We present an exact expression for the mean exit time through the cap of a confining sphere for particles alternating phases of surface and of bulk diffusion. The present approach is based on an integral equation which can be solved analytically. In contrast to the statement of Berezhkovskii and Barzykin [J. Chem. Phys. 136, 54115 (2012)], we show that the mean exit time can be optimized with respect to the desorption rate, under analytically determined criteria.

We study the global geometry of the energy landscape of a simple model of a self-gravitating system, the self-gravitating ring (SGR). This is done by endowing the configuration space with a metric such that the dynamical trajectories are identified with geodesics. The average curvature and curvature fluctuations of the energy landscape are computed by means of Monte Carlo simulations and, when possible, of a mean-field method, showing that these global geometric quantities provide a clear geometric characterization of the collapse phase transition occurring in the SGR as the transition from a flat landscape at high energies to a landscape with mainly positive but fluctuating curvature in the collapsed phase. Moreover, curvature fluctuations show a maximum in correspondence with the energy of a possible further transition, occurring at lower energies than the collapsed one, whose existence had been previously conjectured on the basis of a local analysis of the energy landscape and whose effect on the usual thermodynamic quantities, if any, is extremely weak. We also estimate the largest Lyapunov exponent $\lambda$ of the SGR using the geometric observables. The geometric estimate always gives the correct order of magnitude of $\lambda$ and is also quantitatively correct at small energy densities and, in the limit $N\to \infty$, in the whole homogeneous phase.

We map the problem of diffusion in the quenched trap model onto a different stochastic process: Brownian motion that is terminated at the coverage time ${\mathcal{S}}_{\alpha }={\sum }_{x=-\infty }^{\infty }{\left(n_x\right)}^{\alpha }$, with $n_x$ being the number of visits to site $x$. Here $0<\alpha =T/T_g<1$ is a measure of the disorder in the original model. This mapping allows us to treat the intricate correlations in the underlying random walk in the random environment. The operational time ${\mathcal{S}}_{\alpha }$ is changed to laboratory time $t$ with a Lévy time transformation. Investigation of Brownian motion stopped at time ${\mathcal{S}}_{\alpha }$ yields the diffusion front of the quenched trap model, which is favorably compared with numerical simulations. In the zero-temperature limit of $\alpha \to 0$ we recover the renormalization group solution obtained by Monthus [Phys. Rev. E 68, 036114 (2003)]. Our theory surmounts the critical slowing down that is found when $\alpha \to 1$. Above the critical dimension 2, mapping the problem to a continuous time random walk becomes feasible, though still not trivial.

There is growing evidence that slow dynamics and dynamic heterogeneity possess structural signatures in glass-forming liquids. However, even in the weakly frustrated glass-forming liquids, whether or not the dynamic heterogeneity has a structural origin is a matter of debate. Via molecular dynamics simulation, we present a study of examining the connection between dynamic heterogeneity and bond orientational order in a weakly frustrated glass-forming liquid in two dimensions by taking advantage of assessing the effect of attractions on the correlation length scales. We find that attractions can strongly affect relaxation dynamics, dynamic heterogeneity, and the associated dynamic correlation length of the liquid, but their influence on bond orientational order and the associated static correlation length shows a manner reminiscent of the effect of attractions on the thermodynamics of liquids. This implies that the growth of bond orientational order and static correlation length scale might be merely a manifestation of favoring the configurational entropy in weakly frustrated glass-forming liquids. Thus, our results provide strong evidence that bond orientational order cannot provide a complete description of dynamic heterogeneity even in weakly frustrated glass-forming systems.

