We present an experimental study of the kinetics of orbitally-shaken, sliding macroscopic particles confined to a two-dimensional space bounded by walls. Discounting the forcing action of the external periodic actuation, the particles undergo a qualitative transition from a ballistic to a diffusive motion regime with time. Despite the deterministic input of kinetic energy provided by the shaker, the particles show translational velocities and diffusivity consistent with a confined random walk model. Such experimental system may therefore represent a suitable macroscopic analog to investigate aspects of molecular dynamics and self-assembly.

We analyze a large system of heterogeneous quadratic integrate-and-fire (QIF) neurons with time delayed, all-to-all synaptic coupling. The model is exactly reduced to a system of firing rate equations that is exploited to investigate the existence, stability, and bifurcations of fully synchronous, partially synchronous, and incoherent states. In conjunction with this analysis we perform extensive numerical simulations of the original network of QIF neurons, and determine the relation between the macroscopic and microscopic states for partially synchronous states. The results are summarized in two phase diagrams, for homogeneous and heterogeneous populations, which are obtained analytically to a large extent. For excitatory coupling, the phase diagram is remarkably similar to that of the Kuramoto model with time delays, although here the stability boundaries extend to regions in parameter space where the neurons are not self-sustained oscillators. In contrast, the structure of the boundaries for inhibitory coupling is different, and already for homogeneous networks unveils the presence of various partially synchronized states not present in the Kuramoto model: Collective chaos, quasiperiodic partial synchronization (QPS), and a novel state which we call modulated-QPS (M-QPS). In the presence of heterogeneity partially synchronized states reminiscent to collective chaos, QPS and M-QPS persist. In addition, the presence of heterogeneity greatly amplifies the differences between the incoherence stability boundaries of excitation and inhibition. Finally, we compare our results with those of a traditional (Wilson Cowan-type) firing rate model with time delays. The oscillatory instabilities of the traditional firing rate model qualitatively agree with our results only for the case of inhibitory coupling with strong heterogeneity.

We study the relationship between the modularity of scale-free excitable networks and their ability to support self-sustained oscillation patterns. We find that the probability for a network of given degree-distribution exponent to be able to support self-sustained oscillations is strongly affected by its modularity. In addition, both high- and low-modularity networks are more likely to exhibit long-period oscillation patterns than those with intermediate modularity, but the degrees of complexity and correlation in these two cases are different. The long-period oscillations cannot be explained by a minimum-length Winfree loop, but instead arise from the interplay between two or more propagating waves. Finally, we introduce an improved method that can be used to analyze the structure of the self-sustained oscillation sources at different levels of detail and show that the period of the oscillation pattern is statistically correlated with the fraction of modules that are part of the oscillation source.

We explore a new simulation scheme for partial differential equations (PDE's) called information field dynamics (IFD). Information field dynamics is a probabilistic numerics method that seeks to preserve the maximum amount of information about the field being simulated. It rests on Bayesian field inference and therefore allows the incorporation of prior knowledge on the field. This makes IFD attractive to address the closure problem of simulations—how to incorporate knowledge about subgrid dynamics into a scheme on a grid with limited resolution. Here we analytically prove that a restricted subset of simulation schemes in IFD are consistent and thus deliver valid predictions in the limit of high resolutions. This has not previously been done for any IFD schemes. This restricted subset is roughly analogous to traditional fixed-grid numerical PDE solvers, given the additional restriction of translational symmetry. Furthermore, given an arbitrary IFD scheme modeling a PDE, it is a priori not obvious to what order the scheme is accurate in space and time. For this subset of models, we also derive an easy rule of thumb for determining the order of accuracy of the simulation. As with all analytic consistency analysis, an analysis for nontrivial systems is intractable; thus these results are intended as a general indicator of the validity of the approach, and it is hoped that the results will generalize.

To capture the dynamic behaviors of reaction-subdiffusion in linear flow fields, in the present paper we analyze a simple monomolecular conversion $A\to B$. We derive the corresponding master equations for the distribution of $A$ and $B$ particles in continuous time random walk schemes. The results are then used to obtain the generalizations of the advection-diffusion reaction equation in which the diffusion and advection operators both depend on the reaction rate.

