The evolution of random nonlinear waves (high-intensity noise) in a dissipative and dispersive media is studied. To describe wave processes, the mathematical model in the form of a nonlinear integrodifferential equation is used. The concrete integrand kernels are determined by both frequency-dependent absorption and velocity of the wave. The nonlinear energy loss of broadband noise is considered in two limiting cases: (i) at the initial stage of propagation, when the wave profile contains a small number of shock fronts, and (ii) at the later stage when the wave reshapes to a sawtoothlike one with randomly located shocks.

We consider a fluid dynamo model generated by the flow on both sides of a moving layer. The magnetic permeability of the layer is larger than that of the flow. We show that there exists an optimum value of magnetic permeability for which the critical magnetic Reynolds number for dynamo onset is smaller than for a nonmagnetic material and also smaller than for a layer of infinite magnetic permeability. We present a mechanism that provides an explanation for recent experimental results. A similar effect occurs when the electrical conductivity of the layer is large.

A self-consistent relativistic two-fluid model is proposed for electron-ion plasma dynamics. A one-dimensional geometry is adopted. Electrons are treated as a relativistically degenerate fluid, governed by an appropriate equation of state. The ion fluid is also allowed to be relativistic, but is cold, nondegenerate, and subject only to an electrostatic potential. Exact stationary-profile solutions are sought, at the ionic scale, via the Sagdeev pseudopotential method. The analysis provides the pulse existence region, in terms of characteristic relativistic parameters, associated with the (ultrahigh) particle density.

A method is developed to calculate the drag and torque coefficients of an isolated particle in a Stokes flow. The method is based on solving the time-independent lattice-Boltzmann equation. The advantage of this method is that the algorithm is easy to code, the method can be applied to any shape of the particle without complicated implementation, and the computational cost is independent of the shape of the particle. This method is validated and shown to be accurate by comparing with analytical solutions for certain problems.

We investigate the hydrodynamic properties of a spherical colloid model, which is composed of a shell of point particles by hybrid mesoscale simulations, which combine molecular dynamics simulations for the sphere with the multiparticle collision dynamics approach for the fluid. Results are presented for the center-of-mass and angular velocity correlation functions. The simulation results are compared with theoretical results for a rigid colloid obtained as a solution of the Stokes equation with no-slip boundary conditions. Similarly, analytical results of a point-particle model are presented, which account for the finite size of the simulated system. The simulation results agree well with both approaches on appropriative time scales; specifically, the long-time correlations are quantitatively reproduced. Moreover, a procedure is proposed to obtain the infinite-system-size diffusion coefficient based on a combination of simulation results and analytical predictions. In addition, we present the velocity field in the vicinity of the colloid and demonstrate its close agreement with the theoretical prediction. Our studies show that a point-particle model of a sphere is very well suited to describe the hydrodynamic properties of spherical colloids, with a significantly reduced numerical effort.

The concept of work is basic for statistical thermodynamics. To gain a fuller understanding of work and its (quantum) features, it needs to be represented as an average of a fluctuating quantity. Here I focus on the work done between two moments of time for a thermally isolated quantum system driven by a time-dependent Hamiltonian. I formulate two natural conditions needed for the fluctuating work to be physically meaningful for a system that starts its evolution from a nonequilibrium state. The existing definitions do not satisfy these conditions due to issues that are traced back to noncommutativity. I propose a definition of fluctuating work that is free of previous drawbacks and that applies for a wide class of nonequilibrium initial states. It allows the deduction of a generalized work-fluctuation theorem that applies for an arbitrary (out-of-equilibrium) initial state.

