We report on the derivation of the discrete complex Ginzburg-Landau equation with first- and second-neighbor couplings using a nonlinear electrical network. Furthermore, we discuss theoretically and numerically modulational instability of plane carrier waves launched through the line. It is pointed out that the underlying analysis not only spells out the discrete Lange-Newell criterion by the means of the linear stability analysis at which the modulational instability occurs for the generation of a train of ultrashort pulses, but also characterizes the long-time dynamical behavior of the system when the instability grows.

We present a description of fluorescence decay kinetics in complex environments based on gamma functions rather than the conventional approach using exponentials. The gamma function description is tested in measurements on the temperature dependence of the protein human serum albumin (HSA), N-acetyl tryptophanamide (NATA), and 2, 5-dipenyl oxazole (PPO). The monitoring of macromolecular structure and dynamics is demonstrated by means of distinct tryptophan (Trp) rotamer populations and their interconversion in HSA.

The pair contact process with diffusion (PCPD) with modulo 2 conservation (PCPD2) [$2A\to 4A$, $2A\to 0$] is studied in one dimension, focused on the crossover to other well established universality classes: the directed Ising (DI) and the directed percolation (DP). First, we show that the PCPD2 shares the critical behaviors with the PCPD, both with and without directional bias. Second, the crossover from the PCPD2 to the DI is studied by including a parity-conserving single-particle process $\left(A\to 3A\right)$. We find the crossover exponent $1/{\varphi }_1=0.57\left(3\right)$, which is argued to be identical to that of the PCPD-to-DP crossover by adding $A\to 2A$. This suggests that the PCPD universality class has a well-defined fixed point distinct from the DP. Third, we study the crossover from a hybrid-type reaction-diffusion process belonging to the DP [$3A\to 5A$, $2A\to 0$] to the DI by adding $A\to 3A$. We find $1/{\varphi }_2=0.73\left(4\right)$ for the DP-to-DI crossover. The inequality of ${\varphi }_1$ and ${\varphi }_2$ further supports the non-DP nature of the PCPD scaling. Finally, we introduce a symmetry-breaking field in the dual spin language to study the crossover from the PCPD2 to the DP. We find $1/{\varphi }_3=1.23\left(10\right)$, which is associated with a new independent route from the PCPD to the DP.

A recent advance in colloidal technology [Zerrouki et al., Nature (London) 455, 380 (2008)] uses magnetic aggregation to enable the formation of micron-scale particle clusters with helical symmetry. The basic building blocks of these aggregates are doublets composed of two micron-scale beads of different radii bonded together by a magnetic cement. Such self-assembled structures offer potential for controllable transport and separation in a low Reynolds number environment using externally applied magnetic or electric fields. Establishing the hydrodynamic properties of the aggregates, in particular the coupling between rotation and translation afforded by the cluster geometry, is an essential initial step toward the design of microfluidic devices employing these aggregates. To quantify this coupling, we first determine parametrized expressions that describe the positions of the beads in an aggregate as a function of size ratio of the two beads composing the doublets. With the geometry of the structure known, we perform hydrodynamic calculations to ascertain entries of the mobility matrix for the aggregate and establish the relationship between the applied torque about the helical axis and translations parallel to this direction. We find that for larger values of the particle radius ratio the coupling between rotations and translations changes sign as the number of doublets in the aggregate increases. This feature indicates that the clusters possess a more complex superhelical structure.

Ordering phenomena on surfaces or in monolayers can be successfully studied by model systems as binary hard-disk mixtures, the influence of a substrate being modeled by an external potential. For the field-free case the thermodynamic stability of space-filling lattice structures for binary hard-disk mixtures is studied by Monte Carlo computer simulations. As these structures prove to be thermodynamically stable only in high pressure environments, the phase behavior of an equimolar binary mixture with a diameter ratio of ${\sigma }_B/{\sigma }_A=0.414$ exposed to an external, one-dimensional, periodic potential is analyzed in detail. The underlying ordering mechanisms and the resulting order differ considerably, depending on which components of the mixture interact with the external potential. The simulations show that slight deviations in the concentration of large particles $x_A$ or the diameter ratio ${\sigma }_B/{\sigma }_A$ have no impact on the occurrence of the various field-induced phenomena as long as the mixture stays in the relevant regime of the packing fraction $\eta$. Furthermore the importance of the commensurability of the external potential to the $S_1\left(AB\right)$ square lattice for the occurrence of the induced ordering is discussed.

