International Journal of Modern Physics B

Driven granular media — assemblies of discrete, macroscopic elements exposed to a source of mechanical energy — represent inherently out-of-equilibrium systems. Although granular media are ubiquitous in both nature and industry, due to their dissipative nature and resultant complex behaviors they remain startlingly poorly understood as compared to classical, thermodynamic systems. Nonetheless, in recent years it has been observed that the behaviors of granular media can, under certain circumstances, closely resemble those of equilibrium systems. One of the most important contemporary questions in the field of granular physics is whether these similarities are merely superficial, or whether the parallels run deep enough that the behaviors of these nonequilibrium systems can in fact be successfully captured using analogs to existing theoretical models developed for classical systems. In this review, we draw together the findings of a variety of recent studies where this question has been addressed, comparing and contrasting the results and conclusions presented. We focus our attention on vibrated and vibrofluidized granular beds, which provide a canonical system representative of various equilibrium and nonequilibrium physical systems, and whose simple dynamics offer a valuable testing ground for exploring the fundamental physics of the granular state.

We examine the long-term behavior of nonintegrable, energy-conserved, 1D systems of macroscopic grains interacting via a contact-only generalized Hertz potential and held between stationary walls. Existing dynamical studies showed the absence of energy equipartitioning in such systems, hence their long-term dynamics was described as quasi-equilibrium. Here, we show that these systems do in fact reach thermal equilibrium at sufficiently long times, as indicated by the calculated heat capacity. This phase is described by equilibrium statistical mechanics, opening up the possibility that the machinery of nonequilibrium statistical mechanics may be used to understand the behavior of these systems away from equilibrium.

A quantum impurity system out of equilibrium comprises two metallic reservoirs with different chemical potentials and one mediating spin impurity between them. We study the highly nonlinear tunneling conductance of this system, and clarify that three coherent peaks, one zero-bias peak and two side peaks, naturally appear in the tunneling conductance. We use the Liouvillian approach, in which a complete set of basis operators is available, and construct a Liouville matrix to obtain Green’s function at the mediating site. We show that the two coherent side peaks are the outcome of steady-state nonequilibrium combined with strong electron correlation at the mediating site. Tunneling in the quantum impurity system out of equilibrium is performed by an entangled state which is a linear combination of two Kondo singlets formed by the spin at the mediating site and the coherent spins in each reservoir. The fluctuation by incoherent spins is also included. The spectral weights and positions of the three coherent peaks are analytically discussed via atomic limit analysis. Our theoretical results well fit experimental data obtained for quantum point contacts with symmetric and asymmetric Kondo couplings.

The Trump phenomenon is argued to depart from current populist rise in Europe. According to a model of opinion dynamics from sociophysics the machinery of Trump’s amazing success obeys well-defined counter-intuitive rules. Therefore, his success was in principle predictable from the start. The model uses local majority rule arguments and obeys a threshold dynamics. The associated tipping points are found to depend on the leading collective beliefs, cognitive biases and prejudices of the social group which undertakes the public debate. And here comes the open sesame of the Trump campaign, which develops along two successive steps. During a first moment, Trump’s statement produces a majority of voters against him. But at the same time, according to the model the shocking character of the statement modifies the prejudice balance. In case the prejudice is present even being frozen among voters, the tipping point is lowered at Trump’s benefit. Nevertheless, although the tipping point has been lowered by the activation of frozen prejudices it is instrumental to preserve enough support from openly prejudiced people to be above the threshold. Then, as infuriated voters launch intense debate, occurrence of ties will drive progressively hostile people to shift their voting intention without needing to endorse the statement which has infuriated them. The ongoing debate does drive towards a majority for Trump. The possible Trump victory at November Presidential election is discussed. In particular, the model shows that to eventually win the Presidential election, Trump must not modify his past shocking attitude but to appeal to a different spectrum of frozen prejudices, which are common to both Democrats and Republicans.

In this paper, we expand upon our recent studies of an agent-based model of a battle between an intelligent army and an insurgent army to explore the role of modifying strategy according to the state of the battle (adaptive strategy) on battle outcomes. This model leads to surprising complexity and rich possibilities in battle outcomes, especially in battles between two well-matched sides. We contend that the use of adaptive strategies may be effective in winning battles.

