We consider the dynamics of systems undergoing parameter sweeps through bifurcation points in the presence of noise. Of interest here are local codimension-one bifurcations that result in large excursions away from an operating point that is transitioning from stable to unstable during the sweep, since information about these “escape events” can be used for system identification, sensing, and other applications. The analysis is based on stochastic normal forms for the dynamic saddle-node and subcritical pitchfork bifurcations with a time-varying bifurcation parameter and additive noise. The results include formulation and numerical solution for the distribution of escape events in the general case and analytical approximations for delayed bifurcations for which escape occurs well beyond the corresponding quasistatic bifurcation points. These bifurcations result in amplitude jumps encountered during parameter sweeps and are particularly relevant to nano- and microelectromechanical systems, for which noise can play a significant role.

Numerical data on the heat transfer constants in two-dimensional Yukawa systems were obtained. Numerical study of the thermal conductivity and diffusivity was carried out for the equilibrium systems with parameters close to conditions of laboratory experiments with dusty plasma. For calculations of heat transfer constants the Green-Kubo formulas were used. The influence of dissipation (friction) on the heat transfer processes in nonideal systems was investigated. The approximation of the coefficient of thermal conductivity is proposed. Comparison of the obtained results to the existing experimental and numerical data is discussed.

Anomalous diffusion of a particle in a cosine periodic potential is treated using fractional diffusion equations in both phase and configuration space. Exact solutions of two distinct forms of the fractional Klein-Kramers (Fokker-Planck) equation for the distribution function in phase space are obtained via matrix continued fractions yielding the average velocity, the velocity autocorrelation function, its spectrum, etc. In the overdamped limit, the results yielded by both equations agree with those from a fractional probability density diffusion equation in configuration space. A simple analytic solution for the spectrum of the velocity correlation function is also given using the effective eigenvalue approximation. The results represent generalizations of the conventional solutions for the normal diffusion of a Brownian particle in a cosine potential to fractional dynamics (giving rise to anomalous diffusion).

We study a nanoscale system composed of one corrugated plate (rack) and one corrugated cylinder (pinion) coupled via the lateral Casimir force. We assume that the rack moves uniformly. The axle of the pinion experiences frictional torque and random torque due to a surrounding fluid. We show that even in the presence of thermal noise, the device can work against external loads: The pinion rotates with a nonzero average velocity. The device operation becomes less influenced by the noise as the gap between rack and pinion decreases.

We present a general framework to describe the evolutionary dynamics of an arbitrary number of types in finite populations based on stochastic differential equations (SDEs). For large, but finite populations this allows us to include demographic noise without requiring explicit simulations. Instead, the population size only rescales the amplitude of the noise. Moreover, this framework admits the inclusion of mutations between different types, provided that mutation rates $\mu$ are not too small compared to the inverse population size $1/N$. This ensures that all types are almost always represented in the population and that the occasional extinction of one type does not result in an extended absence of that type. For $\mu N\ll 1$ this limits the use of SDEs, but in this case there are well established alternative approximations based on time scale separation. We illustrate our approach by a rock-scissors-paper game with mutations, where we demonstrate excellent agreement with simulation based results for sufficiently large populations. In the absence of mutations the excellent agreement extends to small population sizes.

Adenosine-triphosphate (ATP)-dependent chromatin remodeling enzymes (CREs) are biomolecular motors in eukaryotic cells. These are driven by a chemical fuel, namely, ATP. CREs actively participate in many cellular processes that require accessibility of specific segments of DNA which are packaged as chromatin. The basic unit of chromatin is a nucleosome where 146 bp $\sim$ 50 nm of a double-stranded DNA (dsDNA) is wrapped around a spool formed by histone proteins. The helical path of histone-DNA contact on a nucleosome is also called “footprint.” We investigate the mechanism of footprint traversal by a CRE that translocates along the dsDNA. Our two-state model of a CRE captures effectively two distinct chemical (or conformational) states in the mechanochemical cycle of each ATP-dependent CRE. We calculate the mean time of traversal. Our predictions on the ATP dependence of the mean traversal time can be tested by carrying out in vitro experiments on mononucleosomes.

Stochastic modeling of reaction-diffusion kinetics has emerged as a powerful theoretical tool in the study of biochemical reaction networks. Two frequently employed models are the particle-tracking Smoluchowski framework and the on-lattice reaction-diffusion master equation (RDME) framework. As the mesh size goes from coarse to fine, the RDME initially becomes more accurate. However, recent developments have shown that it will become increasingly inaccurate compared to the Smoluchowski model as the lattice spacing becomes very fine. Here we give a general and simple argument for why the RDME breaks down. Our analysis reveals a hard limit on the voxel size for which no local RDME can agree with the Smoluchowski model and lets us quantify this limit in two and three dimensions. In this light we review and discuss recent work in which the RDME has been modified in different ways in order to better agree with the microscale model for very small voxel sizes.

