We calculate the efficiency of a rejection-free dynamic Monte Carlo method for $d$-dimensional off-lattice homogeneous particles interacting through a repulsive power-law potential $r^{-p}$. Theoretically we find the algorithmic efficiency in the limit of low temperatures and∕or high densities is asymptotically proportional to ${\rho }^{(p+2)∕2}{T}^{-d∕2}$ with the particle density $\rho$ and the temperature $T$. Dynamic Monte Carlo simulations are performed in one-, two-, and three-dimensional systems with different powers $p$, and the results agree with the theoretical predictions.

Counting the number of ground states for a spin-glass or nondeterministic polynomial-complete combinatorial optimization problem is even more difficult than the already hard task of finding a single ground state. In this paper the entropy of minimum vertex covers of random graphs is estimated through a set of iterative equations based on the cavity method of statistical mechanics. During the iteration both the cavity entropy contributions and cavity magnetizations for each vertex are updated. This approach overcomes the difficulty of iterative divergence encountered in the zero-temperature first-step replica-symmetry-breaking (1RSB) spin-glass theory. It is still applicable when the 1RSB mean-field theory is no longer stable. The method can be extended to compute the entropies of ground states and metastable minimal-energy states for other random-graph spin-glass systems.

We develop a nonequilibrium mode-coupling theory for uniformly sheared systems starting from microscopic, thermostatted Sllod equations of motion. Our theory aims at describing stationary-state properties including rheological ones of sheared systems, and this is accomplished via two steps. First, a set of self-consistent equations is formulated based on the projection-operator formalism and on the mode-coupling approach for the transient density correlators which measure the correlations between the density fluctuations in the initial equilibrium state and the ones at later times after the shearing force is turned on. The transient time-correlation function formalism is then used which, combined with the mode-coupling approximation, expresses stationary-state properties in terms of the transient density correlators. A detailed comparison of our theory is also presented with the related mode-coupling theory which is based on the Smoluchowski equation for Brownian particles under stationary shearing.

Using discrete simulations, we investigate the behavior of a model granular material within an annular shear cell. Specifically, two-dimensional assemblies of disks are placed between two circular walls, the inner one rotating with prescribed angular velocity, while the outer one may expand or shrink and maintains a constant radial pressure. Focusing on steady state flows, we delineate in parameter space the range of applicability of the recently introduced constitutive laws for sheared granular materials (based on the inertial number). We discuss the two origins of the stronger strain rates observed near the inner boundary, the vicinity of the wall and the heteregeneous stress field in a Couette cell. Above a certain velocity, an inertial region develops near the inner wall, to which the known constitutive laws apply, with suitable corrections due to wall slip, for small enough stress gradients. Away from the inner wall, slow, apparently unbounded creep takes place in the nominally solid material, although its density and shear to normal stress ratio are on the jammed side of the critical values. In addition to rheological characterizations, our simulations provide microscopic information on the contact network and velocity fluctuations that is potentially useful to assess theoretical approaches.

We investigate the influence of directional or bonding interactions on the structure and phase diagram of complex fluids. Using a generalization of the theory of associating fluids we study the interplay between the self-assembly process, driven by the bonding interactions, and the isotropic-nematic transition, driven by the anisotropic shape of the equilibrium clusters, for a model consisting of particles with two bonding sites and discrete orientational degrees of freedom. The theory is applied over a wide range of temperature and density in two dimensions and the results are compared with Monte Carlo simulations on the square lattice. The specific heat is shown to exhibit pronounced structure at the onset of self-assembly and at the nematic-isotropic transition that occur over a narrow range of temperature, at fixed density. The results reveal that bonding is enhanced by the nematic ordering, although a bonding temperature still occurs in the isotropic phase at low densities. The average rod length is described quantitatively in both phases, while the location of the ordering transition, which was found to be continuous, is predicted semiquantitatively by the theory.

By use of a reduced model focusing on the in-plane dependence of plane Couette flow, it is shown that the $\text{turbulent}\to \text{laminar}$ relaxation process can be understood as a nucleation problem similar to that occurring at a thermodynamic first-order phase transition. The approach, apt to deal with the large extension of the system considered, challenges the current interpretation in terms of chaotic transients typical of temporal chaos. The study of the distribution of the sizes of laminar domains embedded in turbulent flow proves that an abrupt transition from sustained spatiotemporal chaos to laminar flow can take place at some given value of the Reynolds number $R_{\text{low}}$, whether or not the local chaos lifetime, as envisioned within low-dimensional dynamical systems theory, diverges at finite $R$ beyond $R_{\text{low}}$.

We study complex systems or networks in which each node holds an internal dynamics and interacts with other nodes through some kinds of topologies. Collective behavior with dynamical fluctuations is analyzed in complex systems. The dynamical fluctuations of a node can be divided into two parts: one is the internal dynamical fluctuation of the node and the other is the external dynamical fluctuation caused by other nodes. Based on a theoretical analysis, a hidden feedback mechanism is identified in complex systems, which is illustrated in a macroeconomic network and in a city-population network. Furthermore, we study the effect of the topology of the networks on the feedback mechanism. The feedback mechanism is preserved for hub nodes in the networks with a scale-free topology as well as in the networks with an evolving topology. By the hidden feedback mechanism, the observation data can be utilized to judge directly whether the system of each node is with positive feedback or with negative feedback even without knowing its dynamical model.

