We study the dynamics of Brownian particles in a heterogeneous one-dimensional medium with a spatially dependent diffusion coefficient of the form $D\left(x\right)\sim {\left|x\right|}^c$, at constant temperature. The particle's probability distribution function (PDF) is calculated both analytically, by solving Fick's diffusion equation, and from numerical simulations of the underdamped Langevin equation. At long times, the PDFs calculated by both approaches yield identical results, corresponding to subdiffusion for $c<0$ and superdiffusion for $0<c<1$. For $c>1$, the diffusion equation predicts that the particles accelerate. Here we show that this phenomenon, previously considered in several works as an illustration for the possible dramatic effects of spatially dependent thermal noise, is unphysical. We argue that in an isothermal medium, the motion cannot exceed the ballistic limit ($〈x^2〉\sim t^2$). The ballistic limit is reached when the friction coefficient drops sufficiently fast at large distances from the origin and is correctly captured by Langevin's equation.

A large variety of rhythms are observed in nature. Rhythms such as electroencephalogram signals in the brain can often be regarded as interacting. In this study, we investigate the dynamical properties of rhythmic systems in two populations of phase oscillators with different frequency distributions. We assume that the average frequency ratio between two populations closely approximates some small integer. Most importantly, we adopt a specific coupling function derived from phase reduction theory. Under some additional assumptions, the system of two populations of coupled phase oscillators reduces to a low-dimensional system in the continuum limit. Consequently, we find chimera states in which clustering and incoherent states coexist. Finally, we confirm consistent behaviors of the derived low-dimensional model and the original model.

We study the modeling and prediction of dynamical systems based on conventional models derived from measurements. Such algorithms are highly desirable in situations where the underlying dynamics are hard to model from physical principles or simplified models need to be found. We focus on symbolic regression methods as a part of machine learning. These algorithms are capable of learning an analytically tractable model from data, a highly valuable property. Symbolic regression methods can be considered as generalized regression methods. We investigate two particular algorithms, the so-called fast function extraction which is a generalized linear regression algorithm, and genetic programming which is a very general method. Both are able to combine functions in a certain way such that a good model for the prediction of the temporal evolution of a dynamical system can be identified. We illustrate the algorithms by finding a prediction for the evolution of a harmonic oscillator based on measurements, by detecting an arriving front in an excitable system, and as a real-world application, the prediction of solar power production based on energy production observations at a given site together with the weather forecast.

We consider a class of biologically motivated stochastic processes in which a unicellular organism divides its resources (volume or damaged proteins, in particular) symmetrically or asymmetrically between its progeny. Assuming the final amount of the resource is controlled by a growth policy and subject to additive and multiplicative noise, we derive the recursive integral equation describing the evolution of the resource distribution over subsequent generations and use it to study the properties of stable resource distributions. We find conditions under which a unique stable resource distribution exists and calculate its moments for the class of affine linear growth policies. Moreover, we apply an asymptotic analysis to elucidate the conditions under which the stable distribution (when it exists) has a power-law tail. Finally, we use the results of this asymptotic analysis along with the moment equations to draw a stability phase diagram for the system that reveals the counterintuitive result that asymmetry serves to increase stability while at the same time widening the stable distribution. We also briefly discuss how cells can divide damaged proteins asymmetrically between their progeny as a form of damage control. In the appendixes, motivated by the asymmetric division of cell volume in Saccharomyces cerevisiae, we extend our results to the case wherein mother and daughter cells follow different growth policies.

Allostery is indispensable for a protein to work, where a locally applied stimulus is transmitted to a distant part of the molecule. While the allostery due to chemical stimuli such as ligand binding has long been studied, the growing interest in mechanobiology prompts the study of the mechanically stimulated allostery, the physical mechanism of which has not been established. By molecular dynamics simulation of a motor protein myosin, we found that a locally applied mechanical stimulus induces electrostatic potential change at distant regions, just like the piezoelectricity. This novel allosteric mechanism, “piezoelectric allostery”, should be of particularly high value for mechanosensor/transducer proteins.