The stability of a free surface under electro-osmotic flow in thin liquid films is investigated where the film thickness can be varied over the scale of a thick to thin electrical double layer while considering the relative contribution from the van der Waals forces. The role of interfacial Maxwell stress on thin film stability is highlighted. This configuration gives some interesting insights into the physics of free surface stability at a scale where various competing forces such as the Coulombic force, van der Waals force, and surface tension come into play. The effects of the mentioned forces are incorporated in the Navier-Stokes equations and a linear stability analysis of the resulting governing equations is performed to obtain the Orr-Sommerfeld equations. The characteristic stability curve of the system is obtained through an asymptotic analysis of the Orr-Sommerfeld equations in the long wave limit. In this study, special focus is given to the effect of the interfacial zeta potential on the free surface stability. It is found that when the free surface and the substrate zeta potential have the same polarity the system is unstable. Since the strength of the free surface potential depends upon the nature of the fluid substrate interaction, this study can help in choosing a proper combination of fluid and substrate to design microfluidic and nanofluidic channels with a desired flow rate without triggering the interfacial instability.

We consider axisymmetric traveling waves propagating on the gravity-driven flow of a liquid down a vertical fiber. Our starting point is the two-equation model for the flow derived in the study by Ruyer-Quil et al. [J. Fluid Mech. 603, 431 (2008)]. The speed, amplitude, and shape of the traveling waves are obtained for a wide range of parameters by using asymptotic analysis and elements from dynamical systems theory. Four different regimes are identified corresponding to the predominance of four different physical effects: advection by the flow, azimuthal curvature, inertia, and viscous dispersion. Construction of the traveling-wave branches of solutions reveals complex transitions from one regime to another. A phase diagram of the different regimes in the parameter space is constructed.

The hydrodynamic force on a particle oscillating in a viscous fluid near a wall with partial-slip boundary condition is studied on the basis of the linearized Navier-Stokes equations. Both incompressible and compressible fluids are considered. It is assumed that the slip length characterizing the partial-slip boundary condition depends on frequency. The consequences of this assumption for the spectrum of Brownian motion near a wall are investigated and compared with a recent experiment.

An electron injection regime in laser wake-field acceleration, namely electron bow-wave injection, is investigated by two- and three-dimensional particle-in-cell simulation as well as analytical model. In this regime electrons in the intense electron bow wave behind the first bubble catch up with the bubble tail and are trapped by the bubble finally, resulting in considerable enhancement of the total trapped electron number. For example, with the increase of the laser intensity from 2 × 10${}^{19}$ to 1 × 10${}^{20}$ W/cm${}^2$, the electron trapping changes from normal self-injection to bow-wave injection and the trapped electron number is enhanced by two orders of magnitude. An analytical model is proposed to explain the numerical observation.

The physically based, benchmarked fluid model developed by Robson et al. [R. E. Robson, R. D. White, and Z. Lj. Petrovic, Rev. Mod. Phys. 77, 1303 (2005)] and extended to analyze electron swarms in a spatially homogeneous electric field under conditions corresponding to the Franck-Hertz experiment [P. Nicoletopoulos and R. E. Robson, Phys. Rev. Lett. 100, 124502 (2008)] is generalized to investigate the nonlocal response and resonance phenomena associated with electrons subject to an externally prescribed, spatially varying electrostatic field. Analytic expressions are first derived for the mean velocity, energy, and heat flux of electrons in a harmonically varying field, and expressions are then given for fields with more general spatial dependences. Numerical examples are given for both benchmark model cross sections and a real gas.

We show that the notion of generalized Lenard chains naturally allows formulation of the theory of multiseparable and superintegrable systems in the context of bi-Hamiltonian geometry. We prove that the existence of generalized Lenard chains generated by a Hamiltonian function defined on a four-dimensional $\omega N$ manifold guarantees the separation of variables. As an application, we construct such chains for the Hénon-Heiles systems and for the classical Smorodinsky-Winternitz systems. New bi-Hamiltonian structures for the Kepler potential are found.

A mixture of hard-sphere particles and model emulsion droplets is studied with a Brownian dynamics simulation. We find that the addition of nonwetting emulsion droplets to a suspension of pure hard spheres can lead to both gas-liquid and fluid-solid phase separations. Furthermore, we find a stable fluid of hard-sphere clusters. The stability is due to the saturation of the attraction that occurs when the surface of the droplets is completely covered with colloidal particles. At larger emulsion droplet densities a percolation transition is observed. The resulting networks of colloidal particles show dynamical and mechanical properties typical of a colloidal gel. The results of the model are in good qualitative agreement with recent experimental findings [E. Koos and N. Willenbacher, Science 331, 897 (2011)] in a mixture of colloidal particles and two immiscible fluids.

