We use a well-known model [T. Vicsek et al., Phys. Rev. Lett. 15, 1226 (1995)] for flocking, to test mutual information as a tool for detecting order-disorder transitions, in particular when observations of the system are limited. We show that mutual information is a sensitive indicator of the phase transition location in terms of the natural dimensionless parameters of the system which we have identified. When only a few particles are tracked and when only a subset of the positional and velocity components is available, mutual information provides a better measure of the phase transition location than the susceptibility of the data.

We compare eigenvalue densities of Wigner random matrices whose elements are independent identically distributed random numbers with a Lévy distribution and maximally random matrices with a rotationally invariant measure exhibiting a power law spectrum given by stable laws of free random variables. We compute the eigenvalue density of Wigner-Lévy matrices using (and correcting) the method by Bouchaud and Cizeau, and of free random Lévy (FRL) rotationally invariant matrices by adapting results of free probability calculus. We compare the two types of eigenvalue spectra. Both ensembles are spectrally stable with respect to the matrix addition. The discussed ensemble of FRL matrices is maximally random in the sense that it maximizes Shannon’s entropy. We find a perfect agreement between the numerically sampled spectra and the analytical results already for matrices of dimension $N=100$. The numerical spectra show very weak dependence on the matrix size $N$ as can be noticed by comparing spectra for $N=400$. After a pertinent rescaling, spectra of Wigner-Lévy matrices and of symmetric FRL matrices have the same tail behavior. As we discuss towards the end of the paper the correlations of large eigenvalues in the two ensembles are, however, different. We illustrate the relation between the two types of stability and show that the addition of many randomly rotated Wigner-Lévy matrices leads by a matrix central limit theorem to FRL spectra, providing an explicit realization of the maximal randomness principle.

Photopolymerization-induced crystallization has been demonstrated in blends of polyethylene oxide–diacrylate at temperatures above the depressed melting temperature of the crystalline component. Upon exposure to ultraviolet irradiation, the melting transition curve moves upward and eventually surpasses the reaction temperature, thereby inducing phase separation as well as crystallization. The present paper demonstrates the occurrence of directionally solidified interface morphologies of polymer crystals subjected to a photointensity gradient. The epitaxially grown seaweed or degenerate structures were observed at the circumference (low-intensity region) while the dense branched spherulites developed at the core high-intensity region.

We study an individual-based predator-prey model of biological coevolution, using linear stability analysis and large-scale kinetic Monte Carlo simulations. The model exhibits approximate $1∕f$ noise in diversity and population-size fluctuations, and it generates a sequence of quasisteady communities in the form of simple food webs. These communities are quite resilient toward the loss of one or a few species, which is reflected in different power-law exponents for the durations of communities and the lifetimes of species. The exponent for the former is near $-1$, while the latter is close to $-2$. Statistical characteristics of the evolving communities, including degree (predator and prey) distributions and proportions of basal, intermediate, and top species, compare reasonably with data for real food webs.

Lipid aggregation into fluid bilayers is an essential process for sustaining life. Simplified models of lipid structure, which allow for long time scales or large length scales not obtainable with all-atom simulations, have recently been developed and show promise for describing lipid dynamics in biological systems. Here, we describe two simplified models, a reduced-lipid model and a bola-lipid model for thermophilic bacterial membranes, developed for use with the rapid discrete molecular dynamics simulation method. In the reduced-lipid model, we represent the lipid chain by a series of three beads interacting through pairwise discrete potentials that model hydrophobic attractions between hydrocarbon tails in implicit solvent. Our phase diagram recapitulates those produced by continuous potential models with similar coarse-grained lipid representations. We also find that phase transition temperatures for our reduced-lipid model are dependent upon the flexibility of the lipid chain, giving an insight into archaebacterial membrane stability and prompting development of a bola-lipid model specific for archaebacteria lipids. With both the reduced-lipid and bola-lipid model, we find that the reduced flexibility inherent in archaebacteria lipids yields more stable bilayers as manifested by increased phase transition temperatures. The results of these studies provide a simulation methodology for lipid molecules in biological systems and show that discrete molecular dynamics is applicable to lipid aggregation and dynamics.