The effect of the torsional potential on several thermodynamic and structural properties of a system of polyethylene chains has been analyzed. To this end, molecular dynamics simulations of a coarse-grained model, whose sites interact through a force field with bending, torsional, and nonbonded terms, have been considered. The torsional potential has three stable configurations: gauche-, trans, and $gauche+$. It has been modeled using a simple functional form with only two parameters: the trans-gauche and the gauche-gauche energy barriers. In order to analyze the influence of these parameters on the properties considered in this work, five models with different values of the torsional barriers have been considered. We have observed that the glass transition temperature, the intrachain radial distribution function, the radius of gyration, and the end to end distribution functions are very sensitive to the changes in the trans-gauche torsional barrier. Moreover, at low temperatures, the interchain radial distribution function, the orientational correlation function, and the volume distribution functions of the Voronoi polyhedra, that surround every site of the polymeric chains, also depend on the trans-gauche torsional barrier. On the contrary, the gauche-gauche energy barrier has a minor influence in the properties considered in this work.

We show how single-molecule unzipping experiments can provide strong evidence that the zero-force melting transition of long molecules of natural dsDNA should be classified as a phase transition of the higher-order type (continuous). Toward this end, we study a statistical-mechanics model for the fluctuating structure of a long molecule of dsDNA, and compute the equilibrium phase diagram for the experiment in which the molecule is unzipped under applied force. We consider a perfect-matching dsDNA model, in which the loops are volume-excluding chains with arbitrary loop exponent $c$. We include stacking interactions, hydrogen bonds, and main-chain entropy. We include sequence heterogeneity at the level of random sequences; in particular, there is no correlation in the base-pairing (bp) energy from one sequence position to the next. We present heuristic arguments to demonstrate that the low-temperature macrostate does not exhibit degenerate ergodicity breaking. We use this claim to understand the results of our replica-theoretic calculation of the equilibrium properties of the system. As a function of temperature, we obtain the minimal force at which the molecule separates completely. This critical-force curve is a line in the temperature-force phase diagram that marks the regions where the molecule exists primarily as a double helix versus the region where the molecule exists as two separate strands. We compare our random-sequence model to magnetic tweezer experiments performed on the $48502$ bp genome of bacteriophage $\lambda$. We find good agreement with the experimental data, which is restricted to temperatures between 24 and $50°\text{C}$. At higher temperatures, the critical-force curve of our random-sequence model is very different for that of the homogeneous-sequence version of our model. For both sequence models, the critical force falls to zero at the melting temperature $T_c$ like ${\left|T-T_c\right|}^{\alpha }$. For the homogeneous-sequence model, $\alpha =1/2$ almost exactly, while for the random-sequence model, $\alpha \approx 0.9$. Importantly, the shape of the critical-force curve is connected, via our theory, to the manner in which the helix fraction falls to zero at $T_c$. The helix fraction is the property that is used to classify the melting transition as a type of phase transition. In our calculation, the shape of the critical-force curve holds strong evidence that the zero-force melting transition of long natural dsDNA should be classified as a higher-order (continuous) phase transition. Specifically, the order is 3rd or greater.

We study the stochastic dynamics of doubly clamped micron-scale beams in a viscous fluid driven by Brownian motion. We use a thermodynamic approach to compute the equilibrium fluctuations in beam displacement that requires only deterministic calculations. From calculations of the autocorrelations and noise spectra we quantify the beam dynamics by the quality factor and resonant frequency of the fundamental flexural mode over a wide range of experimentally accessible geometries. We consider beams with uniform rectangular cross section and explore the increased quality factor and resonant frequency as a baseline geometry is varied by increasing the width, increasing the thickness, and decreasing the length. The quality factor is nearly doubled by tripling either the width or the height of the beam. Much larger improvements are found by decreasing the beam length, however this is limited by the appearance of additional modes of fluid dissipation. Overall, the stochastic dynamics of the wider and thicker beams are well predicted by a two-dimensional approximate theory beyond what may be expected based upon the underlying assumptions, whereas the shorter beams require a more detailed analysis.

We investigate Painlevé integrability of a generalized nonautonomous one-dimensional nonlinear Schrödinger (NLS) equation with time- and space-dependent dispersion, nonlinearity, and external potentials. Through the Painlevé analysis some explicit requirements on the dispersion, nonlinearity, dissipation/gain, and the external potential as well as the constraint conditions are identified. It provides an explicit way to engineer integrable nonautonomous NLS equations at least in the sense of Painlevé integrability. Furthermore analytical solutions of this class of integrable nonautonomous NLS equations can be obtained explicitly from the solutions of the standard NLS equation by a general transformation. The result provides a significant way to control coherently the soliton dynamics in the corresponding nonlinear systems, as that in Bose-Einstein condensate experiments. We analyze explicitly the soliton dynamics under the nonlinearity management and the external potentials and discuss its application in the matter-wave dynamics. Some comparisons with the previous works have also been discussed.