We study the effect of network size on synchronized activity in living neuronal networks. Dissociated cortical neurons form synaptic connections in culture and generate synchronized spontaneous activity within 10 days in vitro. Using micropatterned surfaces to extrinsically control the size of neuronal networks, we show that synchronized activity can emerge in a network as small as 12 cells. Furthermore, a detailed comparison of small (∼20 cells), medium (∼100 cells), and large (∼400 cells) networks reveal that synchronized activity becomes destabilized in the small networks. A computational modeling of neural activity is then employed to explore the underlying mechanism responsible for the size effect. We find that the generation and maintenance of the synchronized activity can be minimally described by: (1) the stochastic firing of each neuron in the network, (2) enhancement in the network activity in a positive feedback loop of excitatory synapses, and (3) Ca-dependent suppression of bursting activity. The model further shows that the decrease in total synaptic input to a neuron that drives the positive feedback amplification of correlated activity is a key factor underlying the destabilization of synchrony in smaller networks. Spontaneous neural activity plays a critical role in cortical information processing, and our work constructively clarifies an aspect of the structural basis behind this.

We present a microfield approach for studying the dependence of the orientational polarization of the water in aqueous electrolyte solutions upon the salt concentration and temperature. The model takes into account the orientation of the solvent dipoles due to the electric field created by ions, and the effect of thermal fluctuations. The model predicts a dielectric functional dependence of the form $\varepsilon \left(c\right)={\varepsilon }_w-\beta L(3\alpha c/\beta ),\beta ={\varepsilon }_w-{\varepsilon }_{\mathrm{ms}}$, where $L$ is the Langevin function, $c$ is the salt concentration, ${\varepsilon }_w$ is the dielectric of pure water, ${\varepsilon }_{\mathrm{ms}}$ is the dielectric of the electrolyte solution at the molten salt limit, and $\alpha$ is the total excess polarization of the ions. The functional form gives a remarkably accurate description of the dielectric constant for a variety of salts and a wide range of concentrations.

This paper addresses the void fraction of polydisperse particles with a Weibull (or Rosin-Rammler) size distribution. It is demonstrated that the governing parameters of this distribution can be uniquely related to those of the lognormal distribution. Hence, an existing closed-form expression that predicts the void fraction of particles with a lognormal size distribution can be transformed into an expression for Weibull distributions. Both expressions contain the contraction coefficient β. Likewise the monosized void fraction ${\phi }_1$, it is a physical parameter which depends on the particles’ shape and their state of compaction only. Based on a consideration of the scaled binary void contraction, a linear relation for $(1-{\phi }_1)\beta$ as function of ${\phi }_1$ is proposed, with proportionality constant B, depending on the state of compaction only. This is validated using computational and experimental packing data concerning random close and random loose packing arrangements. Finally, using this β, the closed-form analytical expression governing the void fraction of Weibull distributions is thoroughly compared with empirical data reported in the literature, and good agreement is found. Furthermore, the present analysis yields an algebraic equation relating the void fraction of monosized particles at different compaction states. This expression appears to be in good agreement with a broad collection of random close and random loose packing data.

We propose a nonlocal Poisson-Fermi model for ionic solvent that includes ion size effects and polarization correlations among water molecules in the calculation of electrostatic potential. It includes the previous Poisson-Fermi models as special cases, and its solution is the convolution of a solution of the corresponding nonlocal Poisson dielectric model with a Yukawa-like kernel function. The Fermi distribution is shown to be a set of optimal ionic concentration functions in the sense of minimizing an electrostatic potential free energy. Numerical results are reported to show the difference between a Poisson-Fermi solution and a corresponding Poisson solution.

In this article, the noise-assisted correlation integral (NCI) is proposed. The purpose of the NCI is to estimate the invariants of a dynamical system, namely the correlation dimension $\left(D\right)$, the correlation entropy $\left(K_2\right)$, and the noise level $\left(\sigma \right)$. This correlation integral is induced by using random noise in a modified version of the correlation algorithm, i.e., the noise-assisted correlation algorithm. We demonstrate how the correlation integral by Grassberger et al. and the Gaussian kernel correlation integral (GCI) by Diks can be thought of as special cases of the NCI. A third particular case is the $U$-correlation integral proposed herein, from which we derived coarse-grained estimators of the correlation dimension $\left(D_m^U\right)$, the correlation entropy $\left(K_m^U\right)$, and the noise level $\left({\sigma }_m^U\right)$. Using time series from the Henon map and the Mackey-Glass system, we analyze the behavior of these estimators under different noise conditions and data lengths. The results show that the estimators $D_m^U$ and ${\sigma }_m^U$ behave in a similar manner to those based on the GCI. However, for the calculation of $K_2$, the estimator $K_m^U$ outperforms its GCI-based counterpart. On the basis of the behavior of these estimators, we have proposed an automatic algorithm to find $D,K_2$, and $\sigma$ from a given time series. The results show that by using this approach, we are able to achieve statistically reliable estimations of those invariants.