An experimental study of magnetic colloidal particles cluster formation induced by an external electric field in a ferrofluid based on transformer oil is presented. Using frequency domain isothermal dielectric spectroscopy, we study the influence of a test cell electrode separation distance on a low-frequency relaxation process. We consider the relaxation process to be associated with an electric double layer polarization taking place on the particle surface. It has been found that the relaxation maximum considerably shifts towards lower frequencies when conducting the measurements in the test cells with greater electrode separation distances. As the electric field intensity was always kept at a constant value, we propose that the particle cluster formation induced by the external ac electric field accounts for that phenomenon. The increase in the relaxation time is in accordance with the Schwarz theory of electric double layer polarization. In addition, we analyze the influence of a static electric field generated by dc bias voltage on a similar shift in the relaxation maximum position. The variation of the dc electric field for the hysteresis measurements purpose provides understanding of the development of the particle clusters and their decay. Following our results, we emphasize the utility of dielectric spectroscopy as a simple, complementary method for detection and study of clusters of colloidal particles induced by external electric field.

Chronotaxic systems represent deterministic nonautonomous oscillatory systems which are capable of resisting continuous external perturbations while having a complex time-dependent dynamics. Until their recent introduction in Phys. Rev. Lett. 111, 024101 (2013) chronotaxic systems had often been treated as stochastic, inappropriately, and the deterministic component had been ignored. While the previous work addressed the case of the decoupled amplitude and phase dynamics, in this paper we develop a generalized theory of chronotaxic systems where such decoupling is not required. The theory presented is based on the concept of a time-dependent point attractor or a driven steady state and on the contraction theory of dynamical systems. This simplifies the analysis of chronotaxic systems and makes possible the identification of chronotaxic systems with time-varying parameters. All types of chronotaxic dynamics are classified and their properties are discussed using the nonautonomous Poincaré oscillator as an example. We demonstrate that these types differ in their transient dynamics towards a driven steady state and according to their response to external perturbations. Various possible realizations of chronotaxic systems are discussed, including systems with temporal chronotaxicity and interacting chronotaxic systems.

We present the fifth-order equation of the nonlinear Schrödinger hierarchy. This integrable partial differential equation contains fifth-order dispersion and nonlinear terms related to it. We present the Lax pair and use Darboux transformations to derive exact expressions for the most representative soliton solutions. This set includes two-soliton collisions and the degenerate case of the two-soliton solution, as well as beating structures composed of two or three solitons. Ultimately, the new quintic operator and the terms it adds to the standard nonlinear Schrödinger equation (NLSE) are found to primarily affect the velocity of solutions, with complicated flow-on effects. Furthermore, we present a new structure, composed of coincident equal-amplitude solitons, which cannot exist for the standard NLSE.

Subcritical Turing bifurcations of reaction-diffusion systems in large domains lead to spontaneous onset of well-developed localized patterns via the homoclinic snaking mechanism. This phenomenon is shown to occur naturally when balancing source and loss effects are included in a typical reaction-diffusion system, leading to a super- to subcritical transition. Implications are discussed for a range of physical problems, arguing that subcriticality leads to naturally robust phase transitions to localized patterns.

Fluctuating hydrodynamics provides a model for fluids at mesoscopic scales where thermal fluctuations can have a significant impact on the behavior of the system. Here we investigate a model for fluctuating hydrodynamics of a single-component, multiphase flow in the neighborhood of the critical point. The system is modeled using a compressible flow formulation with a van der Waals equation of state, incorporating a Korteweg stress term to treat interfacial tension. We present a numerical algorithm for modeling this system based on an extension of algorithms developed for fluctuating hydrodynamics for ideal fluids. The scheme is validated by comparison of measured structure factors and capillary wave spectra with equilibrium theory. We also present several nonequilibrium examples to illustrate the capability of the algorithm to model multiphase fluid phenomena in a neighborhood of the critical point. These examples include a study of the impact of fluctuations on the spinodal decomposition following a rapid quench, as well as the piston effect in a cavity with supercooled walls. The conclusion in both cases is that thermal fluctuations affect the size and growth of the domains in off-critical quenches.