The damped harmonic oscillator under symmetric Lévy white noise shows inhomogeneous phase space, which is in contrast to the homogeneous phase space of the same oscillator under the Gaussian white noise, as shown in a recent paper [Sokolov, Ebeling, and Dybiec, Phys. Rev. E 83, 041118 (2011)]. The inhomogeneity of the phase space shows correlation between the coordinate and the velocity of the damped oscillator under symmetric Lévy white noise. In the present work we further explore the physical origin of these distinguished features and find that it is due to the combination of the damped effect and heavy tail of the noise. We directly demonstrate this in the reduced coordinate $\tilde{x}$ versus velocity $\tilde{v}$ plots and identify the physics of the antiassociation of the coordinate and velocity.

This paper investigates the effect of network topology on the fair allocation of network resources among a set of agents, an all-important issue for the efficiency of transportation networks all around us. We analyze a generic mechanism that distributes network capacity fairly among existing flow demands. The problem can be solved by semianalytical methods on a nearest-neighbor graph with one source and sink pair, when transport occurs over shortest paths. For this setup, we uncover a broad range of patterns of intersecting shortest paths as a function of the distance between the source and the sink. When the number of intersections is the maximum and the distance between the source and the sink is large, we find that a fair allocation implies a decrease of at least $50\%$ from the maximum throughput. We also find that the histogram of the flow allocations assigned to the agents decays as a power law with exponent $-1$. Our semianalytical framework suggests possible explanations for the well-known reduction of the throughput in fair allocations. It also suggests that the combination of network topology and routing rules can lead to highly uneven (but fair) distributions of resources, a remark of caution to network designers.

Using large-scale parallel numerical simulations we explore spatiotemporal chaos in Rayleigh-Bénard convection in a cylindrical domain with experimentally relevant boundary conditions. We use the variation of the spectrum of Lyapunov exponents and the leading-order Lyapunov vector with system parameters to quantify states of high-dimensional chaos in fluid convection. We explore the relationship between the time dynamics of the spectrum of Lyapunov exponents and the pattern dynamics. For chaotic dynamics we find that all of the Lyapunov exponents are positively correlated with the leading-order Lyapunov exponent, and we quantify the details of their response to the dynamics of defects. The leading-order Lyapunov vector is used to identify topological features of the fluid patterns that contribute significantly to the chaotic dynamics. Our results show a transition from boundary-dominated dynamics to bulk-dominated dynamics as the system size is increased. The spectrum of Lyapunov exponents is used to compute the variation of the fractal dimension with system parameters to quantify how the underlying high-dimensional strange attractor accommodates a range of different chaotic dynamics.

The nonlinear phenomenon of synchronization is characterized experimentally for dust density waves, i.e., dust acoustic waves, which are self-excited due to an ion streaming instability. The waves propagate in a dust cloud with a natural frequency of 22 Hz. We synchronize these waves to a different frequency using a driving electrode that sinusoidally modulates the ion density. We study four synchronized states, with frequencies that are multiples of 1, 2, 3, and $1/2$ of the driving frequency. Comparing to phenomena that are typical of the van der Pol paradigm, we find that synchronization of our waves exhibit the signature of the suppression mechanism but not that of the phaselocking mechanism. Additionally, synchronization of our waves exhibits three characteristics that differ from the van der Pol paradigm: a threshold amplitude that can be seen in the Arnold tongue diagram, a branching of the 1:1 harmonic tongue at its lower extremity, and a nonharmonic state. The latter state appears to be a nonlinear oscillation; it is neither at the natural frequency nor a synchronized state.

The excitation of nonlinear electrostatic waves, such as shock and solitons, by ultraintense laser interaction with overdense plasmas and related ion acceleration are investigated by numerical simulations. Stability of solitons and formation of shock waves is strongly dependent on the velocity distribution of ions. Monoenergetic components in ion spectra are produced by “pulsed” reflection from solitary waves. Possible relevance to recent experiments on “shock acceleration” is discussed.

A simple one-dimensional lattice model is suggested to describe the experimentally observed plateau in force-stretching diagrams for some macromolecules. This chain model involves the nearest-neighbor interaction of a Morse-like potential (required to have a saturation branch) and a harmonic second-neighbor coupling. Under an external stretching applied to the chain ends, the intersite Morse-like potential results in the appearance of a double-well potential within each chain monomer, whereas the interaction between the second neighbors provides a homogeneous bistable (degenerate) ground state, at least within a certain part of the chain. As a result, different conformational changes occur in the chain under the external forcing. The transition regions between these conformations are described as topological solitons. With a strong second-neighbor interaction, the solitons describe the transition between the bistable ground states. However, the key point of the model is the appearance of a heterogenous structure, when the second-neighbor coupling is sufficiently weak. In this case, a part of the chain has short bonds with a single-well potential, whereas the complementary part admits strongly stretched bonds with a double-well potential. This case allows us to explain the existence of a plateau in the force-extension diagram for DNA and $\alpha$-helix protein. Finally, the soliton dynamics are studied in detail.