We discuss the possibility to introduce geometrical constraints in shell models of turbulence in order to mimic the turbulent dynamics that takes place in fluid layers with large aspect ratio. By using a scale-dependent set of coupling parameters, we are able to resolve both scales larger and smaller than a geometrical dimension of the flow. The proposed model is able to resolve with high accuracy the split energy cascade phenomenon recently observed in such flows, and allows us to investigate in detail the scaling properties of turbulent convection confined in narrow convective cells.

We analyze the electromagnetic field of a charge crossing a boundary between a vacuum and cold plasma in a waveguide. We obtain exact expressions for the field components and the spectral density of the transition radiation. With the steepest descent technique, we investigate the field components. We show that the electromagnetic field has a different structure in a vacuum than in cold plasma. We also develop an algorithm for the computation of the field based on a certain transformation of the integration path. The behavior of the field depending on distance and time and the spectral density depending on frequency are explored for different charge velocities. Some important physical effects are noted. A considerable increase and concentration of the field near the wave front in the plasma is observed for the case of ultrarelativistic particles. In the plasma, the mode envelopes and spectral density show zero points when the charge velocity is within certain limits.

The polygon construction method of Glaser and Clark is used to characterize melting and crystallization in a two-dimensional (2D) strongly coupled dusty plasma. Using particle positions measured by video microscopy, bonds are identified by triangulation, and unusually long bonds are deleted. The resulting polygons have three or more sides. Geometrical defects, which are polygons with more than three sides, are found to proliferate during melting. Pentagons are found in liquids, where they tend to cluster with other pentagons. Quadrilaterals are a less severe defect, so that disorder can be characterized by the ratio of quadrilaterals to pentagons. This ratio is found to be less in a liquid than in a solid or a superheated solid. Another measure of disorder is the abundance of different kinds of vertices, according to the type of polygons that adjoin there. Unexpectedly, spikes are observed in the abundance of certain vertex types during rapid temperature changes. Hysteresis, revealed by a plot of a disorder parameter vs temperature, is examined to study sudden heating. The hysteresis diagram also reveals features suggesting a possibility of latent heat in the melting and rapid cooling processes.

We study the elastic theory of amorphous solids made of particles with finite range interactions in the thermodynamic limit. For the elastic theory to exist, one requires all the elastic coefficients, linear and nonlinear, to attain a finite thermodynamic limit. We show that for such systems the existence of nonaffine mechanical responses results in anomalous fluctuations of all the nonlinear coefficients of the elastic theory. While the shear modulus exists, the first nonlinear coefficient $B_2$ has anomalous fluctuations and the second nonlinear coefficient $B_3$ and all the higher order coefficients (which are nonzero by symmetry) diverge in the thermodynamic limit. These results call into question the existence of elasticity (or solidity) of amorphous solids at finite strains, even at zero temperature. We discuss the physical meaning of these results and propose that in these systems elasticity can never be decoupled from plasticity: the nonlinear response must be very substantially plastic.

We study reversible polymerization of rings. In this stochastic process, two monomers bond and, as a consequence, two disjoint rings may merge into a compound ring or a single ring may split into two fragment rings. This aggregation-fragmentation process exhibits a percolation transition with a finite-ring phase in which all rings have microscopic length and a giant-ring phase where macroscopic rings account for a finite fraction of the entire mass. Interestingly, while the total mass of the giant rings is a deterministic quantity, their total number and their sizes are stochastic quantities. The size distribution of the macroscopic rings is universal, although the span of this distribution increases with time. Moreover, the average number of giant rings scales logarithmically with system size. We introduce a card-shuffling algorithm for efficient simulation of the ring formation process and we present numerical verification of the theoretical predictions.

We use a phase-field model for the growth of dendrites in dilute binary alloys under coupled thermosolutal control to explore the dependence of the dendrite tip velocity and radius of curvature upon undercooling, Lewis number (ratio of thermal to solutal diffusivity), alloy concentration, and equilibrium partition coefficient. Constructed in the quantitatively valid thin-interface limit, the model uses advanced numerical techniques such as mesh adaptivity, multigrid, and implicit time stepping to solve the nonisothermal alloy solidification problem for material parameters that are realistic for metals. From the velocity and curvature data we estimate the dendrite operating point parameter σ${}^{*}$. We find that σ${}^{*}$ is nonconstant and, over a wide parameter space, displays first a local minimum followed by a local maximum as the undercooling is increased. This behavior is contrasted with a similar type of behavior to that predicted by simple marginal stability models to occur in the radius of curvature, on the assumption of constant σ${}^{*}$.

