We propose a quantitative function for community partition—i.e., modularity density or $D$ value. We demonstrate that this quantitative function is superior to the widely used modularity $Q$ and also prove its equivalence with the objective function of the kernel $k$ means. Both theoretical and numerical results show that optimizing the new criterion not only can resolve detailed modules that existing approaches cannot achieve, but also can correctly identify the number of communities.

We consider the financial market as a model system and study empirically how agents strategically adjust the properties of large orders in order to meet their preference and minimize their impact. We quantify this strategic behavior by detecting scaling relations between the variables characterizing the trading activity of different institutions. We also observe power-law distributions in the investment time horizon, in the number of transactions needed to execute a large order, and in the traded value exchanged by large institutions, and we show that heterogeneity of agents is a key ingredient for the emergence of some aggregate properties characterizing this complex system.

We report the results of an extensive set of direct numerical simulations of forced, incompressible, magnetohydrodynamic (MHD) turbulence with a strong guide field. The aim is to resolve the controversy regarding the power-law exponent ($\alpha$, say) of the field-perpendicular energy spectrum $E\left(k_{\perp }\right)\propto k_{\perp }^{\alpha }$. The two main theoretical predictions $\alpha =-3/2$ and $\alpha =-5/3$ have both received some support from differently designed numerical simulations. Our calculations have a resolution of ${512}^3$ mesh points, a strong guide field, and an anisotropic simulation domain and implement a broad range of large-scale forcing routines, including those previously reported in the literature. Our findings indicate that the spectrum of well-developed, strong incompressible MHD turbulence with a strong guide field is $E\left(k_{\perp }\right)\propto k_{\perp }^{-3/2}$.

Many nonlinear lattice systems exhibit high-amplitude localized structures, or discrete breathers. Such structures emerge in the discrete nonlinear Schrödinger equation when the energy is above a critical threshold. This paper studies the statistical mechanics at the transition and constructs the probability distribution in the regime where breathers emerge. The entropy as a function of the energy is nonanalytic at the transition. The entropy is independent of the energy in the regime of breathers above the transition.

Models undergoing a phase transition to an absorbing state weakly broken by the addition of a very low spontaneous nucleation rate are shown to exhibit hysteresis loops whose width $\mathrm{\Delta }\lambda$ depends algebraically on the ramp rate $r$. Analytical arguments and numerical simulations show that $\mathrm{\Delta }\lambda \sim r^{\kappa }$ with $\kappa =1∕({\beta }^{\prime }+1)$, where ${\beta }^{\prime }$ is the critical exponent governing the survival probability of a seed near threshold. These results explain similar hysteresis scaling observed before in liquid crystal convection experiments. This phenomenon is conjectured to occur in a variety of other experimental systems.

A Brownian pump of particles powered by a stochastic flashing ratchet mechanism is studied. The pumping device is embedded in a finite region and bounded by particle reservoirs. In the steady state, we exactly calculate the spatial density profile, the concentration ratio between both reservoirs and the particle flux. We propose a simulation framework for the consistent evaluation of such observable quantities.

General microscopic mechanism of ferroelectric ordering in chiral smectic-$C^{*}$ liquid crystals is considered. It is shown that if the mesogenic molecules have a sufficiently low symmetry, the spontaneous polarization is proportional to one of the biaxial vector order parameters of the smectic-$C$ phase. This order parameter may be determined by intermolecular interactions which are not sensitive to molecular chirality. At the same time, the polarization is also proportional to a pseudoscalar parameter which vanishes if the molecules are nonchiral. The general statistical theory of ferroelectric ordering is illustrated by two particular models. The first model is based on electrostatic quadrupole-quadrupole interactions, and it enables one to obtain explicit analytical expressions for the spontaneous polarization. In the second model, the molecular chirality and polarity are determined by a pair of off-center nonparallel dipoles. For this case, the spontaneous polarization is calculated numerically as a function of temperature. The theory provides a more general interpretation of the previous approaches including the classical Boulder model.

We consider particles transport in the Sinai billiard with infinite horizon. The simulation shows that the transport is superdiffusive in both continuous and discrete time. Also, it is shown that the moments do not converge to the Gaussian moments even in the logarithmically renormalized time scale, at least for a fairly long computational time. These results are discussed with respect to the existent rigorous theorems. Similar results are obtained for the stadium billiard.