The phenomenon of the glass transition is an unresolved problem in condensed matter physics. Its prominent feature, the super-Arrhenius temperature dependence of the transport coefficients, remains a challenge to be described over the full temperature range. For a series of molecular glass formers, we combined τ($T$) collected from dielectric spectroscopy and dynamic light scattering covering a range 10${}^{-12}$ s < τ($T$) < 10${}^2$ s. Describing the dynamics in terms of an activation energy $E$($T$), we distinguish a high-temperature regime characterized by an Arrhenius law with a constant activation energy $E_{\infty }$ and a low-temperature regime for which $E_{\mathrm{coop}}$($T$) ≡ $E$($T$)–$E_{\infty }$ increases exponentially while cooling. A scaling is introduced, specifically $E_{\mathrm{coop}}$($T$)/$E_{\infty }$ ∝ exp[–λ($T$/$T_{\mathrm{A}}$–1)], where λ is a fragility parameter and $T_{\mathrm{A}}$ a reference temperature proportional to $E_{\infty }$. In order to describe τ($T$) still the attempt time ${\tau }_{\infty }$ has to be specified. Thus, a single interaction parameter $E_{\infty }$ describing the high-temperature regime together with λ controls the temperature dependence of low-temperature cooperative dynamics.

The Sznajd model is a sociophysics model that is used to model opinion propagation and consensus formation in societies. Its main feature is that its rules favor bigger groups of agreeing people. In a previous work, we generalized the bounded confidence rule in order to model biases and prejudices in discrete opinion models. In that work, we applied this modification to the Sznajd model and presented some preliminary results. The present work extends what we did in that paper. We present results linking many of the properties of the mean-field fixed points, with only a few qualitative aspects of the confidence rule (the biases and prejudices modeled), finding an interesting connection with graph theory problems. More precisely, we link the existence of fixed points with the notion of strongly connected graphs and the stability of fixed points with the problem of finding the maximal independent sets of a graph. We state these results and present comparisons between the mean field and simulations in Barabási-Albert networks, followed by the main mathematical ideas and appendices with the rigorous proofs of our claims and some graph theory concepts, together with examples. We also show that there is no qualitative difference in the mean-field results if we require that a group of size $q>2$, instead of a pair, of agreeing agents be formed before they attempt to convince other sites (for the mean field, this would coincide with the $q$-voter model).

We have performed numerical simulations of boundary-driven dynamos using a three-dimensional nonlinear magnetohydrodynamical model in a spherical shell geometry. A conducting fluid of magnetic Prandtl number $\mathrm{Pm}=0.01$ is driven into motion by the counter-rotation of the two hemispheric walls. The resulting flow is of von Kármán type, consisting of a layer of zonal velocity close to the outer wall and a secondary meridional circulation. Above a certain forcing threshold, the mean flow is unstable to non-axisymmetric motions within an equatorial belt. For fixed forcing above this threshold, we have studied the dynamo properties of this flow. The presence of a conducting outer wall is essential to the existence of a dynamo at these parameters. We have therefore studied the effect of changing the material parameters of the wall (magnetic permeability, electrical conductivity, and thickness) on the dynamo. In common with previous studies, we find that dynamos are obtained only when either the conductivity or the permeability is sufficiently large. However, we find that the effect of these two parameters on the dynamo process are different and can even compete to the detriment of the dynamo. Our self-consistent approach allow us to analyze in detail the dynamo feedback loop. The dynamos we obtain are typically dominated by an axisymmetric toroidal magnetic field and an axial dipole component. We show that the ability of the outer shear layer to produce a strong toroidal field depends critically on the presence of a conducting outer wall, which shields the fluid from the vacuum outside. The generation of the axisymmetric poloidal field, on the other hand, occurs in the equatorial belt and does not depend on the wall properties.

We evaluate various molecular tagging velocimetry (MTV) techniques for application in turbulent flows of gases where the smallest length scales must be resolved. We argue that tracer diffusion dictates the use of large complex molecules and discuss a few candidate molecules. The accuracy of MTV is determined by the profile of written lines which widen due to molecular dynamics, including both diffusion and chemical reaction. We evaluate these profiles for tagging with phosphorescing biacetyl molecules, which is a commonly used probe in MTV. For relatively large laser power, these profiles are determined not by molecular diffusion, but by the triplet-triplet annihilation reaction of excited biacetyl molecules. We identify a new reaction pathway, and present a model for the observed line shapes. The rapid widening of tagged lines of biacetyl molecules due to chemical reaction restricts this MTV technique to large-scale turbulent motion in gases of comparable molecular weight.