This paper extends and applies a self-consistent field theory of compressible, fully hydrated phospholipid bilayers to binary mixtures. The mixtures contain two kinds of lipids, which have identical head groups but different acyl chain lengths. The formalism we develop allows for the calculation of a range of thermodynamic and structural properties for systems at equilibrium, including the compatibility of the two components, the equilibrium phase, the bilayer thickness, the orientational order parameter profiles, and the ordering environment as a function of depth within the layer. Our focus in this paper is on the mutual effects of the different chains on each other, and the effects of the local environment on the order parameter of each segment in the chains, for systems in the fluid phase. We include an analysis of a feature of the order parameter profile identified as a “second plateau” by Morrow and co-workers [Biochemistry 32, 290 (1993)].

We show that introducing periodic planar fronts with long excitation duration can lead to spiral attenuation. The attenuation occurs periodically over cycles of several planar fronts, forming a variety of complex spatiotemporal patterns. We find that these attenuation patterns occur only at specific phases of the descending fronts relative to the rotational phase of the spiral. These patterns fall into two general classes, each defined by a specific expression for the number of attenuated spirals per cycle of planar fronts, and represented by a structured diagram in parameter space. The spiral attenuation patterns we observe remain stable in time and do not change during the evolution of the system.

A mathematical model describing the coupling between two independent amplification mechanisms in auditory hair cells is proposed and analyzed. Hair cells are cells in the inner ear responsible for translating sound-induced mechanical stimuli into an electrical signal that can then be recorded by the auditory nerve. In nonmammals, two separate mechanisms have been postulated to contribute to the amplification and tuning properties of the hair cells. Models of each of these mechanisms have been shown to be poised near a Hopf bifurcation. Through a weakly nonlinear analysis that assumes weak periodic forcing, weak damping, and weak coupling, the physiologically based models of the two mechanisms are reduced to a system of two coupled amplitude equations describing the resonant response. The predictions that follow from an analysis of the reduced equations, as well as performance benefits due to the coupling of the two mechanisms, are discussed and compared with published experimental auditory nerve data.

We study a six-species Lotka-Volterra-type system on different two-dimensional lattices when each species has two superior and two inferior partners. The invasion rates from predator sites to a randomly chosen neighboring prey site depend on the predator-prey pair, whereby cyclic symmetries within the two three-species defensive alliances are conserved. Monte Carlo simulations reveal an unexpected nonmonotonous dependence of alliance survival on the difference of alliance-specific invasion rates. This behavior is qualitatively reproduced by a four-point mean-field approximation. The study addresses fundamental problems of stability for the competition of two defensive alliances and thus has important implications in natural and social sciences.

Mateos [Phys. Rev. Lett. 84, 258 (2000)] conjectured that current reversal in a classical deterministic ratchet is associated with bifurcations from chaotic to periodic regimes. This is based on the comparison of the current and the bifurcation diagram as a function of a given parameter for a periodic asymmetric potential. Barbi and Salerno [Phys. Rev. E 62, 1988 (2000)] have further investigated this claim and argue that, contrary to Mateos’ claim, current reversals can occur also in the absence of bifurcations. Barbi and Salerno’s studies are based on the dynamics of one particle rather than the statistical mechanics of an ensemble of particles moving in the chaotic system. The behavior of ensembles can be quite different, depending upon their characteristics, which leaves their results open to question. In this paper we present results from studies showing how the current depends on the details of the ensemble used to generate it, as well as conditions for convergent behavior (that is, independent of the details of the ensemble). We are then able to present the converged current as a function of parameters, in the same system as Mateos as well as Barbi and Salerno. We show evidence for current reversal without bifurcation, as well as bifurcation without current reversal. We conjecture that it is appropriate to correlate abrupt changes in the current with bifurcation, rather than current reversals, and show numerical evidence for our claims.

The large-scale dynamics of an interface separating two immiscible fluids in a channel is studied in the case of large viscosity contrasts. A long-wave analysis in conjunction with the Kármán-Polhausen method to approximate the velocity profile in the less viscous fluid is used to derive a single equation for the interface. This equation accounts for the presence of interfacial stress, capillarity, and viscous retardation as well as inertia in the less viscous fluid layer where the flow is considered to be quasistatic; the equation is shown to reduce to a Benney-type equation and the Kuramoto-Sivashinskiy equation in the relevant limits. The solutions of this equation are parametrized by an initial thickness ratio $h_0$ and a dimensionless parameter $\mathcal{S}$, which measures the relative significance of inertial to capillary forces. A parametric continuation technique is employed, which reveals that nonuniqueness of periodic solutions is possible in certain regions of $(h_0,\mathcal{S})$ space. Transient numerical simulations are also reported, whose results demonstrate good agreement with the bifurcation structure obtained from the parametric continuation results.