Some properties of the annular billiard under the presence of weak dissipation are studied. We show, in a dissipative system, that the average energy of a particle acquires higher values than its average energy of the conservative case. The creation of attractors, associated with a chaotic dynamics in the conservative regime, both in appropriated regions of the phase space, constitute a generic mechanism to increase the average energy of dynamical systems.

We experimentally investigate the propagation of chemical fronts in steady laminar cellular flows at large Péclet numbers and large Damköhler numbers. Fronts are generated in an aqueous solution by an autocatalytic oxydoreduction reaction. They propagate in a channel in which a chain of counter-rotative parallel vortices is induced by electroconvection. We first accurately determine the form, the dynamics and the mean velocity of these fronts in the whole Hele-Shaw regime of the flow. We then address the modeling of the evolution of their mean velocity with the flow amplitude. The structure of the front wakes yields us to reject an effective reaction-diffusion wave as a relevant model for large-scale front propagation. On the other hand, analysis of the role of front heads brings us to introduce a kinematic model at the vortex scale for uncovering the front dynamics. This model addresses the propagation of the front leading point in a chain of vortices whose field is modeled by a two-dimensional solid rotation complemented by a boundary layer. Interestingly, it sensitively relies on the effective trajectory followed by the front leading point. To account for this, a competition is worked out among a one-parameter family of potential trajectories. The actual trajectory is then selected as the fastest one with quite a good agreement with measurements and observations. In particular, the measured effective front velocities are well recovered from the model, including their intrinsic dependence on the boundary layer width. Accordingly, effective front propagation in a laminar steadily stirred medium is thus understood from an optimization principle similar to the Fermat principle of ray propagation in heterogeneous media.

A classical particle oscillating in an arbitrary high-frequency or static field effectively exhibits a modified rest mass $m_{\mathrm{eff}}$ derived from the particle averaged Lagrangian. Relativistic ponderomotive and diamagnetic forces, as well as magnetic drifts, are obtained from the $m_{\mathrm{eff}}$ dependence on the guiding center location and velocity. The effective mass is not necessarily positive and can result in backward acceleration when an additional perturbation force is applied. As an example, adiabatic dynamics with $m_{\parallel }>0$ and $m_{\parallel }<0$ is demonstrated for a wave-driven particle along a dc magnetic field, $m_{\parallel }$ being the effective longitudinal mass derived from $m_{\mathrm{eff}}$. Multiple energy states are realized in this case, yielding up to three branches of $m_{\parallel }$ for a given magnetic moment and parallel velocity.

Results of a comprehensive dynamical analysis are reported for several fundamental species of bright solitons in the one-dimensional lattice modeled by the discrete nonlinear Schrödinger equation with the cubic-quintic nonlinearity. Staggered solitons, which were not previously considered in this model, are studied numerically, through the computation of the eigenvalue spectrum for modes of small perturbations, and analytically, by means of the variational approximation. The numerical results confirm the analytical predictions. The mobility of discrete solitons is studied by means of direct simulations, and semianalytically, in the framework of the Peierls-Nabarro barrier, which is introduced in terms of two different concepts, free energy and mapping analysis. It is found that persistently moving localized modes may only be of the unstaggered type.

Two Darboux transformations of the $\left(1+1\right)$-dimensional Wu-Zhang (WZ) equation and the two-component Camassa-Holm (2CH) system with the reciprocal transformation are obtained. One-loop and two-loop soliton solutions and multisoliton(like) solutions of the 2CH system are obtained by using the Darboux transformations and selecting different seed solutions of the corresponding equations. The bidirectional soliton solutions of the $\left(1+1\right)$-dimensional WZ equation are also obtained. The interactions of two-soliton head-on and overtaking collisions for the WZ equation and the evolution of the two-soliton(-like) solutions for the 2CH system are studied.

We present and verify a multicomponent lattice Boltzmann simulation scheme for two immiscible and incompressible fluids with a large density contrast. Our method is constructed from a continuum approximation description of a single inhomogeneous, and essentially incompressible, fluid. The equations that arise from this analysis are mapped onto an established multicomponent lattice Boltzmann method. The approach avoids the computational expense of a numerical solution of the fluid pressure field in a separate step. We present results obtained with our model which validate the initial assumptions and verify correct static and dynamic operation of the model up to a fluid density contrast ratio of more than 500. The paper concludes with an example that illustrates the potential utility of the approach by modeling a gas bubble rising under gravity and breaking through a free surface.