The combined effect of two modes of electroconvection, i.e., (a) the electro-osmotic flow of the second kind induced by a curved membrane surface and (b) electrokinetic instability, is studied numerically. Both physical mechanisms are responsible for electric current enhancement to the surface, and these modes are strongly nonlinearly coupled. For the limiting regimes, their resonant interaction near the threshold of instability with a corresponding resonantly amplified current enhancement is found. For the overlimiting regimes, inside the unstable region, their interaction becomes more complex with negative “sideband” and positive “subharmonic” resonant interactions. Wall corrugation can still be in resonance with the unstable modes. At some wave numbers of corrugation, these two mechanisms compete and electrokinetic instability can even be completely suppressed by the wall corrugation.

We show that patterned surfaces can promote bistable configurations of nematics for reasons other than the symmetry of the surface. Numerical and analytical calculations reveal that a nematic liquid crystal in contact with a striped surface is subject to the competing aligning influences of elastic anisotropy, differing energy cost of various types of deformation, and flexoelectricity, curvature-induced spontaneous polarization. These effects favor opposing ground states where the azimuthal alignment is, respectively, parallel or perpendicular to the stripes. Material parameters for which the effect might be observed lie within the range measured for bent-core nematogens.

Every lattice for which the bond percolation critical probability can be found exactly possesses a critical polynomial, with the root in $[0,1]$ providing the threshold. Recent work has demonstrated that this polynomial may be generalized through a definition that can be applied on any periodic lattice. The polynomial depends on the lattice and on its decomposition into identical finite subgraphs, but once these are specified, the polynomial is essentially unique. On lattices for which the exact percolation threshold is unknown, the polynomials provide approximations for the critical probability with the estimates appearing to converge to the exact answer with increasing subgraph size. In this paper, I show how this generalized critical polynomial can be viewed as a graph invariant, similar to the Tutte polynomial. In particular, the critical polynomial is computed on a finite graph and may be found using the recursive deletion-contraction algorithm. This allows calculation on a computer, and I present such results for the kagome lattice using subgraphs of up to 36 bonds. For one of these, I find the prediction $p_c=0.52440572\cdots$, which differs from the numerical value, $p_c=0.52440503\left(5\right)$, by only $6.9\times {10}^{-7}$.

We study systems with a crossover parameter $\lambda$, such as the temperature $T$, which has a threshold value ${\lambda }_{*}$ across which the correlation function changes from exhibiting fixed wavelength (or time period) modulations to continuously varying modulation lengths (or times). We introduce a hitherto unknown exponent ${\nu }_L$ characterizing the universal nature of this crossover and compute its value in general instances. This exponent, similar to standard correlation length exponents, is obtained from motion of the poles of the momentum (or frequency) space correlation functions in the complex $k$-plane (or $\omega$-plane) as the parameter $\lambda$ is varied. Near the crossover (i.e., for $\lambda \to {\lambda }_{*}$), the characteristic modulation wave vector $K_R$ in the variable modulation length “phase” is related to that in the fixed modulation length “phase” $q$ via $|K_R-q|\propto |T-T_{*}{|}^{{\nu }_L}$. We find, in general, that ${\nu }_L=1/2$. In some special instances, ${\nu }_L$ may attain other rational values. We extend this result to general problems in which the eigenvalue of an operator or a pole characterizing general response functions may attain a constant real (or imaginary) part beyond a particular threshold value ${\lambda }_{*}$. We discuss extensions of this result to multiple other arenas. These include the axial next-nearest-neighbor Ising (ANNNI) model. By extending our considerations, we comment on relations pertaining not only to the modulation lengths (or times), but also to the standard correlation lengths (or times). We introduce the notion of a Josephson time scale. We comment on the presence of aperiodic “chaotic” modulations in “soft-spin” and other systems. These relate to glass-type features. We discuss applications to Fermi systems, with particular application to metal to band insulator transitions, change of Fermi surface topology, divergent effective masses, Dirac systems, and topological insulators. Both regular periodic and glassy (and spatially chaotic behavior) may be found in strongly correlated electronic systems.