Thermodynamic Casimir forces of film systems in the $\mathrm{O}\left(n\right)$ universality classes with Dirichlet boundary conditions are studied below bulk criticality. Substantial progress is achieved in resolving the long-standing problem of describing analytically the pronounced minimum of the scaling function observed experimentally in ${}^4\mathrm{He}$ films $(n=2)$ by Garcia and Chan [Phys. Rev. Lett. 83, 1187 (1999)] and in Monte Carlo simulations for the three-dimensional Ising model $\left(n=1\right)$ by O. Vasilyev et al. [Europhys. Lett. 80, 60009 (2007)]. Our finite-size renormalization-group approach describes the film systems as the limit of finite-slab systems with vanishing aspect ratio. This yields excellent agreement with the depth and the position of the minimum for $n=1$ and semiquantitative agreement with the minimum for $n=2$. Our theory also predicts a pronounced minimum for the $n=3$ Heisenberg universality class.

We devise a nonlinear dynamical model of the growth of strained islands on a pattern. We study the morphological instability of a thin film that develops with a characteristic wavelength in the presence of an external forcing due to an underlying patterned substrate with another wavelength. We find in some conditions that the islands can form in well-organized arrays located on either the peaks or valleys of the pattern depending on the film thickness and ratio of the two characteristic wavelengths. These results are rationalized by a kinetic phase diagram and correlated with the morphology when the islands and the wetting layer grow. We find that the islands may be ordered and homogeneous when their coarsening is significantly slowed down, in agreement with experimental observations reported in the literature.

Two types of stable, highly localized Faraday resonant standing waves with multiple crests and troughs are observed in an ethanol-water solution partly filled in a Hele-Shaw cell vertically oscillated with a single frequency. Systematical experiments are performed to investigate the properties of these oscillons. It is found that the wave height of these oscillons is independent of fluid depth from 1 to 5 cm. In particular, some experiments are performed to indicate the high localization of the oscillons, which suggests that these oscillons may be regarded as a combination of the two elementary oscillons discovered by Rajchenbach et al. [Phys. Rev. Lett. 107, 024502 (2011)], for instance, $(2,3)=(1,1)+(1,2)$, where $\left(m,n\right)$ denotes an oscillon with $m$ crests and $n$ troughs. So, our experiments also reveal an elegant “arithmetic” of these oscillons. These experimental phenomena are helpful to deepen and enrich our understanding about Faraday waves.

A numerical linked-cluster algorithm was recently introduced to study quantum quenches in the thermodynamic limit starting from thermal initial states [M. Rigol, Phys. Rev. Lett. 112, 170601 (2014)]. Here, we tailor that algorithm to quenches starting from ground states. In particular, we study quenches from the ground state of the antiferromagnetic Ising model to the $XXZ$ chain. Our results for spin correlations are shown to be in excellent agreement with recent analytical calculations based on the quench action method. We also show that they are different from the correlations in thermal equilibrium, which confirms the expectation that thermalization does not occur in general in integrable models even if they cannot be mapped to noninteracting ones.

We consider the subdiffusion-reaction process with reactions of a type $A+B\to B$ (in which particles $A$ are assumed to be mobile, whereas $B$ are assumed to be static) in comparison to the subdiffusion-reaction process with $A\to B$ reactions which was studied by Sokolov, Schmidt, and Sagués [Phys. Rev. E 73, 031102 (2006)]. In both processes a rule that reactions can only occur between particles which continue to exist is taken into account. Although in both processes a probability of the vanishing of particle $A$ due to a reaction is independent of both time and space variables (assuming that in the system with the $A+B\to B$ reactions, particles $B$ are distributed homogeneously), we show that subdiffusion-reaction equations describing these processes as well as their Green's functions are qualitatively different. The reason for this difference is as follows. In the case of the former reaction, particles $A$ and $B$ have to meet with some probability before the reaction occurs in contradiction with the case of the latter reaction. For the subdiffusion process with the $A+B\to B$ reactions we consider three models which differ in some details concerning a description of the reactions. We base the method considered in this paper on a random walk model in a system with both discrete time and discrete space variables. Then the system with discrete variables is transformed into a system with both continuous time and continuous space variables. Such a method seems to be convenient in analyzing subdiffusion-reaction processes with partially absorbing or partially reflecting walls. The reason is that within this method we can determine Green's functions without a necessity of solving a fractional differential subdiffusion-reaction equation with boundary conditions at the walls. As an example, we use the model to find the Green's functions for a subdiffusive reaction system (with the reactions mentioned above), which is bounded by a partially absorbing wall. This example shows how the model can be used to analyze the subdiffusion-reaction process in a system with partially absorbing or reflecting thin membranes. Employing a simple phenomenological model, we also derive equations related to the reaction parameters used in the considered models.