We consider a nontrivial one-species population dynamics model with finite and infinite carrying capacities. Time-dependent intrinsic and extrinsic growth rates are considered in these models. Through the model per capita growth rate we obtain a heuristic general procedure to generate scaling functions to collapse data into a simple linear behavior even if an extrinsic growth rate is included. With this data collapse, all the models studied become independent from the parameters and initial condition. Analytical solutions are found when time-dependent coefficients are considered. These solutions allow us to perceive nontrivial transitions between species extinction and survival and to calculate the transition’s critical exponents. Considering an extrinsic growth rate as a cancer treatment, we show that the relevant quantity depends not only on the intensity of the treatment, but also on when the cancerous cell growth is maximum.

We propose a simple mechanical model of locomotion induced by internal force and friction. We first construct a system of two elements as an analog of the bipedal motion. The internal force does not induce a directional motion by itself because of the action-reaction law, but a directional motion becomes possible by the control of the frictional force. The efficiency of these model systems is studied using an analogy to the heat engine. As a modified version of the two-element model, we construct a model that exhibits a bipedal motion similar to kinesin’s motion of molecular motor. Next, we propose a linear chain model and a ladder model as an extension of the original two-element model. We find a transition from a straight to a snake-like motion in a ladder model by changing the strength of the internal force.

While it is well recognized that realistic networks are typically growing with time, the dynamical features of their growing processes remain to be explored. In the present paper, incorporating the requirement of synchronization stability into the conventional models of network growth, we will investigate how the growing process of a complex network is influenced by, and also will influence, the network collective dynamics. Our study shows that, constrained by the synchronization stability, the network will be growing in a selective and dynamical fashion. In particular, we find that the chance for a new node to be accepted by the growing network could have a large variation, i.e., it follows roughly a power-law distribution. Furthermore, we find that, with the dynamical growth, the network is always developed into structures of clear scale-free features, despite the form of the link attachment (preferential or random). The dynamical properties of network growth are studied using the method of eigenvalue analysis, and they are verified by direct simulations of coupled chaotic oscillators. Our study implies that, driven by the network collective dynamics, network growth could also be highly dynamical.

We combine a discrete-element-method simulation with a stochastic process to model the movement of spherical particles in a turbulent channel flow. With this model we investigate the mixing properties of two species of particles flowing through the channel. We find a linear increase of the mixing zone with the length of the pipe. Flows at different Reynolds number are studied. Below a critical Reynolds number at the Taylor microscale of around $R_c\approx 300$ the mixing rate is strongly dependent on the Reynolds number. Above $R_c$ the mixing rate stays nearly constant.

In recent experiments [M. Dubois, B. Demé, T. Gulik-Krzywicki, J.-C. Dedieu, C. Vautrin, S. Désert, E. Perez, and T. Zemb, Nature (London) 411, 672 (2001)] the spontaneous formation of hollow bilayer vesicles with polyhedral symmetry has been observed. On the basis of the experimental phenomenology it was suggested [M. Dubois, V. Lizunov, A. Meister, T. Gulik-Krzywicki, J. M. Verbavatz, E. Perez, J. Zimmerberg, and T. Zemb, Proc. Natl. Acad. Sci. USA 101, 15082 (2004)] that the mechanism for the formation of bilayer polyhedra is minimization of elastic bending energy. Motivated by these experiments, we study the elastic bending energy of polyhedral bilayer vesicles. In agreement with experiments, and provided that excess amphiphiles exhibiting spontaneous curvature are present in sufficient quantity, we find that polyhedral bilayer vesicles can indeed be energetically favorable compared to spherical bilayer vesicles. Consistent with experimental observations we also find that the bending energy associated with the vertices of bilayer polyhedra can be locally reduced through the formation of pores. However, the stabilization of polyhedral bilayer vesicles over spherical bilayer vesicles relies crucially on molecular segregation of excess amphiphiles along the ridges rather than the vertices of bilayer polyhedra. Furthermore, our analysis implies that, contrary to what has been suggested on the basis of experiments, the icosahedron does not minimize elastic bending energy among arbitrary polyhedral shapes and sizes. Instead, we find that, for large polyhedron sizes, the snub dodecahedron and the snub cube both have lower total bending energies than the icosahedron.

Using fundamental-measure density functional theory we investigate entropic wetting in an asymmetric binary mixture of hard spheres with positive nonadditivity. We consider a general planar hard wall, where preferential adsorption is induced by a difference in closest approach of the different species and the wall. Close to bulk fluid-fluid coexistence, the phase rich in the minority component adsorbs either through a series of first-order layering transitions, where an increasing number of liquid layers adsorbs sequentially, or via a critical wetting transition, where a thick film grows continuously.