We examine the effectiveness of assuming an equal probability for states far from equilibrium. For this aim, we propose a method to construct a master equation for extensive variables describing nonstationary nonequilibrium dynamics. The key point of the method is the assumption that transient states are equivalent to the equilibrium state that has the same extensive variables, i.e., an equal probability holds for microscopic states in nonequilibrium. We demonstrate an application of this method to the critical relaxation of the two-dimensional Potts model by Monte Carlo simulations. While the one-variable description, which is adequate for equilibrium, yields relaxation dynamics that are very fast, the redundant two-variable description well reproduces the true dynamics quantitatively. These results suggest that some class of the nonequilibrium state can be described with a small extension of degrees of freedom, which may lead to an alternative way to understand nonequilibrium phenomena.

Whether or not there is growing static order accompanying the dynamical heterogeneity and increasing relaxation times seen in glassy systems is a matter of dispute. An obstacle to resolving this issue is that the order is expected to be amorphous and so not amenable to simple order parameters. We use mutual information to provide a general measurement of order that is sensitive to multiparticle correlations. We apply this to two glass-forming systems (two-dimensional binary mixtures of hard disks with different size ratios to give varying amounts of hexatic order) and show that there is little growth of amorphous order in the system without crystalline order. In both cases we measure the dynamical length with a four-point correlation function and find that it increases significantly faster than the static lengths in the system as density is increased. We further show that we can recover the known scaling of the dynamic correlation length in a kinetically constrained model, the two-vacancy-assisted-hopping triangular lattice gas.

We investigate the effect of a homogeneous high-frequency stimulation (HFS) on a one-dimensional chain of coupled excitable elements governed by the FitzHugh-Nagumo equations. We eliminate the high-frequency term by the method of averaging and show that the averaged dynamics depends on the parameter $A=a/\omega$ equal to the ratio of the amplitude $a$ to the frequency $\omega$ of the stimulating signal, so that for large frequencies an appreciable effect from the HFS is attained only at sufficiently large amplitudes. The averaged equations are analyzed by an asymptotic theory based on the different time scales of the recovery and excitable variables. As a result, we obtain the main characteristics of a propagating pulse as functions of the parameter $A$ and derive an analytical criterion for the propagation failure. We show that depending on the parameter $A$, the HFS can either enhance or suppress pulse propagation and reveal the mechanism underlying these effects. The theoretical results are confirmed by numerical simulations of the original system with and without noise.

The nonlinear dynamics of Taylor-Couette flow in a small-aspect-ratio wide-gap annulus in the counterrotating regime is investigated by solving the full three-dimensional Navier-Stokes equations. The system is invariant under arbitrary rotations about the axis, reflection about the annulus midplane, and time translations. A systematic investigation is presented both in terms of the flow physics elucidated from the numerical simulations and from a dynamical system perspective provided by equivariant normal form theory. The dynamics are primarily associated with the behavior of the jet of angular momentum that emerges from the inner cylinder boundary layer at about the midplane. The sequence of bifurcations as the differential rotation is increased consists of an axisymmetric Hopf bifurcation breaking the reflection symmetry of the basic state leading to an axisymmetric limit cycle with a half-period-flip spatiotemporal symmetry. This undergoes a Hopf bifurcation breaking axisymmetry, leading to quasiperiodic solutions evolving on a 2-torus that is setwise symmetric. These undergo a further Hopf bifurcation, introducing a third incommensurate frequency leading to a 3-torus that is also setwise symmetric. On the 3-torus, as the differential rotation is further increased, a saddle-node-invariant-circle bifurcation takes place, destroying the 3-torus and leaving a pair of symmetrically related 2-tori states on which all symmetries of the system have been broken.

