We present a mean-field model for the phase diagram of a community of micro-organisms, interacting through their metabolism so that they are, in effect, engaging in a cooperative social game. We show that as a function of the concentration of the nutrients glucose and histidine, the community undergoes a phase transition separating a state in which one strain is dominant to a state which is characterized by coexisting populations. Our results are in good agreement with recent experimental results, correctly reproducing quantitative trends and predicting the phase diagram.

We report external noise-induced threshold shift and pattern formation in ac-driven electroconvection (EC) in nematic liquid crystals. We investigate the noise intensity dependence of the threshold and determine the relationship $f_c^{*}=h{f}_{cd}^a$ (α ∼ 1.4, $h$ ∼ 0.1) between the characteristic cutoff frequency ($f_c^{*}$) of the noise and the characteristic ac frequency ($f_{cd}$) of EC. For $f_c>f_c^{*}$, the noise contributes to stabilizing EC (i.e., upward threshold shift), whereas for $f_c<f_c^{*}$, it contributes to destabilizing EC (i.e., downward threshold shift). For $f_c=f_c^{*}$, the noise makes no contribution to the threshold shift. However, the characteristic wavelength of EC strongly depends on the noise intensity but not on $f_c$.

Some effects in the soliton dynamics governed by higher-order Korteweg–de Vries (KdV) type equations are discussed. This is done based on the exact explicit solutions of the equations derived in the paper. It is shown that some higher order KdV equations possessing multisoliton solutions also admit steady state solutions in terms of algebraic functions describing localized patterns. Solutions including both those static patterns and propagating KdV-like solitons are combinations of algebraic and hyperbolic functions. It is shown that the localized structures behave like static solitons upon collisions with regular moving solitons, with their shape remaining unchanged after the collision and only the position shifted. These phenomena are not revealed in common multisoliton solutions derived using inverse scattering or Hirota’s method. The solutions of the higher-order KdV type equations were obtained using a method devised for obtaining soliton solutions of nonlinear evolution equations. This method can be combined with Hirota’s method with a modified representation of the solution which allows the results to be extended to multisoliton solutions. The prospects for applying the methods to soliton equations not of KdV type are discussed.

The theoretical formulation of the smoothed particle hydrodynamics (SPH) method deserves great care because of some inconsistencies occurring when considering free-surface inviscid flows. Actually, in SPH formulations one usually assumes that (i) surface integral terms on the boundary of the interpolation kernel support are neglected, (ii) free-surface conditions are implicitly verified. These assumptions are studied in detail in the present work for free-surface Newtonian viscous flow. The consistency of classical viscous weakly compressible SPH formulations is investigated. In particular, the principle of virtual work is used to study the verification of the free-surface boundary conditions in a weak sense. The latter can be related to the global energy dissipation induced by the viscous term formulations and their consistency. Numerical verification of this theoretical analysis is provided on three free-surface test cases including a standing wave, with the three viscous term formulations investigated.

The statistical mechanics of particles with shapes on a one-dimensional lattice is investigated in the context of the $s=1$ Ising chain with uniform nearest-neighbor coupling, quadratic single-site potential, and a magnetic field, which supports four distinct ground states: $|\uparrow \downarrow \uparrow \downarrow \cdots \rangle$, $|\circ \circ \cdots \rangle$, $|\uparrow \uparrow \cdots \rangle$, $|\uparrow \circ \uparrow \circ \cdots \rangle$. The complete spectrum is generated from each ground state by particles from a different set of six or seven species. Particles and elements of the pseudovacuum are characterized by motifs (patterns of several consecutive site variables). Particles are floating objects that can be placed into open slots on the lattice. Open slots are recognized as permissible links between motifs. The energy of a particle varies between species but is independent of where it is placed. Placement of one particle changes the open-slot configuration for particles of all species. This statistical interaction is encoded in a generalized Pauli principle, from which the multiplicity of states for a given particle combination is determined and used for the exact statistical mechanical analysis. Particles from all species belong to one of four categories: compacts, hosts, tags, or hybrids. Compacts and hosts find open slots in segments of pseudovacuum. Tags find open slots inside hosts. Hybrids are tags with hosting capability. In the taxonomy of particles proposed here, “species” is indicative of structure and “category” indicative of function. The hosting function splits the Pauli principle into exclusion and accommodation parts. Near phase boundaries, the state of the Ising chain at low temperature is akin to that of miscible or immiscible liquids with particles from one species acting as surfactant molecules.