We study the transport of information between two complex systems with similar properties. Both systems generate non-Poisson renewal fluctuations with a power-law spectrum $1/f^{3-\mu }$, the case $\mu =2$ corresponding to ideal $1/f$ noise. We denote by ${\mu }_S$ and ${\mu }_P$ the power-law indexes of the system of interest $S$ and the perturbing system $P$, respectively. By adopting a generalized fluctuation-dissipation theorem (FDT) we show that the ideal condition of $1/f$ noise for both systems corresponds to maximal information transport. We prove that to make the system $S$ respond when ${\mu }_S<2$ we have to set the condition ${\mu }_P<2$. In the latter case, if ${\mu }_P<{\mu }_S$, the system $S$ inherits the relaxation properties of the perturbing system. In the case where ${\mu }_P>2$, no response and no information transmission occurs in the long-time limit. We consider two possible generalizations of the fluctuation dissipation theorem and show that both lead to maximal information transport in the condition of $1/f$ noise.

We inquire into the possible coexistence of macroscopic and microstructured phases in random $Q$-block copolymers built of incompatible monomer types $A$ and $B$ with equal average concentrations. In our microscopic model, one block comprises $M$ identical monomers. The block-type sequence distribution is Markovian and characterized by the correlation $\lambda$. Upon increasing the incompatibility $\chi$ (by decreasing temperature) in the disordered state, the known ordered phases form: for $\lambda >{\lambda }_c$, two coexisting macroscopic $A$- and $B$-rich phases, for $\lambda <{\lambda }_c$, a microstructured (lamellar) phase with wave number $k(\lambda )$. In addition, we find a fourth region in the $\lambda -\chi$ plane where these three phases coexist, with different, non-Markovian sequence distributions (fractionation). Fractionation is revealed by our analytically derived multiphase free energy, which explicitly accounts for the exchange of individual sequences between the coexisting phases. The three-phase region is reached, either from the macroscopic phases, via a third lamellar phase that is rich in alternating sequences, or, starting from the lamellar state, via two additional homogeneous, homopolymer-enriched phases. These incipient phases emerge with zero volume fraction. The four regions of the phase diagram meet in a multicritical point $({\lambda }_c,{\chi }_c)$, at which $A$-$B$ segregation vanishes. The analytical method, which for the lamellar phase assumes weak segregation, thus proves reliable particularly in the vicinity of $({\lambda }_c,{\chi }_c)$. For random triblock copolymers, $Q=3$, we find the character of this point and the critical exponents to change substantially with the number $M$ of monomers per block. The results for $Q=3$ in the continuous-chain limit $M\to \infty$ are compared to numerical self-consistent field theory (SCFT), which is accurate at larger segregation.

Based on counting statistics and Bogoliubov theory, we present a recurrence relation for the microcanonical partition function for a weakly interacting Bose gas with a finite number of particles in a cubic box. According to this microcanonical partition function, we calculate numerically the distribution function, condensate fraction, and condensate fluctuations for a finite and isolated Bose-Einstein condensate. For ideal and weakly interacting Bose gases, we compare the condensate fluctuations with those in the canonical ensemble. The present approach yields an accurate account of the condensate fluctuations for temperatures close to the critical region. We emphasize that the interactions between excited atoms turn out to be important for moderate temperatures.

We provide numerical constructions of one-dimensional hyperuniform many-particle distributions that exhibit unusual clustering and asymptotic local number density fluctuations growing more slowly than the volume of an observation window but faster than the surface area. Hyperuniformity, defined by vanishing infinite-wavelength local density fluctuations, provides a quantitative metric of global order within a many-particle configuration and signals the onset of an “inverted” critical point in which the direct correlation function becomes long ranged. By targeting a specified form of the structure factor at small wavenumbers ($S(k)~k^{\alpha }$ for $0<\alpha <1$) using collective density variables, we are able to tailor the form of asymptotic local density fluctuations while simultaneously measuring the effect of imposing weak and strong constraints on the available degrees of freedom within the system. This procedure is equivalent to finding the (possibly disordered) classical ground state of an interacting many-particle system with up to four-body interactions. Even in one dimension, the long-range effective interactions induce clustering and nontrivial phase transitions in the resulting ground-state configurations. We provide an analytical connection between the fraction of constrained degrees of freedom within the system and the disorder-order phase transition for a class of target structure factors by examining the realizability of the constrained contribution to the pair correlation function. Our results explicitly demonstrate that disordered hyperuniform many-particle ground states, and therefore also point distributions, with substantial clustering can be constructed. We directly relate the local coordination structure of our point patterns to the distribution of the void space external to the particles, and we provide a scaling argument for the configurational entropy (analogous to spin-frustated system) of the disordered ground states. By emphasizing the intimate connection between geometrical constraints on the particle distribution and structural regularity, our work has direct implications for higher-dimensional systems, including an understanding of the appearance of hyperuniformity and quasi-long-range pair correlations in maximally random strictly jammed packings of hard spheres.