We show that model-based Bayesian clustering, the probabilistically most systematic approach to the partitioning of data, can be mapped into a statistical physics problem for a gas of particles and as a result becomes amenable to a detailed quantitative analysis. A central role in the resulting statistical physics framework is played by an entropy function. We demonstrate that there is a relevant parameter regime where mean-field analysis of this function is exact and that, under natural assumptions, the lowest entropy state of the hypothetical gas corresponds to the optimal clustering of data. The by-product of our analysis is a simple but effective clustering algorithm, which infers both the most plausible number of clusters in the data and the corresponding partitions. Describing Bayesian clustering in statistical mechanical terms is found to be natural and surprisingly effective.

This paper reports results of a study of cyclization kinetics in finite random graphs. The relationship between the generating function for the number of edges in the graph and the generating function of linked components applies for deriving the exact expression for the spectrum of linked components in finite random graphs with a given number of edges. The spectrum of cycled linked components of the random graph is shown to be a product of two terms one of which is independent of the component structures and sensitive to the total size of the graph while the second one depends on the fine structure of the graph and does not include any dependence on the global structure of the graph. The latter term is expressed via quite analyzable integrals the asymptotics of which at large orders $g$ of the components defines the critical properties of the spectra of $k$-cycled linked components. Cyclization dominates after emergence of the giant component, with the $k$-cycled components being algebraically distributed as $U_k\left(g\right)\propto g^{(3k-5)/2}$ at the critical point. The spectra are shown to be modulated by a factor independent of the complexity of the graph.

Based on the connection between the spectral form factor and the probability to return, the origin of the energy level fluctuation $1/f^{\alpha }$ noise in fully chaotic and fully integrable systems is traced to the quantum interference between invariant manifolds of the classical dynamics and the dimensionality of those invariant manifolds. This connection and the order-to-chaos transition are analyzed in terms of the statistics of Floquet's quasienergies of a classically chaotic driving nonlinear system. An immediate prediction of the connection established here is that in the presence of decoherence, the spectral exponent $\alpha$ takes the same value, $\alpha =2$, for both fully chaotic and fully integrable systems.

Drying of colloidal drops on solid surfaces is the widely known method to form self-assembled patterns. The underlying principle of this method is the phenomenon known as the coffee-ring effect. Here, we report a phenomenon of pattern formation involving not drying but conversely wetting and spreading of drops on a solid surface containing a thin layer of dispersed particulates. Fascinating ringlike patterns are formed in a subsecond timescale by the interplay between the dynamics of spreading and imbibition. Occasionally, such patterns can be observed when, say, water spills on dusted floors and they are often misidentified as those formed by the coffee-ring effect. In the highly wetting scenario, we found that this pattern formation is independent of the liquid properties, however it is strongly dependent on the powder properties. Our findings have both fundamental and technological importance.

The macroscopic heterojunction materials under Fourier law can present a thermal rectification effect due to their asymmetrical property, where the opposite temperature dependence of thermal conductivity for two junctions plays a important role. Here we investigate the relation between the rectification ratio and the coefficients of the temperature-dependent thermal conductivities of the junctions. For heterojunction materials with linear temperature-dependent thermal conductivities, the maximal rectification ratio is predicted to be 0.5. For the quadratic temperature-dependent–thermal-conductivity case, the maximal rectification ratio can be improved to about 0.86. The thermal rectification can asymptotically approach a maximum value of 1 for high-order–temperature-dependent cases. To obtain the optimal thermal rectification, we present a temperature-dependent thermal-conductivity function law which gives practical guidance for searching an ideal thermal diode.