Cluster formation of oppositely charged colloidal particles is studied numerically. A simple Brownian dynamics method with a screened-Coulomb (Yukawa) potential is employed for numerical simulations. An equilibrium phase which consists of clusters and unassociated particles is obtained. It is shown that the equilibrium association number of clusters and their shapes are determined by charge numbers and charge ratio of the binary particles. The phase diagram of cluster formation for various charge numbers and their ratios is obtained. A simple relation between the association number and the charge ratio is found. It is demonstrated that in the case of high charge ratio the cluster takes a multilayer structure which is highly symmetric. It is also pointed out that the cluster-particle interaction changes dynamically in the cluster formation process, which is involved in the selection of final cluster structure.

We build up a phenomenological picture in terms of the effective dynamics of a tracer confined in a cage experiencing random hops to capture some characteristics of glassy systems. This minimal description exhibits scale invariance properties for the small-displacement distribution that echo experimental observations. We predict the existence of exponential tails as a crossover between two Gaussian regimes. Moreover, we demonstrate that the onset of glassy behavior is controlled only by two dimensionless numbers: the number of hops occurring during the relaxation of the particle within a local cage and the ratio of the hopping length to the cage size.

Flows of hard granular materials depend strongly on the interparticle friction coefficient ${\mu }_p$ and on the inertial number $\mathcal{I}$, which characterizes proximity to the jamming transition where flow stops. Guided by numerical simulations, we derive the phase diagram of dense inertial flow of spherical particles, finding three regimes for ${10}^{-4}\lesssim \mathcal{I}\lesssim {10}^{-1}$: frictionless, frictional sliding, and rolling. These are distinguished by the dominant means of energy dissipation, changing from collisional to sliding friction, and back to collisional, as ${\mu }_p$ increases from zero at constant $\mathcal{I}$. The three regimes differ in their kinetics and rheology; in particular, the velocity fluctuations and the stress ratio both display nonmonotonic behavior with ${\mu }_p$, corresponding to transitions between the three regimes of flow. We rationalize the phase boundaries between these regimes, show that energy balance yields scaling relations between microscopic properties in each of them, and derive the strain scale at which particles lose memory of their velocity. For the frictional sliding regime most relevant experimentally, we find for $\mathcal{I}\ge {10}^{-2.5}$ that the growth of the macroscopic friction $\mu \left(\mathcal{I}\right)$ with $\mathcal{I}$ is induced by an increase of collisional dissipation. This implies in that range that $\mu \left(\mathcal{I}\right)-\mu \left(0\right)\sim {\mathcal{I}}^{1-2b}$, where $b\approx 0.2$ is an exponent that characterizes both the dimensionless velocity fluctuations $\mathcal{L}\sim {\mathcal{I}}^{-b}$ and the density of sliding contacts $\chi \sim {\mathcal{I}}^b$.

The dielectric properties of complex plasma containing either metal or dielectric spherical inclusions (macroparticles, dust) are investigated. We focus on surface plasmon resonances on the macroparticle surfaces and their effect on electromagnetic wave propagation. It is demonstrated that the presence of surface plasmon oscillations can significantly modify plasma electromagnetic properties by resonances and cutoffs in the effective permittivity. This leads to related branches of electromagnetic waves and to the wave band gaps. The conditions necessary to observe the band-gap structure in laboratory dusty plasma and/or space (cosmic) dusty plasmas are discussed.

The ground state of an elastic interface in a disordered medium undergoes collective jumps upon variation of external parameters. These mesoscopic jumps are called shocks, or static avalanches. Submitting the interface to a parabolic potential centered at $w$, we study the avalanches which occur as $w$ is varied. We are interested in the correlations between the avalanche sizes $S_1$ and $S_2$ occurring at positions $w_1$ and $w_2$. Using the functional renormalization group (FRG), we show that correlations exist for realistic interface models below their upper critical dimension. Notably, the connected moment ${\langle S_1{S}_2\rangle }^c$ is up to a prefactor exactly the renormalized disorder correlator, itself a function of $|w_2-w_1|$. The latter is the universal function at the center of the FRG; hence, correlations between shocks are universal as well. All moments and the full joint probability distribution are computed to first nontrivial order in an $\epsilon$ expansion below the upper critical dimension. To quantify the local nature of the coupling between avalanches, we calculate the correlations of their local jumps. We finally test our predictions against simulations of a particle in random-bond and random-force disorder, with surprisingly good agreement.