Although the statistical mechanics of periodically driven (“Floquet”) systems in contact with a heat bath has some formal analogy with the traditional statistical mechanics of undriven systems, closer examination reveals radical differences. In Floquet systems all quasienergies ${\epsilon }_j$ can be placed in a finite frequency interval $0\le {\epsilon }_j<\omega$ (with $\omega$ the driving frequency and $\hslash =1$). Therefore, if we describe a Floquet system approximately by restricting its available state space to be spanned by a finite number $N$ of basis states, then the number of near degeneracies ($\left|{\epsilon }_j-{\epsilon }_k\right|\le \delta$ for arbitrarily small fixed $\delta$) in this interval grows without limit as $N$ is increased. As we noted in a previous paper, this leads to pathologies, including drastic changes in Floquet states, as $N$ tends to infinity. In earlier work on Floquet systems in contact with a heat bath these difficulties were often put aside by fixing $N$ while taking the coupling to the bath to be smaller than any quasienergy difference. This led to a simple explicit theory for the reduced density matrix, with some major differences from the usual time-independent statistical mechanics. We show that, for weak but finite coupling between system and heat bath, the accuracy of a calculation within the truncated space spanned by the $N$ lowest energy eigenstates of the undriven system is limited, as $N$ increases indefinitely, only by the usual Born-Markov approximation, which neglects bath memory effects. As we seek higher accuracy by increasing $N$, we inevitably encounter quasienergy differences smaller than the system-bath coupling. We therefore derive here the equations for the steady-state reduced density matrix without restriction on the size of quasienergy splittings. In general, this matrix is no longer diagonal in the Floquet states. We analyze, in particular, the behavior near a weakly avoided crossing, where near degeneracies of quasienergies appear. In spite of the Floquet state pathologies, the explicit form of our results for the density matrix gives a consistent prescription for the statistical mechanics of periodically driven systems in the limit as $N$ approaches infinity.

The phase-field-crystal model is by now widely used in order to predict crystal nucleation and growth. For colloidal solidification with completely overdamped individual particle motion, we show that the phase-field-crystal dynamics can be derived from the microscopic Smoluchowski equation via dynamical density-functional theory. The different underlying approximations are discussed. In particular, a variant of the phase-field-crystal model is proposed which involves less approximations than the standard phase-field-crystal model. We finally test the validity of these phase-field-crystal models against dynamical density-functional theory. In particular, the velocities of a linear crystal front from the undercooled melt are compared as a function of the undercooling for a two-dimensional colloidal suspension of parallel dipoles. Good agreement is only obtained by a drastic scaling of the free energies in the phase-field-crystal model in order to match the bulk freezing transition point.

In our recent Brownian dynamics (BD) simulation study, the structure and dynamics of anionic polyelectrolyte xanthan in bulk solution as well as confined spaces of slitlike channel were examined by applying a coarse-grained model with nonlinear bead-spring discretization of a whole chain [J. Jeon and M.-S. Chun, J. Chem. Phys. 126, 154904 (2007)]. This model goes beyond other simulations as they did not consider both long-range electrostatic and hydrodynamic interactions between pairs of beads. Simulation parameters are obtained from the viscometric method of rheology data on the native and sonicated xanthan polysaccharides, which have a contour length less than $1\mu \text{m}$. The size of the semiflexible polyelectrolyte can be well described by the wormlike chain model once the electrostatic effects are taken into account by the persistence length measured at a long length scale. For experimental verifications, single molecule visualization was performed on fluorescein-labeled xanthan using an inverted fluorescence microscope, and the motion of an individual molecule was quantified. Experimental results on the conformational changes in xanthan chain in the electrolyte solution have a reasonable trend to agree with the prediction by BD simulations. In the translational diffusion induced by the Debye screening effect, the simulation prediction reveals slightly higher values compared to those of our measurements, although it agrees with the literature data. Considering the experimental restrictions, our BD simulations are verified to model the single polyelectrolyte well.

The cytoskeleton (CSK) is a nonequilibrium polymer network that uses hydrolyzable sources of free energy such as adenosine triphosphate (ATP) to remodel its internal structure. As in inert nonequilibrium soft materials, CSK remodeling has been associated with structural rearrangements driven by energy-activated processes. We carry out particle tracking and traction microscopy measurements of alveolar epithelial cells at various temperatures and ATP concentrations. We provide the first experimental evidence that the remodeling dynamics of the CSK is driven by structural rearrangements over free-energy barriers induced by thermally activated forces mediated by ATP. The measured activation energy of these forces is $\sim 40k_B{T}_r$ ($k_B$ being the Boltzmann constant and $T_r$ being the room temperature). Our experiments provide clues to understand the analogy between the dynamics of the living CSK and that of inert nonequilibrium soft materials.