We study numerically the properties of (statistically) homogeneous soliton gas depending on soliton density (proportional to number of solitons per unit length) and soliton velocities, in the framework of the focusing one-dimensional nonlinear Schrödinger (NLS) equation. To model such gas we use $N$-soliton solutions ($N$-SS) with $N\sim 100$, which we generate with specific implementation of the dressing method combined with 100-digits arithmetics. We examine the major statistical characteristics, in particular the kinetic and potential energies, the kurtosis, the wave-action spectrum and the probability density function (PDF) of wavefield intensity. We show that in the case of small soliton density the kinetic and potential energies, as well as the kurtosis, are very well described by the analytical relations derived without taking into account soliton interactions. With increasing soliton density and velocities, soliton interactions enhance, and we observe increasing deviations from these relations leading to increased absolute values for all of these three characteristics. The wave-action spectrum is smooth, decays close to exponentially at large wavenumbers and widens with increasing soliton density and velocities. The PDF of wave intensity deviates from the exponential (Rayleigh) PDF drastically for rarefied soliton gas, transforming much closer to it at densities corresponding to essential interaction between the solitons. Rogue waves emerging in soliton gas are multisoliton collisions, and yet some of them have spatial profiles very similar to those of the Peregrine solutions of different orders. We present example of three-soliton collision, for which even the temporal behavior of the maximal amplitude is very well approximated by the Peregrine solution of the second order.

Interactions between solitons and the coherent oscillation structures generated by localized disturbances via modulational instability are studied within the framework of the focusing nonlinear Schrödinger equation. Two main interaction regimes are identified based on the relative value of the velocity of the incident soliton compared to the amplitude of the background: soliton transmission and soliton trapping. Specifically, when the incident soliton velocity exceeds a certain threshold, the soliton passes through the coherent structure and emerges on the other side with its velocity unchanged. Conversely, when the incident soliton velocity is below the threshold, once the soliton enters the coherent structure, it remains confined there forever. It is demonstrated that the soliton is not destroyed, but its velocity inside the coherent structure is different from its initial one. Moreover, it is also shown that, depending on the location of the discrete eigenvalue associated to the soliton, these phenomena can also be accompanied by the generation of additional, localized propagating waves in the coherent structure, akin to a soliton-generated wake.

We study the stability and bifurcation structure of spatially extended patterns arising in nonlinear optical resonators with a Kerr-type nonlinearity and anomalous group velocity dispersion, as described by the Lugiato-Lefever equation. While there exists a one-parameter family of patterns with different wavelengths, we focus our attention on the pattern with critical wave number $k_c$ arising from the modulational instability of the homogeneous state. We find that the branch of solutions associated with this pattern connects to a branch of patterns with wave number $2k_c$. This next branch also connects to a branch of patterns with double wave number, this time $4k_c$, and this process repeats through a series of 2:1 spatial resonances. For values of the detuning parameter approaching $\theta =2$ from below the critical wave number $k_c$ approaches zero and this bifurcation structure is related to the foliated snaking bifurcation structure organizing spatially localized bright solitons. Secondary bifurcations that these patterns undergo and the resulting temporal dynamics are also studied.

We study the light scattering by localized quasiplanar excitations of a cholesteric liquid crystal known as spherulites. We quantitatively evaluate the cross section of the axis rotation for polarized light by taking into account the anisotropic optical properties of the medium and the peculiar shape of the excitations. Because of the complexity of the system under consideration, we exploit also a simplified analytical description of the spherulite and evaluate the scattering cross sections in the Born approximation. We compare the scattering data by the analytical and the numerical skyrmion solutions for several choices of the model paramenters. The effects of changing values of the driving external static electric (or magnetic) field is also considered. Possible applications of the phenomenon are envisaged.

We show how orbital-free molecular dynamics can be regarded as the limit of quantum molecular dynamics at high temperature for the calculation of the equation of state of carbon. As a result, we construct an approach that allows to reproduce the quantum molecular dynamics equation of state of carbon at high temperature by combining quantum and orbital-free molecular dynamics. We apply this approach to a wide range of density and temperature conditions of the liquid or plasma state: $30\ge \rho \ge 0.8\mathrm{g}/{\mathrm{cm}}^3, T\ge 1$ eV. Then we test the sensitivity of the equation of state obtained to the choice of the exchange-correlation functional. We retain the generalized gradient approximation of this functional, corrected to approximately include the influence of temperature, and compare the results obtained to published first-principles calculations and to experimental shock data.