A network analysis is used to investigate the low connections of natural river channels. At the basin scale, the river networks are analyzed according to the Horton-Strahler hierarchy. We propose a quantitative criterion for the average junction degree as a function of a fixed hierarchical order of the network and independent of the usual scaling laws. The numerical results of this analysis are compared with exact results of the Peano river network, showing differences of the order of ${10}^{-3}$. This aspect is especially relevant for the characterization of transport and diffusion processes at the basin scale.

In this paper, we examine the dynamics of reaction-diffusion systems with fractional time derivatives. It is shown that in these conditions diffusion is anomalous, in the sense that the mean-square displacement $⟨r^2⟩\sim t^{\gamma }$, where $\gamma <1$, a situation known as subdiffusion. We study the conditions for the appearance of a diffusion-driven instability and show that the restrictive conditions for a Turing instability are relaxed. This implies that systems whose kinetics are not of the activator-inhibitor kind can have a Turing instability and a modulated final state. We demonstrate our results with numerical calculations in two dimensions using a generic Turing model.

Soft biological materials can exhibit various active motion such as limit-cycle oscillation. The limit-cycle oscillation of active matter might be used for a mechanism of locomotion. We propose a simple model of an active elastic chain as an extension of the van der Pol equation. The uniform state is unstable, and exhibits a limit cycle of breathing motion. If the breathing motion is mirror symmetric, the elastic chain does not move as a whole. However, the breathing motion becomes a caterpillar motion and a unidirectional motion is induced, if an additional heterogeneity is involved or the chain is set on a spatially periodic sawtooth potential. We also analyze the model equation with coupled mode equations, and try to understand the bifurcation to the collective oscillation and the directional motion.

We report on spatiotemporal evolution of relativistically intense longitudinal electron plasma waves in a cold homogeneous plasma, using the physically appealing Dawson sheet model. Calculations presented here in the weakly relativistic limit clearly show that under very general initial conditions, a relativistic wave will always phase mix and eventually break at arbitrarily low amplitudes, in a time scale ${\omega }_{\mathrm{pe}}{\tau }_{\text{mix}}\sim {\{\frac{3}{64}({\omega }_{\mathrm{pe}}^2{\delta }^3∕c^2{k}^2)[\mid \Delta k∕k\mid ∕\left(\mid 1+\Delta k∕k\mid \right)](1+1∕\mid 1+\Delta k∕k\mid )\}}^{-1}$. We have verified this scaling with respect to amplitude of perturbation $\delta$ and width of the spectrum $(\Delta k∕k)$ using numerical simulations. This result may be of relevance to ultrashort, ultraintense laser pulse-plasma interaction experiments where relativistically intense waves are excited.

We argue that the classical theory of electromagnetism is based on Maxwell’s macroscopic equations, an energy postulate, a momentum postulate, and a generalized form of the Lorentz law of force. These seven postulates constitute the foundation of a complete and consistent theory, thus eliminating the need for actual (i.e., physical) models of polarization $\mathbf{P}$ and magnetization $\mathbf{M}$, these being the distinguishing features of Maxwell’s macroscopic equations. In the proposed formulation, $\mathbf{P}(\mathbf{r},t)$ and $\mathbf{M}(\mathbf{r},t)$ are arbitrary functions of space and time, their physical properties being embedded in the seven postulates of the theory. The postulates are self-consistent, comply with the requirements of the special theory of relativity, and satisfy the laws of conservation of energy, linear momentum, and angular momentum. One advantage of the proposed formulation is that it sidesteps the long-standing Abraham-Minkowski controversy surrounding the electromagnetic momentum inside a material medium by simply “assigning” the Abraham momentum density $\mathbf{E}(\mathbf{r},t)\times \mathbf{H}(\mathbf{r},t)∕c^2$ to the electromagnetic field. This well-defined momentum is thus taken to be universal as it does not depend on whether the field is propagating or evanescent, and whether or not the host medium is homogeneous, transparent, isotropic, dispersive, magnetic, linear, etc. In other words, the local and instantaneous momentum density is uniquely and unambiguously specified at each and every point of the material system in terms of the $\mathbf{E}$ and $\mathbf{H}$ fields residing at that point. Any variation with time of the total electromagnetic momentum of a closed system results in a force exerted on the material media within the system in accordance with the generalized Lorentz law.

We perform a numerical study of the three-dimensional random-field Ising model at $T=0$. We compare work distributions along metastable trajectories obtained with the single-spin flip dynamics with the distribution of the internal energy change along equilibrium trajectories. The goal is to investigate the possibility of extending the Crooks fluctuation theorem [G. E. Crooks, Phys. Rev. E. 60, 2721 (1999)] to zero temperature when, instead of the standard ensemble statistics, one considers the ensemble generated by the quenched disorder. We show that a simple extension of Crooks fails close to the disordered induced equilibrium phase transition due to the fact that work and internal energy distributions are very asymmetric.