We introduce the heterogeneous $k$-core, which generalizes the $k$-core, and contrast it with bootstrap percolation. Vertices have a threshold $r_i$, that may be different at each vertex. If a vertex has fewer than $r_i$ neighbors it is pruned from the network. The heterogeneous $k$-core is the subgraph remaining after no further vertices can be pruned. If the thresholds $r_i$ are 1 with probability $f$, or $k⩾3$ with probability $1-f$, the process can be thought of as a pruning process counterpart to ordinary bootstrap percolation, which is an activation process. We show that there are two types of transitions in this heterogeneous $k$-core process: the giant heterogeneous $k$-core may appear with a continuous transition and there may be a second discontinuous hybrid transition. We compare critical phenomena, critical clusters, and avalanches at the heterogeneous $k$-core and bootstrap percolation transitions. We also show that the network structure has a crucial effect on these processes, with the giant heterogeneous $k$-core appearing immediately at a finite value for any $f>0$ when the degree distribution tends to a power law $P(q)~q^{-\gamma }$ with $\gamma <3$.

Transport of point-size Brownian particles under the influence of a constant and uniform force field through a planar three-dimensional channel with smoothly varying, axis-symmetric periodic side walls is investigated. Here we employ an asymptotic analysis in the ratio between the difference of the widest and the most narrow constriction divided through the period length of the channel geometry. We demonstrate that the leading-order term is equivalent to the Fick-Jacobs approximation. By use of the higher-order corrections to the probability density we show that in the diffusion-dominated regime the average transport velocity is obtained as the product of the zeroth-order Fick-Jacobs result and the expectation value of the spatially dependent diffusion coefficient $D(x)$, which substitutes the constant diffusion coefficient in the common Fick-Jacobs equation. The analytic findings are corroborated with the precise numerical results of a finite element calculation of the Smoluchowski diffusive particle dynamics occurring in a reflection symmetric sinusoidal-shaped channel.

Hyperuniform many-particle distributions possess a local number variance that grows more slowly than the volume of an observation window, implying that the local density is effectively homogeneous beyond a few characteristic length scales. Previous work on maximally random strictly jammed sphere packings in three dimensions has shown that these systems are hyperuniform and possess unusual quasi-long-range pair correlations decaying as $r^{-4}$, resulting in anomalous logarithmic growth in the number variance. However, recent work on maximally random jammed sphere packings with a size distribution has suggested that such quasi-long-range correlations and hyperuniformity are not universal among jammed hard-particle systems. In this paper, we show that such systems are indeed hyperuniform with signature quasi-long-range correlations by characterizing the more general local-volume-fraction fluctuations. We argue that the regularity of the void space induced by the constraints of saturation and strict jamming overcomes the local inhomogeneity of the disk centers to induce hyperuniformity in the medium with a linear small-wave-number nonanalytic behavior in the spectral density, resulting in quasi-long-range spatial correlations scaling with $r^{-(d+1)}$ in $d$ Euclidean space dimensions. A numerical and analytical analysis of the pore-size distribution for a binary maximally random jammed system in addition to a local characterization of the $n$-particle loops governing the void space surrounding the inclusions is presented in support of our argument. This paper is the first part of a series of two papers considering the relationships among hyperuniformity, jamming, and regularity of the void space in hard-particle packings.

We extend the results from the first part of this series of two papers by examining hyperuniformity in heterogeneous media composed of impenetrable anisotropic inclusions. Specifically, we consider maximally random jammed (MRJ) packings of hard ellipses and superdisks and show that these systems both possess vanishing infinite-wavelength local-volume-fraction fluctuations and quasi-long-range pair correlations scaling as $r^{-(d+1)}$ in $d$ Euclidean dimensions. Our results suggest a strong generalization of a conjecture by Torquato and Stillinger [Phys. Rev. E 68, 041113 (2003)], namely, that all strictly jammed saturated packings of hard particles, including those with size and shape distributions, are hyperuniform with signature quasi-long-range correlations. We show that our arguments concerning the constrained distribution of the void space in MRJ packings directly extend to hard-ellipse and superdisk packings, thereby providing a direct structural explanation for the appearance of hyperuniformity and quasi-long-range correlations in these systems. Additionally, we examine general heterogeneous media with anisotropic inclusions and show unexpectedly that one can decorate a periodic point pattern to obtain a hard-particle system that is not hyperuniform with respect to local-volume-fraction fluctuations. This apparent discrepancy can also be rationalized by appealing to the irregular distribution of the void space arising from the anisotropic shapes of the particles. Our work suggests the intriguing possibility that the MRJ states of hard particles share certain universal features independent of the local properties of the packings, including the packing fraction and average contact number per particle.