Recent particulate flow research using a discrete element simulation-dynamical systems approach is described. The simulation code used is very efficient and the mathematical model is an integro-partial differential equation. Examples are presented to show the effectiveness of the approach.

The emergence of techniques to extract high-frequency high-resolution data introduces a new avenue for modal decomposition to assess the underlying dynamics, especially of complex flows. However, this task requires the differentiation of robust, repeatable flow constituents from noise and other irregular features of a flow. Traditional approaches involving low-pass filtering and principle components analysis have shortcomings. The approach outlined here, referred to as robust-mode analysis, is based on Koopman decomposition. Three applications to (a) a counter-rotating cellular flame state, (b) variations in financial markets, and (c) turbulent injector flows are provided.

This paper is meant as an accessible introduction to/tutorial on the analytical construction and numerical simulation of a class of nonstandard solitary waves termed peakompactons. These peaked compactly supported waves arise as solutions to nonlinear evolution equations from a hierarchy of nonlinearly dispersive Korteweg–de Vries-type models. Peakompactons, like the now-well-known compactons and unlike the soliton solutions of the Korteweg–de Vries equation, have finite support, i.e., they are of finite wavelength. However, unlike compactons, peakompactons are also peaked, i.e., a higher spatial derivative suffers a jump discontinuity at the wave’s crest. Here, we construct such solutions exactly by reducing the governing partial differential equation to a nonlinear ordinary differential equation and employing a phase-plane analysis. A simple, but reliable, finite-difference scheme is also designed and tested for the simulation of collisions of peakompactons. In addition to the peakompacton class of solutions, the general physical features of the so-called $K^{\#}(n,m)$ hierarchy of nonlinearly dispersive Korteweg–de Vries-type models are discussed as well.

The binary collision approximation has been successfully applied to the study of pulse propagation in several types of granular chains. Here we present the theory in a generalized and comprehensive way and discuss and review a number of results found in the literature.

This article details some of the techniques used to derive exact solutions from the discrete equations of motion of strongly and weakly nonlinear discrete systems. The distinction between strongly and weakly nonlinear systems is related to the amplitude of a traveling pulse and the external confining load. Materials with an anomalous strain-softening behavior will be emphasized (i.e., $f\sim {\delta }^n$, $n<1$), though this choice does not preclude applications for strain-hardening systems like those with Hertzian potentials. Discrete materials with tunable acoustic transmission properties and novel impact mitigation capacity have gained interest in recent years due to their practical application across many scientific fields. Wave-guides comprised of discrete materials with a nonlinear interaction potential have been used to investigate the interplay between nonlinearity, dispersion and dissipation.

Frictional constitutive law for very slow vertical withdrawing of a thin rod from a granular bed is experimentally studied. Using a very precise creep meter, geometry-dependent granular frictional constitutive law is particularly examined. In some previous works, a dimensionless number $I=\dot{\gamma }{D}_g/\sqrt{p{\rho }_g}$ has been used to characterize granular frictional constitutive laws, where $\dot{\gamma }$, $D_g$, $p$, and ${\rho }_g$ are the shear strain rate, grain diameter, confining pressure, and bulk density of granular bed, respectively. It has been considered that granular frictional constitutive law expressed by $I$ is universal (almost geometry-independent) in dense flow regime. In this study, however, we find that the geometry of the system is much more crucial to characterize granular friction in a very slow withdrawing regime. Specifically, the ratio between rod and grain diameters must be an essential parameter to describe the granular frictional constitutive law. Physical meaning of the geometry-dependent constitutive law is discussed on the basis of grains-contact-number dependence of granular behavior.