We show how the dynamic renormalization of nonequilibrium systems can be carried out within the general framework of nonequilibrium thermodynamics. Whereas the renormalization of Hamiltonians is well known from equilibrium thermodynamics, the renormalization of dissipative brackets, or friction matrices, is the main new feature for nonequilibrium systems. Renormalization is a reduction rather than a coarse-graining technique; that is, no new dissipative processes arise in the dynamic renormalization procedure. The general ideas are illustrated for dilute polymer solutions where, in renormalizing bead-spring chain models, dissipative hydrodynamic interactions between different smaller beads contribute to the friction coefficient of a single larger bead.

We show that a $(2+1)$-dimensional discrete surface growth model exhibiting Kardar-Parisi-Zhang (KPZ) class scaling can be mapped onto a two-dimensional conserved lattice gas model of directed dimers. The KPZ height anisotropy in the surface model corresponds to a driven diffusive motion of the lattice gas dimers. We confirm by numerical simulations that the scaling exponents of the dimer model are in agreement with those of the $(2+1)$-dimensional KPZ class. This opens up the possibility of analyzing growth models via reaction-diffusion models, which allow much more efficient computer simulations.

We investigate the condensation-driven aggregation model that we recently proposed whereby an initial ensemble of chemically identical Brownian particles are continuously growing by condensation and at the same time undergo aggregation upon collision. We solved the model exactly by using scaling theory for the case when a particle, say of size $x$, grows by an amount $\alpha x$ over the time it takes to collide with another particle of any size. It is shown that the particle size spectra exhibit transition to scaling $c(x,t)\sim t^{-\beta }\varphi (x∕t^z)$ accompanied by the emergence of a fractal of dimension $d_f=1/(1+2\alpha )$. A remarkable feature of this model is that it is governed by a nontrivial conservation law, namely, the $d_f\text{th}$ moment of $c(x,t)$ is time invariant. The reason why it remains conserved is explained. Exact values for the exponents $\beta$, $z$, and $d_f$ are obtained and it is shown that they obey a generalized scaling relation $\beta =(1+d_f)z$.

We present a detailed experimental study of high-temperature self-propagating fronts using image processing techniques. The intrinsic features of the wave propagation are investigated as a function of the combustion temperature $T_C$ for a model system made of titanium and silicon powders. Different front behavior is realized by changing the molar ratio $x$ of the mixture $\mathrm{Ti}+x\mathrm{Si}$. Outside the range $x=[0.3,1.5]$, no thermal front is propagating while inside, three regimes are observed: steady-state combustion which is characterized by a flat front propagating at constant velocity and two unsteady regimes. The combustion temperature (or the corresponding ratio $x$) is thus playing the role of bifurcation parameter leading from stationary state to complex behavior. In the titanium-rich mixture, the position of the front oscillates and hot spots propagate along the external border of the sample. At lower amounts of Ti, localized bright regions appear randomly and deform the front profile. The associated dynamical behavior is a relay-race mechanism which becomes more pronounced close to the combustion limit. Methods are developed to characterize the structural and dynamical properties of thermal waves near instabilities, with a special emphasis on the statistical aspects. It is clearly demonstrated that the mesoscopic scale phenomena interfere significantly with the macroscopic behavior. The experiments reveal front behaviors that cannot be described using the usual macroscopic theories.

We study the classical dynamics of bound state and scattering trajectories of the chlorine atom interacting with the HO molecule using a two-dimensional model in which the HO bond length is held fixed. The bound state system forms the HOCl molecule and at low energies is predominantly integrable. Below dissociation a number of bifurcations are observed, most notably a series of saddle-center bifurcations related to a 2:1 and at higher energies 3:1 resonance between bend and stretch motions. At energies above dissociation the classical phase space becomes dominated by a homoclinic tangle which induces a fractal distribution of singularities in all scattering functions. The structure of the homoclinic tangle is examined directly using Poincaré surfaces of section as well as indirectly through its influence on the time delay of the scattered chlorine atom and the angular momentum of the scattered HO molecule.

We study the electromagnetic behavior of spherical semishell structures that have cloaking material properties proposed by Pendry, Schurig, and Smith [Science 312, 1780 (2006)]. We use three-dimensional full-wave time-harmonic field analysis to evaluate the field and dipolar force distribution produced by these structures in free-space under plane wave illumination. We show that the optical force in proximity to these structures is suitable for active and size-selective manipulation and trapping of neutral nanoscale particles.

Lattices that can be represented in a kagomé-like form are shown to satisfy a universal percolation criticality condition, expressed as a relation between $P_3$, the probability that all three vertices in the triangle connect, and $P_0$, the probability that none connect. A linear approximation for $P_3\left(P_0\right)$ is derived and appears to provide a rigorous upper bound for critical thresholds. A numerically determined relation for $P_3\left(P_0\right)$ gives thresholds for the kagomé, site-bond honeycomb, $\left(3\text{−}{12}^2\right)$ lattice, and “stack-of-triangle” lattices that compare favorably with numerical results.