)}^{\alpha /2}\psi$. Here, $\alpha$ is a real number and $q$ is referred to as the vorticity. In this study, the Green’s function refers to the stream function produced by a delta-functional distribution of $q$, i.e., a point vortex with unit strength. The Green’s function has the form $G^{\left(\alpha \right)}\left(\mathit{r}\right)\propto r^{\alpha -2}$, except when $\alpha$ is an even number, where $r$ is the distance from the point vortex. This functional form is known as the Riesz potential. When $\alpha$ is a positive even number, the logarithmic correction to the Riesz potential has the form $G{\left(\mathit{r}\right)}^{\left(\alpha \right)}\propto r^{\alpha -2}\text{ln}r$. In contrast, when $\alpha$ is a negative even number, $G^{\left(\alpha \right)}$ is given by the higher-order Laplacian of the delta function. The transition of the small-scale behavior of $q$ at $\alpha =2$, a well-known property of forced and dissipative $\alpha$ turbulence, is explained in terms of the Green’s function. Moreover, the azimuthal velocity around the point vortex is derived from the Green’s function. The functional form of the azimuthal velocity indicates that physically realizable systems for the generalized 2D fluid exist only when $\alpha \le 3$. The Green’s function and physically realizable systems for an anisotropic generalized 2D fluid are presented as an application of the present study.

We demonstrate that the modulation instability of the zone-boundary mode in a finite (periodic) Fermi-Pasta-Ulam chain is the necessary but not sufficient condition for the efficient energy transfer by localized excitations. This transfer results from the exclusion of complete energy exchange between spatially different parts of the chain, and the excitation level corresponding to that turns out to be twice more than threshold of zone-boundary mode’s instability. To obtain this result one needs in far going extension of the beating concept to a wide class of finite oscillatory chains. In turn, such an extension leads to description of energy exchange and transition to energy localization and transfer in terms of effective particles and limiting phase trajectories. The effective particles appear naturally when the frequency spectrum crowding ensures the resonance interaction between zone-boundary and two nearby nonlinear normal modes, but there are no additional resonances. We show that the limiting phase trajectories corresponding to the most intensive energy exchange between effective particles can be considered as an alternative to nonlinear normal modes, which describe the stationary process.

The propagation of highly nonlinear signals in a branched two-dimensional granular system was investigated experimentally and numerically for a system composed of chains of spherical beads of different materials. The system studied consists of a double Y-shaped guide in which high- and low-modulus/mass chains of spheres are arranged in various geometries. We observed the transformation of a single or a train of solitary pulses crossing the interface between branches. We report fast splitting of the initial pulse, rapid chaotization of the signal and impulse redirection and bending. Pulse and energy trapping was also observed in the branches. Numerical analysis based on Hertzian interaction between the particles and the side walls of the guide was found in agreement with the experimental data, except for nonsymmetric arrangements of particles excited by a large mass striker.

Statistical properties of the transport coefficient for deterministic subdiffusion are investigated from the viewpoint of infinite ergodic theory. We find that the averaged diffusion coefficient is characterized by the infinite invariant measure of the reduced map. We also show that when the time difference is much smaller than the total observation time, the time-averaged mean square displacement depends linearly on the time difference. Furthermore, the diffusion coefficient becomes a random variable and its limit distribution is characterized by the universal law called the Mittag-Leffler distribution.

We study the evolutionary dynamics of a maladapted population of self-replicating sequences on strongly correlated fitness landscapes. Each sequence is assumed to be composed of blocks of equal length and its fitness is given by a linear combination of four independent block fitnesses. A mutation affects the fitness contribution of a single block leaving the other blocks unchanged and hence inducing correlations between the parent and mutant fitness. On such strongly correlated fitness landscapes, we calculate the dynamical properties such as the number of jumps in the most populated sequence and the temporal distribution of the last jump, which is shown to exhibit an inverse square dependence as in evolution on uncorrelated fitness landscapes. We also obtain exact results for the distribution of records and extremes for correlated random variables.

We consider continuous-time random walk models described by arbitrary sojourn time probability density functions. We find a general expression for the distribution of time-averaged observables for such systems, generalizing some recent results presented in the literature. For the case where sojourn times are identically distributed independent random variables, our results shed some light on the recently proposed transitions between ergodic and weakly nonergodic regimes. On the other hand, for the case of nonidentical trapping time densities over the lattice points, the distribution of time-averaged observables reveals that such systems are typically nonergodic, in agreement with some recent experimental evidences on the statistics of blinking quantum dots. Some explicit examples are considered in detail. Our results are independent of the lattice topology and dimensionality.

In molecular dynamics simulations, temperature is evaluated, via the equipartition principle, by computing the mean kinetic energy of atoms. There is no similar recipe yet for evaluating temperature of a dynamic system of interacting spins. By solving semiclassical Langevin spin-dynamics equations, and applying the fluctuation-dissipation theorem, we derive an equation for the temperature of a spin ensemble, expressed in terms of dynamic spin variables. The fact that definitions for the kinetic and spin temperatures are fully consistent is illustrated using large-scale spin dynamics and spin-lattice dynamics simulations.

The universal quantum work relation connects a functional of an arbitrary observable averaged over the forward process to the free-energy difference and another functional averaged over the time-reversed process. Here, we ask the question if the system is driven out of equilibrium by a different Hamiltonian rather than the original one during the forward process and similarly during the reversed process then how accurate is the quantum work relation. We present an inequality that must be satisfied when the system is driven out by such a trial Hamiltonian. This also answers the issue of accuracy of the Jarzynski relation with a trial Hamiltonian. We have shown that the correction term can be expressed as the averages of the difference operator between the accurate and trial Hamiltonians. This leads to a generalized version of the Bogoliubov inequality for the free-energy differences.

We study the electrostatic contribution to the effective potential between two spherical low-dielectric particles that carry proton-titratable sites within a linearized setting. To evaluate the needed work of charging for each possible proton occupancy configuration, together with its crucial dependence on sphere separation, we numerically solve a coarse-grained linear Debye-Hückel model that incorporates nonuniform dielectric and ionic solution properties at a series of intersphere separations and for chosen titratable charge locations on each sphere. We combine the resulting work-of-charging matrix with site-specific chemical potentials of proton binding to construct the Boltzmann-weighted probabilities of each possible occupancy pattern of the titratable sites as functions of intersphere separation. With the use of these probabilities we find that a nonmonotonic average electrostatic potential can result that is repulsive at larger sphere separations but attractive at close separations. The nonmonotonic potential corresponds to particular choices of site-specific unoccupied charge values and their corresponding proton affinities, and its occurrence is dependent on $p\text{H}$ in relation to the $p{K}_a$ values of the titratable groups. For the chosen titratable groups, we identify the particular change from repulsive to attractive proton occupancy patterns with decreasing intersphere separation that gives rise to the modeled nonmonotonic dependence and derive more general conditions under which such a nonmonotonic dependence can occur. Within the present model we find that stationary points of the charge-regulated average electrostatic potential, considered as a function of intersphere separation, occur when a normalized Boltzmann-averaged intersphere charge number product equals its covariance with an average free energy of charging divided by $k_{B}T$. We derive more general conditions for the location and nature of critical points in the electrostatic intersphere potential, which are not dependent on the validity of the present linear model. Analysis of the present simple prototype model can be a helpful step toward developing a framework for predicting when (i) patterned charge-regulated occupancy patterns, (ii) orientation-dependent attractions due to relatively fixed heterogeneous charging patterns, and (iii) screened net protein charge could separately dominate the electrostatic portion of the interactions between model biological macromolecules and other nanoparticles.