When repelling particles are confined in a quasi-one-dimensional trap by a transverse potential, a configurational phase transition happens. All particles are aligned along the trap axis at large confinement, but below a critical transverse confinement they adopt a staggered row configuration (zigzag phase). This zigzag transition is a subcritical pitchfork bifurcation in extended systems and in systems with cyclic boundary conditions in the longitudinal direction. Among many evidences, phase coexistence is exhibited by localized nonlinear patterns made of a zigzag phase embedded in otherwise aligned particles. We give the normal form at the bifurcation and we show that these patterns can be described as solitary wave envelopes that we call bubbles. They are stable in a large temperature range and can diffuse as quasi-particles, with a diffusion coefficient that may be deduced from the normal form. The potential energy of a bubble is found to be lower than that of the homogeneous bifurcated phase, which explains their stability. We observe also metastable states, that are pairs of solitary wave envelopes which spontaneously evolve toward a stable single bubble. We evidence a strong effect of the discreteness of the underlying particles system and introduce the concept of topological frustration of a bubble pair. A configuration is frustrated when the particles between the two bubbles are not organized in a modulated staggered row. For a nonfrustrated (NF) bubble pair configuration, the bubbles interaction is attractive so that the bubbles come closer and eventually merge as a single bubble. In contrast, the bubbles interaction is found to be repulsive for a frustrated (F) configuration. We describe a model of interacting solitary wave that provides all qualitative characteristics of the interaction force: it is attractive for NF-systems, repulsive for F-systems, and decreases exponentially with the bubbles distance.

We present the detailed dynamics of the particles in the $\beta$-Fermi–Pasta–Ulam–Tsingou (FPUT) chain after the initiation of a localized nonlinear excitation (LNE) by squeezing a central bond in the monodispersed chain at time $t$ = 0 while all other particles remain in their original relaxed positions. In the absence of phonons in the system, the LNE appears to initiate its relaxation process by symmetrically emitting two very weak solitary waves. The next stage involves the spreading of the LNE and the formation of nonsolitary wave-like objects to broaden the excitation region until a stage is reached when many weak solitary wave-like objects can be emitted as the system begins its journey to quasi-equilibrium and then to equilibrium. In addition to being of fundamental interest, these systems may be realized using cantilever systems and could well hold the key to constructing the next generation of broadband energy harvesting systems.

We examine the spectral properties of single and multiple matter-wave dark solitons in Bose–Einstein condensates confined in parabolic traps, where the scattering length is periodically modulated. In addition to the large density limit picture previously established for homogeneous nonlinearities, we explore a perturbative analysis in the vicinity of the linear limit, which provides good agreement with the observed spectral modes. Between these two analytically tractable limits, we use numerical computations to fill in the relevant intermediate regime. We find that the scattering length modulation can cause a variety of features absent for homogeneous nonlinearities. Among them, we note the potential oscillatory instability even of the single dark soliton, the potential absence of instabilities in the immediate vicinity of the linear limit for two dark solitons, and the existence of an exponential instability associated with the in-phase motion of three dark solitons.

We provide a review of propagating traveling waves and solitary pulses in uncompressed one-dimensional ($1d$) ordered granular media. The first such solution in homogeneous granular media was discovered by Nesterenko in the form of a single-hump solitary pulse with energy-dependent profile and velocity. Considering directly the discrete, strongly nonlinear governing equations of motion of these media (i.e., without resorting to continuum approximation or homogenization), we show the existence of countably infinite families of stable multi-hump propagating traveling waves with arbitrary wavelengths. A semi-analytical approach is used to study the dependence of these waves on spatial periodicity (wavenumber) and energy, and to show that in a certain asymptotic limit, these families converge to the single-hump Nesterenko solitary wave. Then the study is extended in dimer granular chains composed of alternating “heavy” and “light” beads. For a set of specific mass ratios between the light and heavy beads, we show the existence of multi-hump solitary waves that propagate faster than the Nesterenko solitary wave in the corresponding homogeneous granular chain composed of only heavy beads. The existence of these waves has interesting implications in energy transmission in ordered granular chains.

Solitary waves in alignment of elastic beads have been an important area of study. An important and rich area has been the behavior of solitary waves at a boundary, where features such as localization, anomalous behavior in scattering and transmission, quasiequilibrium phase, etc. are being studied. An application area of significance is the design of artificial granular alignments for shock decimation and dispersion. In this review article, we first present a summary and background of these unique features, and some designs in 1D which exploit these features. We further discuss some extensions to higher dimensional systems and their impact decimation ability.