We consider the problem of diffusion-controlled evolution of the $A$-particle-island–$B$-particle-island system at propagation of the sharp annihilation front $A+B\to 0$. We show that this general problem, which includes as particular cases the sea-sea and island-sea problems, demonstrates rich dynamical behavior from self-accelerating collapse of one of the islands to synchronous exponential relaxation of both islands. We find a universal asymptotic regime of the sharp-front propagation and reveal the limits of its applicability for the cases of mean-field and fluctuation fronts.

I derive precise results for absorbing-state phase transitions using exact (numerically determined) quasistationary (QS) probability distributions for small systems. Analysis of the contact process on rings of 23 or fewer sites yields critical properties (control parameter, order-parameter ratios, and critical exponents $z$ and $\beta ∕{\nu }_{\perp }$) with an accuracy of 0.06% or better; precise results are also obtained for the pair contact process. The QS distribution yields insights on the statistical entropy of these models. Preliminary application to a model in the stochastic sandpile class is also described.

We show that a time series $x_t$ evolving by a nonlocal update rule $x_t=f\left(x_{t-n},x_{t-k}\right)$ with two different delays $k<n$ can be mapped onto a local process in two dimensions with special time-delayed boundary conditions, provided that $n$ and $k$ are coprime. For certain stochastic update rules exhibiting a nonequilibrium phase transition, this mapping implies that the critical behavior does not depend on the short delay $k$. In these cases, the autocorrelation function of the time series is related to the critical properties of the corresponding two-dimensional model.

Noise and time delay act simultaneously on real ecological systems. The Lotka-Volterra model of symmetric two-species competition with noise and time delay was investigated in this paper. By means of stochastic simulation, we find that (i) the time delay induces the densities of the two species to periodically oscillate synchronously; (ii) the stationary probability distribution function of the two-species densities exhibits a transition from multiple to single stability as the delay time increases; (iii) the characteristic correlation time for the sum of the two-species densities squared exhibits a nonmonotonic behavior as a function of delay time. Our results have the implication that the combination of noise and time delay could provide an efficient tool for understanding real ecological systems.

We studied how the inhomogeneity of a sequence affects the phase transition that takes place at DNA melting. Unlike previous works, which considered thermodynamic quantities averaged over many different inhomogeneous sequences, we focused on precise sequences and investigated the succession of local openings that lead to their dissociation. For this purpose, we performed transfer-integral-type calculations with two different dynamical models: namely, the heterogeneous Dauxois-Peyrard-Bishop model and the model based on finite stacking enthalpies we recently proposed. It appears that, for both models, the essential effect of heterogeneity is to let different portions of the investigated sequences open at slightly different temperatures. Besides this macroscopic effect, the local aperture of each portion indeed turns out to be very similar to that of a homogeneous sequence with the same length. Rounding of each local opening transition is therefore merely a size effect. For the Dauxois-Peyrard-Bishop model, sequences with a few thousand base pairs are still far from the thermodynamic limit, so that it is inappropriate, for this model, to discuss the order of the transition associated with each local opening. In contrast, sequences with several hundred to a few thousand base pairs are pretty close to the thermodynamic limit for the model we proposed. The temperature interval where a power law holds is consequently broad enough to enable the estimation of critical exponents. On the basis of the few examples we investigated, it seems that, for our model, disorder does not necessarily induce a decrease of the order of the transition.

We present experimental and theoretical results on the voltage-dependent escape dynamics of DNA hairpins of different orientations threaded into an $\alpha$-hemolysin channel. Using a coarse-grained formulation, we map the motion of the polymer in the pore to that of a biased single-particle random walk along the translocation coordinate. By fitting the escape probability distributions obtained from theory to experimental data, we extract the voltage-dependent diffusion constants and bias-induced velocities. Using our two-parameter theory, we obtain excellent agreement with experimentally measured escape time distributions. Further, we find that the ratio of mean escape times for hairpins of different orientations is strongly voltage dependent, with the ratio of $3^{\prime }$- to $5^{\prime }$-threaded DNA decreasing from $\approx 1.7$ to $\approx 1$ with increasing assisting voltages $V_a$. We also find that our model describes $5^{\prime }$-threaded DNA escape extremely well, while providing inadequate fits for $3^{\prime }$ escape. Finally, we find that the escape times for both orientations are equal for high assisting voltages, suggesting that the interactions of DNA with the $\alpha$-hemolysin channel are both orientation and voltage dependent.