In this paper we extend the subdiffusive Klein-Kramers model, in which the waiting times are modeled by the $\alpha$-stable laws, to the case of waiting times belonging to the class of tempered $\alpha$-stable distributions. We introduce a generalized version of the Klein-Kramers equation, in which the fractional Riemman-Liouville derivative is replaced with a more general integro-differential operator. This allows a transition from the initial subdiffusive character of motion to the standard diffusion for long times to be modeled. Taking advantage of the corresponding Langevin equation, we study some properties of the tempered dynamics, in particular, we approximate solutions of the tempered Klein-Kramers equation via Monte Carlo methods. Also, we study the distribution of the escape time from the potential well and compare it to the classical results in the Kramers escape theory. Finally, we derive the analytical formula for the first-passage-time distribution for the case of free particles. We show that the well-known Sparre Andersen scaling holds also for the tempered subdiffusion.

When many variables are coupled to each other, a single case study could not give us thorough and precise information. When these time series are stationary, different methods of random matrix analysis and complex networks can be used. But, in nonstationary cases, the multifractal–detrended-cross-correlation-analysis (MF-DXA) method was introduced for just two coupled time series. In this article, we have extended the MF-DXA to the method of coupling detrended fluctuation analysis (CDFA) for the case when more than two series are correlated to each other. Here, we have calculated the multifractal properties of the coupled time series, and by comparing CDFA results of the original series with those of the shuffled and surrogate series, we can estimate the source of multifractality and the extent to which our series are coupled to each other. We illustrate the method by selected examples from air pollution and foreign exchange rates.

Starting from a classical Kröener-Rieder kinematic picture for plasticity, we derive a set of dynamical equations describing plastic flow in a Lagrangian formulation. Our derivation is a natural and straightforward extension of simple fluids, elastic, and viscous solids theories. These equations contain the Maxwell model as a special limit. This paper is inspired by the particularly important work of Langer and coworkers. We shall show that our equations bear some resemblance with the shear-transformation zones model developed by Langer and coworkers. We shall point out some important differences. We discuss some results of plasticity, which can be described by the present model. We exploit the model equations for the simple examples: straining of a slab and a rod. We find that necking manifests always itself (not as a result of instability), except if the very special constant-velocity stretching process is imposed.

We present a method for calculating high-order virial expansions of the isotropic-nematic phase transition which we apply here to hard spheroids. Studying a range of aspect ratios, for both oblate and prolate particles, we obtain equations of state, coexistence densities, and nematic order parameters, using expansions truncated at up to eighth virial level. For particles of large aspect ratios our results show rapid convergence, with truncation at sixth order sufficient to give excellent agreement with simulation data. For more spherical particles the convergence is less rapid, with results for up to eighth-order theory approaching but still not reaching simulation data. Our results indicate that high-order viral expansions are better suited to predicting equations of state than coexistence densities. We also test the validity of using the Onsager trial function to approximate the orientational distribution function, finding only small errors when making this approximation.

Regulatory genes called small RNAs (sRNAs) are known to play critical roles in cellular responses to changing environments. For several sRNAs, regulation is effected by coupled stoichiometric degradation with messenger RNAs (mRNAs). The nonlinearity inherent in this regulatory scheme indicates that exact analytical solutions for the corresponding stochastic models are intractable. Here, we present a variational approach to analyze a well-studied stochastic model for regulation by sRNAs via coupled degradation. The proposed approach is efficient and provides accurate estimates of mean mRNA levels as well as higher-order terms. Results from the variational ansatz are in excellent agreement with data from stochastic simulations for a wide range of parameters, including regions of parameter space where mean-field approaches break down. The proposed approach can be applied for quantitative modeling of stochastic gene expression in complex regulatory networks.