The dominant cost of most lattice QCD simulations is the inversion of the Dirac operator required to calculate the force term in the rational hybrid Monte Carlo (RHMC) update. One way to improve this situation is to use multiple pseudofermions, which reduces the size and variance of this force and hence allows a larger integration step size to be used. This means fewer force term calculations are required, but at the cost of having to invert the Dirac operator for each pseudofermion field. This bottleneck can be addressed: recently there has been renewed interest in the use of block Krylov solvers, which can solve multiple right-hand-side vectors with significantly fewer iterations than are required if each vector is solved using a separate Krylov solver. We combine these two ideas, achieving a significant speed-up of RHMC lattice QCD simulations.

Recent studies unveiled the Fickian yet non-Gaussian (FNG) dynamics of many soft matter systems and suggested this phenomenon as a general characteristic of the diffusion in complex fluids. In particular, it was shown that the distribution of particle displacements in Fickian diffusion is not necessarily Gaussian, and thus the Einstein and Smoluchowski theory describing the Brownian motion of individual objects in a fluid would not be applicable. In the present work, we investigate whether the FNG dynamics so far reported in gels, granular materials, biological systems, and active matter systems is also a distinctive feature of colloidal liquid crystals. To this end, we perform Brownian dynamics simulations of oblate and prolate colloidal particles in the nematic phase. We detect a normal and Gaussian dynamics at short and long time scales, whereas, at intermediate time scales, a non-Fickian and non-Gaussian dynamics is found. Additionally, we revisit the nature of the decay of the self–van Hove correlation function, $G_s(r,t)$, which is approximated here with an ellipsoidal, rather than spherical, Gaussian distribution. The expression that we propose is able to correctly assess the Gaussian dynamics in inherently anisotropic systems, such as liquid crystals, where the standard Gaussian approximation of $G_s(r,t)$ would fail.

An asymmetric Brownian particle subjected to an external time-dependent force may acquire a net drift velocity and thus operate as a motor or ratchet, even if the external force is represented by an unbiased time-periodic function or by a zero-centered noise. For an adequate description of such ratchets, a conventional Langevin equation linear in the particle's velocity is insufficient, and one needs to take into account the first nonlinear correction to the dissipation force which emerges beyond the weak coupling limit. We derived microscopically the relevant nonlinear Langevin equation by extending the standard projection operation technique beyond the weak coupling limit. The particle is modeled as a rigid cluster of atoms and its asymmetry may be geometrical, compositional (when a cluster is composed of atoms of different types), or due to a combination of both factors. The drift velocity is quadratic in the external force's amplitude and increases with decreasing the force's frequency (for a periodic force) and inverse correlation time (for a fluctuating force). The maximum value of the drift velocity is independent on the particle's mass and achieved in the adiabatic limit, i.e., for an infinitely slow change of the external field.

The combination of linear and nonlinear potentials, both shaped as a single well, enables competition between the confinement and expulsion induced by the former and latter potentials, respectively. We demonstrate that this setting leads to spontaneous symmetry breaking (SSB) of the ground state in the respective generalized nonlinear Schrödinger (Gross–Pitaevskii) equation, through a spontaneous off-center shift of the trapped mode. Two different SSB bifurcation scenarios are possible, depending on the shape of the nonlinearity-modulation profile, which determines the nonlinear potential. If the profile is bounded (remaining finite at $\left|x\right|\to \infty$), at a critical value of the integral norm the spatially symmetric state loses its stability, giving rise to a pair of mutually symmetric stable asymmetric ones via a direct pitchfork bifurcation. On the other hand, if the nonlinear potential is unbounded, two unstable asymmetric modes merge into the symmetric metastable mode and destabilize it via an inverted pitchfork bifurcation. Parallel to a systematic numerical investigation, basic results are obtained in an analytical form. The settings can be realized in Bose–Einstein condensates and nonlinear optical waveguides.