The relationships between noise and complex dynamic behaviors of neuronal ensembles are key questions in computational neuroscience, particularly in understanding some basic signal transmission mechanisms of the brain. Here we systemically investigate both the stochastic resonance (SR) and coherence resonance (CR) in the triple-neuron feed-forward-loop (FFL) network motifs by computational modeling. We use the Izhikevich neuron model as well as the chemical coupling to build the FFL motifs, and consider all possible motif types. The simulation results demonstrate that these motifs can exploit noise to enrich its dynamic performance. With a proper choice of noise intensities, both the SR and CR can be exhibited in many types of the FFLs. On the other hand, our results also indicate that the coupling strength serves as a control parameter, which has great impacts on the stochastic dynamics of the FFL motifs. Additionally, biological implications of presented results in the field of neuroscience are outlined.

We study vibrational thermodynamic stability of small-world oscillator networks by relating the average mean-square displacement $S$ of oscillators to the eigenvalue spectrum of the Laplacian matrix of networks. We show that the cross-links suppress $S$ effectively and there exist two phases on the small-world networks: (1) an unstable phase: when $p⪡1/N$, $S\sim N$; (2) a stable phase: when $p⪢1/N$, $S\sim p^{-1}$, i.e., $S/N\sim E_{cr}^{-1}$. Here, $p$ is the parameter of small-world, $N$ is the number of oscillators, and $E_{cr}=pN$ is the number of cross-links. The results are exemplified by various real protein structures that follow the same scaling behavior $S/N\sim E_{cr}^{-1}$ of the stable phase. We also show that it is the “small-world” property that plays the key role in the thermodynamic stability and is responsible for the universal scaling $S/N\sim E_{cr}^{-1}$, regardless of the model details.

Static packings of frictional rigid particles are investigated by means of discrete element simulations. We explore the ensemble of allowed force realizations in the space of contact forces for a given packing structure. We estimate the extent of force indeterminacy with different methods. The indeterminacy exhibits a nonmonotonic dependence on the interparticle friction coefficient. We verify directly that larger force indeterminacy is accompanied by a more robust behavior against local perturbations. We also investigate the local indeterminacy of individual contact forces. The probability distribution of local indeterminacy changes its shape depending on friction. We find that local indeterminacy tends to be larger on force chains for intermediate friction. This correlation disappears in the large friction limit.

This study explores the diffusive properties of human eye movements during a language comprehension task. In this task, adults are given auditory instructions to locate named objects on a computer screen. Although it has been convention to model visual search as standard Brownian diffusion, we find evidence that eye movements are hyperdiffusive. Specifically, we use comparisons of maximum-likelihood fit as well as standard deviation analysis and diffusion entropy analysis to show that visual search during language comprehension exhibits Lévy-like rather than Gaussian diffusion.

Elastic interfaces embedded in (quenched) random media exhibit metastability and stick-slip dynamics. These nontrivial dynamical features have been shown to be associated with cusp singularities of the coarse-grained disorder correlator. Here we show that annealed systems with many absorbing states and a conservation law but no quenched disorder exhibit identical cusps. On the other hand, similar nonconserved systems in the directed percolation class are also shown to exhibit cusps but of a different type. These results are obtained both by a recent method to explicitly measure disorder correlators and by defining an alternative new protocol inspired by self-organized criticality, which opens the door to easily accessible experimental realizations.

A decade ago, a macroscopic theory for closure relations has been proposed for systems out of Onsager’s region. This theory is referred to as the thermodynamic field theory (TFT). The aim of this work was to determine the nonlinear flux-force relations that respect the thermodynamic theorems for systems far from equilibrium. We propose a formulation of the TFT where one of the basic restrictions, namely, the closed-form solution for the skew-symmetric piece of the transport coefficients, has been removed. In addition, the general covariance principle is replaced by the De Donder-Prigogine thermodynamic covariance principle (TCP). The introduction of TCP requires the application of an appropriate mathematical formalism, which is referred to as the entropy-covariant formalism. By geometrical arguments, we prove the validity of the Glansdorff-Prigogine universal criterion of evolution. A new set of closure equations determining the nonlinear corrections to the linear (“Onsager”) transport coefficients is also derived. The geometry of the thermodynamic space is non-Riemannian. However, it tends to be Riemannian for high values of the entropy production. In this limit, we recover the transport equations found by the old theory. Applications of our approach to transport in magnetically confined plasmas, materials submitted to temperature, and electric potential gradients or to unimolecular triangular chemical reactions can be found at references cited herein. Transport processes in tokamak plasmas are of particular interest. In this case, even in the absence of turbulence, the state of the plasma remains close to (but, it is not in) a state of local equilibrium. This prevents the transport relations from being linear.