Uncovering the causal relationship between spike train recordings from different neurons is a key issue for understanding the neural coding. This paper presents a method, called permutation conditional mutual information (PCMI), for characterizing the causality between a pair of neurons. The performance of this method is demonstrated with the spike trains generated by the Poisson point process model and the Izhikevich neuronal model, including estimation of the directionality index and detection of the temporal dynamics of the causal link. Simulations show that the PCMI method is superior to the transfer entropy and causal entropy methods at identifying the coupling direction between the spike trains. The advantages of PCMI are twofold: It is able to estimate the directionality index under the weak coupling and against the missing and extra spikes.

Optical tweezers are used to manipulate the shape of artificial dioleoyl-phosphatidylcholine (DOPC) phospholipid vesicles of around $30\mu \mathrm{m}$ diameter. Using a time-shared trapping system, a complex of traps drives oscillations of the vesicle equator, with a sinusoidal time dependence and over a range of spatial and temporal frequencies. The mechanical response of the vesicle membrane as a function of the frequency and wavelength of the driving oscillation is monitored. A simple model of the vesicles as spherical elastic membranes immersed in a Newtonian fluid, driven by a harmonic trapping potential, describes the experimental data. The bending modulus of the membrane is recovered. The method has potential for future investigation of nonthermally driven systems, where comparison of active and passive rheology can help to distinguish nonthermal forces from equilibrium fluctuations.

We investigate a conjecture on the cover times of planar graphs by means of large Monte Carlo simulations. The conjecture states that the cover time $\tau \left(G_N\right)$ of a planar graph $G_N$ of $N$ vertices and maximal degree $d$ is lower bounded by $\tau \left(G_N\right)⩾C_{d}N{\left(\ln N\right)}^2$ with $C_d=(d/4\pi )\tan (\pi /d)$, with equality holding for some geometries. We tested this conjecture on the regular honeycomb ($d=3$), regular square ($d=4$), regular elongated triangular ($d=5$), and regular triangular ($d=6$) lattices, as well as on the nonregular Union Jack lattice ($d_{\min }=4$, $d_{\max }=8$). Indeed, the Monte Carlo data suggest that the rigorous lower bound may hold as an equality for most of these lattices, with an interesting issue in the case of the Union Jack lattice. The data for the honeycomb lattice, however, violate the bound with the conjectured constant. The empirical probability distribution function of the cover time for the square lattice is also briefly presented, since very little is known about cover time probability distribution functions in general.

A numerical and statistical study is performed to describe the positive and negative local subgrid energy fluxes in the one-dimensional random-force-driven Burgers turbulence (Burgulence). We use a subensemble method to decompose the field into shock wave and rarefaction wave subensembles by group velocity difference. We observe that the shock wave subensemble shows a strong intermittency which dominates the whole Burgulence field, while the rarefaction wave subensemble satisfies the Kolmogorov 1941 (K41) scaling law. We calculate the two subensemble probabilities and find that in the inertial range they maintain scale invariance, which is the important feature of turbulence self-similarity. We reveal that the interconversion of shock and rarefaction waves during the equation’s evolution displays in accordance with a Markov process, which has a stationary transition probability matrix with the elements satisfying universal functions and, when the time interval is much greater than the corresponding characteristic value, exhibits the scale-invariant property.

We investigate the dynamics of traveling oscillating solitons of the cubic nonlinear Schrödinger (NLS) equation under an external spatiotemporal forcing of the form $f(x,t)=a\exp \left[iK\right(t\left)x\right]$. For the case of time-independent forcing, a stability criterion for these solitons, which is based on a collective coordinate theory, was recently conjectured. We show that the proposed criterion has a limited applicability and present a refined criterion which is generally applicable, as confirmed by direct simulations. This includes more general situations where $K\left(t\right)$ is harmonic or biharmonic, with or without a damping term in the NLS equation. The refined criterion states that the soliton will be unstable if the “stability curve” $p\left(v\right)$, where $p\left(t\right)$ and $v\left(t\right)$ are the normalized momentum and the velocity of the soliton, has a section with a negative slope. In the case of a constant $K$ and zero damping, we use the collective coordinate solutions to compute a “phase portrait” of the soliton where its dynamics is represented by two-dimensional projections of its trajectories in the four-dimensional space of collective coordinates. We conjecture, and confirm by simulations, that the soliton is unstable if a section of the resulting closed curve on the portrait has a negative sense of rotation.