We use an “equation-free,” coarse-grained computational approach to accelerate molecular dynamics-based computations of demixing (segregation) of dissimilar particles subject to an upward gas flow (gas-fluidized beds). We explore the coarse-grained dynamics of these phenomena in gently fluidized beds of solid mixtures of different densities, typically a slow process for which reasonable continuum models are currently unavailable.

We propose a Landau–de Gennes variational theory fit to simultaneously describe isotropic, nematic, smectic-$A$, and smectic-$C$ phases of a liquid crystal. The unified description allows us to deal with systems in which one, or all, of the order parameters develop because of the influence of defects, external fields and/or boundary conditions. We derive the complete phase diagram of the system, that is, we characterize how the homogeneous minimizers depend on the value of the constitutive parameters. The coupling between the nematic order tensor and the complex smectic order parameter generates an elastic potential which is a nonconvex function of the gradient of the smectic order parameter. This lack of convexity yields in turn a loss of regularity of the free-energy minimizers. We then consider the effect on an infinitesimal second-order regularization term in the free-energy functional, which fixes the optimal number of defects in the singular configurations.

Incorporating the spike-timing-dependent synaptic plasticity (STDP) into a learning rule, we study spatiotemporal learning in analog neural networks. First, we study learning of a finite number of periodic spatiotemporal patterns by deriving the dynamics of the order parameters. When a pattern is retrieved successfully, the order parameters exhibit periodic oscillation. Analyzing this oscillation of the order parameters, we elucidate the relation of the STDP time window to the properties of the retrieval state; the phase of the Fourier transform of the STDP time window determines the retrieval frequency and the time average of the STDP time window crucially affects the storage capacity. We also evaluate the stability of the order parameter oscillation and identify the retrieval state that is stable in single-pattern learning but unstable in multiple-pattern learning even when the retrieval state is independent of a pattern number. To examine the further applicability of the STDP-based learning rule, we also study learning of nonperiodic spatiotemporal Poisson patterns. Our numerical simulations demonstrate that the Poisson patterns are memorized successfully not only in analog neural networks but also in spiking neural networks.

The representation of the human electroencephalogram (EEG) records by neurophysiologists demands standardized time-amplitude scales for their correct conventional interpretation. In a suite of graphical experiments involving scaling affine transformations we have been able to convert electroencephalogram samples corresponding to any particular sleep phase and relaxed wakefulness into each other. We propound a statistical explanation for that finding in terms of data collapse. As a sequel, we determine characteristic time and amplitude scales and outline a possible physical interpretation. An analysis for characteristic times based on lacunarity is also carried out as well as a study of the synchrony between left and right EEG channels.

A well-defined stochastic theory for neural activity, which permits the calculation of arbitrary statistical moments and equations governing them, is a potentially valuable tool for theoretical neuroscience. We produce such a theory by analyzing the dynamics of neural activity using field theoretic methods for nonequilibrium statistical processes. Assuming that neural network activity is Markovian, we construct the effective spike model, which describes both neural fluctuations and response. This analysis leads to a systematic expansion of corrections to mean field theory, which for the effective spike model is a simple version of the Wilson-Cowan equation. We argue that neural activity governed by this model exhibits a dynamical phase transition which is in the universality class of directed percolation. More general models (which may incorporate refractoriness) can exhibit other universality classes, such as dynamic isotropic percolation. Because of the extremely high connectivity in typical networks, it is expected that higher-order terms in the systematic expansion are small for experimentally accessible measurements, and thus, consistent with measurements in neocortical slice preparations, we expect mean field exponents for the transition. We provide a quantitative criterion for the relative magnitude of each term in the systematic expansion, analogous to the Ginsburg criterion. Experimental identification of dynamic universality classes in vivo is an outstanding and important question for neuroscience.

The response of a granular bed to a punctual, vertically flowing water jet underneath it is studied experimentally, theoretically, and numerically. Experiments show that three regimes depending on the flow rate $Q$ appear to outline the bed’s behavior. For sufficiently small $Q$, the bed remains motionless and acts as a rigid porous medium [regime (i)]. It then becomes deformed when $Q$ is sufficiently increased [regime (ii)]. Finally, the bed “explodes” and a locally fluidized bed limited to a domain above the water jet is observed as $Q$ is increased further [regime (iii)]. This fluidization creates a “chimney” in the bed, roughly cylindrical in shape, inside which the grains are in motion. The flow motion in regime (i) is theoretically modeled as a Darcy flow inside an unbounded granular bed while a numerical model accounting for the boundaries is performed. Results from the theory and computations are compared to experimental data and the effects of the boundaries, the bed’s thickness, and the size of the jet on the flow motion inside the bed are underlined. The onset for fluidization [regime (iii)] is explained by assuming a stick-slip behavior of the chimney. Despite the simplistic model, the comparison with experimental data show very good agreement for bony sand granules and relatively good agreement for spherical glass beads.