We use numerical simulations to study the motion of a large asymmetric tracer immersed in a low-density suspension of self-propelled particles in two dimensions. Specifically, we analyze how the curvature of the tracer affects its translational and rotational motion in an active environment. We find that even very small amounts of curvature are sufficient for the active bath to impart directed motion to the tracer, which results in its effective activation. We propose simple scaling arguments to characterize this induced activity in terms of the curvature of the tracer and the strength of the self-propelling force. Our results suggest new ways of controlling the transport properties of passive tracers in an active medium by carefully tailoring their geometry.

The diffusive arrival of transcription factors at the promoter sites on DNA sets a lower bound on how accurately a cell can regulate its protein levels. Using results from the literature on diffusion-influenced reactions, we derive an analytical expression for the lower bound on the precision of transcriptional regulation. In our theory, transcription factors can perform multiple rounds of one-dimensional (1D) diffusion along the DNA and 3D diffusion in the cytoplasm before binding to the promoter. Comparing our expression for the lower bound on the precision against results from Green's function reaction dynamics simulations shows that the theory is highly accurate under biologically relevant conditions. Our results demonstrate that, to an excellent approximation, the promoter switches between the transcription-factor bound and unbound state in a Markovian fashion. This remains true even in the presence of sliding, i.e., with 1D diffusion along the DNA. This has two important implications: (1) Minimizing the noise in the promoter state is equivalent to minimizing the search time of transcription factors for their promoters; (2) the complicated dynamics of 3D diffusion in the cytoplasm and 1D diffusion along the DNA can be captured in a well-stirred model by renormalizing the promoter association and dissociation rates, making it possible to efficiently simulate the promoter dynamics using Gillespie simulations. Based on the recent experimental observation that sliding can speed up the promoter search by a factor of 4, our theory predicts that sliding can enhance the precision of transcriptional regulation by a factor of 2.

We examine a community structure in random graphs of size $n$ and link probability $p/n$ determined with the Newman greedy optimization of modularity. Calculations show that for $p<1$ communities are nearly identical with clusters. For $p=1$ the average sizes of a community $s_{\mathrm{av}}$ and of the giant community $s_g$ show a power-law increase $s_{\mathrm{av}}\sim n^{{\alpha }^{\prime }}$ and $s_g\sim n^{\alpha }$. From numerical results we estimate ${\alpha }^{\prime }\approx 0.26\left(1\right)$ and $\alpha \approx 0.50\left(1\right)$ and using the probability distribution of sizes of communities we suggest that ${\alpha }^{\prime }=\alpha /2$ should hold. For $p>1$ the community structure remains critical: (i) $s_{\mathrm{av}}$ and $s_g$ have a power-law increase with ${\alpha }^{\prime }\approx \alpha <1$ and (ii) the probability distribution of sizes of communities is very broad and nearly flat for all sizes up to $s_g$. For large $p$ the modularity $Q$ decays as $Q\sim p^{-0.55}$, which is intermediate between some previous estimations. To check the validity of the results, we also determine the community structure using another method, namely, a nongreedy optimization of modularity. Tests with some benchmark networks show that the method outperforms the greedy version. For random graphs, however, the characteristics of the community structure determined using both greedy and nongreedy optimizations are, within small statistical fluctuations, the same.

A general scheme for the construction of dynamical systems able to learn generation of the desired kinds of dynamics through adjustment of their internal structure is proposed. The scheme involves intrinsic time-delayed feedback to steer the dynamics towards the target performance. As an example, a system of coupled phase oscillators, which can, by changing the weights of connections between its elements, evolve to a dynamical state with the prescribed (low or high) synchronization level, is considered and investigated.