Effective stochastic resonance (SR) is numerically studied using a model of two coupled excitable FitzHugh-Nagumo elements exposed to noise of unequal amplitude. Compared to previous SR models of a single excitable element and of two coupled excitable elements exposed to noise of equal amplitude, the present model exhibits a more intense resonance to the external signal. To extract the key dynamics of this effective SR, the idea of triggered excitation is introduced. Triggered excitation means a process in which the noise-free element is excited synchronously to the external signal with the triggering action of another element that randomly fluctuates asynchronously to the external signal. A deterministic simulation simply proving the triggered excitation process supports our idea.

Zipf’s power-law distribution is a generic empirical statistical regularity found in many complex systems. However, rather than universality with a single power-law exponent (equal to 1 for Zipf’s law), there are many reported deviations that remain unexplained. A recently developed theory finds that the interplay between (i) one of the most universal ingredients, namely stochastic proportional growth, and (ii) birth and death processes, leads to a generic power-law distribution with an exponent that depends on the characteristics of each ingredient. Here, we report the first complete empirical test of the theory and its application, based on the empirical analysis of the dynamics of market shares in the product market. We estimate directly the average growth rate of market shares and its standard deviation, the birth rates and the “death” (hazard) rate of products. We find that temporal variations and product differences of the observed power-law exponents can be fully captured by the theory with no adjustable parameters. Our results can be generalized to many systems for which the statistical properties revealed by power-law exponents are directly linked to the underlying generating mechanism.

We study experimentally the influence of the transverse dimension on film flow in relatively wide channels with sidewalls. Large deviations from two-dimensional predictions are observed in the primary instability and in the post-threshold traveling waves, and the deviations are presently shown to depend strongly on fluid physical properties. Measurements for a wide range of fluid properties are found to correlate with the Kapitza number, which represents the ratio of capillary to viscous stresses. These observations point to an unexpected long-range effect of surface tension that provides transverse coherence to the flow.

Sound velocity measurements in dense glass bead packs reveal significant softening effect at large amplitudes, due to the frictional nonlinearity at the grain contacts. Beyond a certain amplitude, the sound-matter interaction becomes irreversible, leaving the medium in a weakened and slightly compacted state. A slow recovery of the initial elastic modulus is observed after acoustic perturbation, revealing the plastic creep growth of microcontacts. The cross-correlation function of configuration-specific acoustic speckles highlights the relationship between the macroscopic elastic weakening and the local change of the contact networks, induced by strong sound vibration, in the absence of appreciable grain motion.

The last decade showed an increased interest in Langevin equations for modeling time series recorded from complex dynamical systems. These equations allow to discriminate between deterministic (drift) and stochastic (diffusion) components of the recorded time series. In practice, the estimation of drift and diffusion is often based on approximations of the models' dynamics that are valid only for high sampling frequencies. Also, model assessment is not or only indirectly performed, potentially leading to false claims. In this study we compare the performance of an asymptotically unbiased estimation method with a generally used approximate method, demonstrating the necessity of using (asymptotically) unbiased estimators. Furthermore, we describe how confidence intervals for the unknown parameters can be constructed and how model assessment can be carried out. We apply the methodology to local field potentials recorded in vitro from mouse hippocampus from eight genetically different strains. The recorded field potentials turn out to be well described by linearly damped Langevin equations with parabolic diffusion. The modeling enables a dynamical interpretation of the spectral power of the field potentials. It reveals that observed spectral power differences in the field potentials across hippocampal regions are associated with differences in the deterministic component of the system, and it reveals transiently active current dipoles, which are not detectable by conventional methods. Also, all estimated parameters have significant heritabilities, which suggests that the Langevin equations capture biological relevant aspects of electrical hippocampal activity.