We consider the critical point of two mean-field disordered models: (i) the random energy model (REM), introduced by Derrida as a mean-field spin-glass model of $N$ spins and (ii) the directed polymer of length $N$ on a Cayley Tree (DPCT) with random bond energies. Both models are known to exhibit a freezing transition between a high-temperature phase where the entropy is extensive and a low-temperature phase of finite entropy, where the weight statistics coincides with the weight statistics of Lévy sums with index $\mu =T∕T_c<1$. In this paper, we study the weight statistics at criticality via the entropy $S=-\sum w_i\ln w_i$ and the generalized moments $Y_k=\sum w_i^k$, where the $w_i$ are the Boltzmann weights of the $2^N$ configurations. In the REM, we find that the critical weight statistics is governed by the finite-size exponent $\nu =2$: the entropy scales as ${\overline{S}}_N\left(T_c\right)\sim N^{1∕2}$, the typical values $e^{\overline{\ln Y_k}}$ decay as $N^{-k∕2}$, and the disorder-averaged values $\overline{Y_k}$ are governed by rare events and decay as $N^{-1∕2}$ for any $k>1$. For the DPCT, we find that the entropy scales similarly as ${\overline{S}}_N\left(T_c\right)\sim N^{1∕2}$, whereas another exponent ${\nu }^{\prime }=1$ governs the $Y_k$ statistics: the typical values $e^{\overline{\ln Y_k}}$ decay as $N^{-k}$, and the disorder-averaged values $\overline{Y_k}$ decay as $N^{-1}$ for any $k>1$. As a consequence, the asymptotic probability distribution ${\overline{\pi }}_{N=\infty }\left(q\right)$ of the overlap $q$, in addition to the delta function $\delta \left(q\right)$, which bears the whole normalization, contains an isolated point at $q=1$, as a memory of the delta peak $(1-T∕T_c)\delta (q-1)$ of the low-temperature phase $T<T_c$. The associated value ${\overline{\pi }}_{N=\infty }(q=1)$ is finite for the DPCT, and diverges as ${\overline{\pi }}_{N=\infty }(q=1)\sim N^{1∕2}$ for the REM.

We study a constrained statistical-mechanical model in two dimensions that has three useful descriptions. They are (i) the Ising model on the honeycomb lattice, constrained to have three up spins and three down spins on every hexagon, (ii) the three-color and fully packed loop model on the links of the honeycomb lattice, with loops around a single hexagon forbidden, and (iii) three Ising models on interleaved triangular lattices, with domain walls of the different Ising models not allowed to cross. Unlike the three-color model, the configuration space on the sphere or plane is connected under local moves. On higher-genus surfaces there are infinitely many dynamical sectors, labeled by a noncontractible set of nonintersecting loops. We demonstrate that at infinite temperature the transfer matrix admits an unusual structure related to a gauge symmetry for the same model on an anisotropic lattice. This enables us to diagonalize the original transfer matrix for up to 36 sites, finding an entropy per plaquette $S∕k_B\approx 0.3661\cdots$ and substantial evidence that the model is not critical. We also find the striking property that the eigenvalues of the transfer matrix on an anisotropic lattice are given in terms of Fibonacci numbers. We comment on the possibility of a topological phase, with infinite topological degeneracy, in an associated two-dimensional quantum model.

Motivated by synchronization problems in noisy environments, we study the Edwards-Wilkinson process on weighted uncorrelated scale-free networks. We consider a specific form of the weights, where the strength (and the associated cost) of a link is proportional to $(k_i{k}_j{)}^{\beta }$ with $k_i$ and $k_j$ being the degrees of the nodes connected by the link. Subject to the constraint that the total edge cost is fixed, we find that in the mean-field approximation on uncorrelated scale-free graphs, synchronization is optimal at ${\beta }^{*}=-1$. Numerical results, based on exact numerical diagonalization of the corresponding network Laplacian, confirm the mean-field results, with small corrections to the optimal value of ${\beta }^{*}$. Employing our recent connections between the Edwards-Wilkinson process and resistor networks, and some well-known connections between random walks and resistor networks, we also pursue a naturally related problem of optimizing performance in queue-limited communication networks utilizing local weighted routing schemes.