We formulate the quasi-Lagrangian fluid transport dynamics of mass density $\rho$ and the projection $q=\mathbf{\omega }·\nabla \rho$ of the vorticity $\mathbf{\omega }$ onto the density gradient, as determined by the three-dimensional compressible Navier-Stokes equations for an ideal gas, although the results apply for an arbitrary equation of state. It turns out that the quasi-Lagrangian transport of $q$ cannot cross a level set of $\rho$. That is, in this formulation, level sets of $\rho$ (isopycnals) are impermeable to the transport of the projection $q$.

A novel discrete model (D model) is presented describing nonlinear wave interactions in systems with small and moderate nonlinearity under narrow frequency band excitation. It integrates in a single theoretical frame two mechanisms of energy transport between modes, namely, intermittency and energy cascade, and gives the conditions under which each regime will take place. Conditions for the formation of a cascade, cascade direction, conditions for cascade termination, etc., are given and depend strongly on the choice of excitation parameters. The energy spectra of a cascade may be computed, yielding discrete and continuous energy spectra. The model does not require statistical assumptions, as all effects are derived from the interaction of distinct modes. In the example given—surface water waves with dispersion function ${\omega }^2=gk$ and small nonlinearity—the D model predicts asymmetrical growth of side-bands for Benjamin-Feir instability, while the transition from discrete to continuous energy spectrum, excitation parameters properly chosen, yields the saturated Phillips’ power spectrum $\sim$$g^2{\omega }^{-5}$. The D model can be applied to the experimental and theoretical study of numerous wave systems appearing in hydrodynamics, nonlinear optics, electrodynamics, plasma, convection theory, etc.

The estimate of free energy changes based on Bennett's acceptance ratio method is examined in several limiting cases and compared with other estimates based on the Jarzynski equality and on the Crooks relation. While the absolute amount of the dissipated work, defined as the surplus of the average work over the free energy difference, limits the practical applicability of Jarzynski's and Crooks’ methods, the reliability of Bennett's approach is restricted by the difference of the dissipated works in the forward and the backward processes. We illustrate these points by considering a Gaussian chain and a hairpin chain which both are extended during the forward and accordingly compressed during the backward protocols. The reliability of the Crooks relation predominantly depends on the sample size; for the Jarzynski estimator the slowness of the work protocol is crucial, and the Bennett method is shown to give precise estimates irrespective of the pulling speed and sample size as long as the dissipated works are the same for the forward and the backward processes as it is the case for Gaussian work distributions. With an increasing dissipated work difference the Bennett estimator also acquires a bias which increases roughly in proportion to this difference. A substantial simplification of the Bennett estimator is provided by the 1/2 formula which expresses the free energy difference by the algebraic average of the Jarzynski estimates for the forward and the backward processes. It agrees with the Bennett estimate in all cases when the Jarzynski and the Crooks estimates fail to give reliable results.

This contribution reports on numerical simulations of two-dimensional granular flows on erodible beds. The broad aim is to investigate whether simple flows of model granular matter exhibit spontaneous oscillatory motion in generic flow conditions, and in this case, whether the frictional properties of the contacts between grains may affect the existence or the characteristics of this oscillatory motion. The analysis of different series of simulations shows that the flow develops an oscillatory motion with a well-defined frequency which increases like the inverse of the velocity's square root. We show that the oscillation is essentially a surface phenomenon. The amplitude of the oscillation is higher for lower volume fractions and can thus be related to the flow velocity and grains' friction properties. The study of the influence of the periodic geometry of the simulation cell shows no significant effect. These results are discussed in relation to sonic sands.