We develop a general framework to study the time and frequency domain characteristics of detrending-operation-based scaling analysis methods, such as detrended fluctuation analysis (DFA) and detrending moving average (DMA) analysis. In this framework, using either the time or frequency domain approach, the frequency responses of detrending operations are calculated analytically. Although the frequency domain approach based on conventional linear analysis techniques is only applicable to linear detrending operations, the time domain approach presented here is applicable to both linear and nonlinear detrending operations. Furthermore, using the relationship between the time and frequency domain representations of the frequency responses, the frequency domain characteristics of nonlinear detrending operations can be obtained. Based on the calculated frequency responses, it is possible to establish a direct connection between the root-mean-square deviation of the detrending-operation-based scaling analysis and the power spectrum for linear stochastic processes. Here, by applying our methods to DFA and DMA, including higher-order cases, exact frequency responses are calculated. In addition, we analytically investigate the cutoff frequencies of DFA and DMA detrending operations and show that these frequencies are not optimally adjusted to coincide with the corresponding time scale.

In this work we explain how to properly use mean-field methods to solve the inverse Ising problem when the phase space is clustered, that is, many states are present. The clustering of the phase space can occur for many reasons, e.g., when a system undergoes a phase transition, but also when data are collected in different regimes (e.g., quiescent and spiking regimes in neural networks). Mean-field methods for the inverse Ising problem are typically used without taking into account the eventual clustered structure of the input configurations and may lead to very poor inference (e.g., in the low-temperature phase of the Curie-Weiss model). In this work we explain how to modify mean-field approaches when the phase space is clustered and we illustrate the effectiveness of our method on different clustered structures (low-temperature phases of Curie-Weiss and Hopfield models).

A fixed-energy Manna sandpile model undergoes an absorbing phase transition at a critical ${\rho }_c$, where an order parameter $\varphi \left(t\right)$ decays as $t^{-\alpha }$ in time $t$. As the prototype of the Manna class, the model has been extensively studied in one dimension. However, the previous estimates of ${\rho }_c$ and some critical exponents are different, depending on the types of initial conditions; random, natural, and regular conditions. The estimates of ${\rho }_c$ for the random and the regular conditions are the lower and the upper bound among currently known estimates, respectively. In this work, for the random conditions, ${\rho }_c$ and $\alpha$ are measured by taking into account finite-size (FS) effects. At the previous estimate of ${\rho }_c$, simulation results show that the temporal decay of $\varphi \left(t\right)$ is strongly affected by the FS effects up to much larger system size ($\sim {10}^6$). For the sizes for which $\varphi \left(t\right)$ is independent up to $t=2\times {10}^7$, we estimate ${\rho }_c=0.8925\left(1\right)$ and $\alpha =0.110\left(5\right)$, which clearly differ from the previous results for the random conditions, ${\rho }_c=0.89199\left(5\right)$ and $\alpha =0.141\left(24\right)$. Instead, the present ${\rho }_c$ agrees with ${\rho }_c=0.89255\left(2\right)$ of the regular conditions. In addition, the present $\alpha$ is substantially distinguishable from the results of the other types of initial conditions, $\alpha =0.159\left(3\right)$ and $0.146\left(2\right)$ for the natural and the regular conditions, respectively, which supports the claim of the initial condition dependence of dynamical exponents in the Manna class.

Network couplings of oscillatory large-scale systems, such as the brain, have a space-time structure composed of connection strengths and signal transmission delays. We provide a theoretical framework, which allows treating the spatial distribution of time delays with regard to synchronization, by decomposing it into patterns and therefore reducing the stability analysis into the tractable problem of a finite set of delay-coupled differential equations. We analyze delay-structured networks of phase oscillators and we find that, depending on the heterogeneity of the delays, the oscillators group in phase-shifted, anti-phase, steady, and non-stationary clusters, and analytically compute their stability boundaries. These results find direct application in the study of brain oscillations.