Chaotic saddles are unstable invariant sets in the phase space of dynamical systems that exhibit transient chaos. They play a key role in mediating transport processes involving scattering and chaotic transients. Here we present evidence (long chaotic transients and fractal basins of attraction) of transient chaos in a “logistic” delay differential equation. We adapt an existing method (stagger-and-step) to numerically construct the chaotic saddle for this system. This is the first such analysis of transient chaos in an infinite-dimensional dynamical system, and in delay differential equations in particular. Using Poincaré section techniques we illustrate approaches to visualizing the saddle set, and confirm that the saddle has the Cantor-like fractal structure consistent with a chaotic saddle generated by horseshoe-type dynamics.

We study numerically the influence of the wettability condition on slow evaporation in two-dimensional pore square networks of aspect ratio 1. We show how evaporation in a hydrophobic network can be simulated numerically by combining imbibition rules and the computation of diffusion transport in the gas phase. Then we conduct a statistical study of drying in hydrophilic or hydrophobic networks based on pore network simulations. We concentrate on the situation where the external transfer resistance and liquid film effect are negligible and the invasion is dominated by capillary effects. It is found that drying in a hydrophilic network is significantly faster than in a hydrophobic one. The dimensionless overall drying time is found to be network size dependent, approaching exponentially a limit for large size hydrophilic networks. The dimensionless average overall drying time is 0.93 and 0.75 in hydrophobic and hydrophilic large networks, respectively. Other properties, such as the overall saturation and the evaporation flux (through the concept of equivalent flat front position) are also studied. In a last part the impact of liquid film flow on the overall drying time fluctuation is briefly investigated for the case of hydrophilic networks. It is found that the films dampen the drying time fluctuations.

Excursion sets of Gaussian random fields (GRFs) have been frequently used in the literature to model two-phase random media with measurable phase autocorrelation functions. The goal of successful modeling is finding the optimal field autocorrelation function that best approximates the prescribed phase autocorrelation function. In this paper, we present a technique which uses convex quadratic programming to find the best admissible field autocorrelation function under a prescribed discretization. Unlike previous methods, this technique efficiently optimizes over all admissible field autocorrelation functions, instead of optimizing only over a predetermined parametrized family. The results from using this technique indicate that the GRF model is significantly more versatile than observed in previous studies. An application to modeling a base-catalyzed tetraethoxysilane aerogel system given small-angle neutron scattering data is also presented

The tumbling dynamics of DNA have been examined via experiments and Brownian dynamics (BD) simulations in mixed flows that vary from pure shear to pure rotation. In shear, tumbling pathways and periods agree well with earlier studies; in rotation-dominated flows, a new tumbling pathway is identified and experimentally observed. Based on these results, we have developed robust scaling laws for DNA tumbling in both shear and rotational flows and have found a critical flow-type parameter for transition from the shearlike flow regime to the rotation-dominated one.

Results of numerical simulation of cluster size distribution in the site percolation problem are presented. These results disagree with the theoretical data obtained on the basis of the standard drop model of finite cluster structure, in particular, they give a different value of exponent $\zeta$ $(\ln n_s\sim -s^{\zeta })$. Therefore, a more precise fractal model for describing the structure of clusters in a percolation system is proposed. The consideration is based on the solution of a kinetic equation for the number of finite clusters. In the framework of the proposed approach (fractal model together with kinetic equation), a correct value of exponent $\zeta$ is obtained and an explanation is given to the dependence of this exponent on the fraction of occupied sites $p$, which was revealed by numerical simulations. Additionally, a relation is established between the characteristics of cluster size distribution and fractal dimension of the Cantor dust constructed on the percolation cluster.

What is the best description that we can construct of a thermodynamic system that is not in equilibrium, given only one, or a few, extra parameters over and above those needed for a description of the same system at equilibrium? Here, we argue the most appropriate additional parameter is the nonequilibrium entropy of the system. Moreover, we should not attempt to estimate the probability distribution of the system directly, but rather the metaprobability (or hyperensemble) that the system is described by a particular probability distribution. The result is an entropic distribution with two parameters, one a nonequilibrium temperature, and the other a measure of distance from equilibrium. This dispersion parameter smoothly interpolates between certainty of a canonical distribution at equilibrium and great uncertainty as to the probability distribution as we move away from equilibrium. We deduce that, in general, large, rare fluctuations become far more common as we move away from equilibrium.

The equilibrium and dynamic properties of fluids confined to mesoporous material have been studied using nuclear magnetic resonance (NMR) methods. Molecular diffusion of $n$-pentane in Vycor porous glass within closed sample tubes has been measured by means of the pulsed field gradient (PFG) NMR method for temperatures notably exceeding the boiling point of the neat liquid. It is found that the temperature dependence of the diffusivity dramatically depends on the state of the fluid surrounding the mesoporous monoliths. In an oversaturated sample, i.e., in a sample containing some amount of the liquid also outside of the porous material, the diffusivity in the mesopores followed the Arrhenius dependence. In samples with only the mesopores saturated by the liquid, i.e., without any excess fluid, with increasing temperature the diffusivity notably deviated from the Arrhenius dependence towards higher diffusivities. The analysis of the intensities of the respective NMR signals from the fluid within the porous material and in the surrounding phase has revealed that this anomaly is accompanied by the formation of a space free of liquid within the pore system. With the measured pore filling factors, the resulting overall diffusivity is estimated by a two-region approach with diffusion occurring in either the liquid phase or the free space within the pore volume. It is shown that this procedure, free of any fitting parameters, yields excellent agreement with the experimental data.

Dripping is usually associated with fluid motion, but here we describe the analogous phenomenon of a ${}^3\mathrm{He}$ crystal growing and melting under the influence of surface tension and gravity. The pinch-off of the crystal is described by a purely geometric equation of motion, viscous dissipation or inertia being negligible. In analogy to fluid pinch-off, the minimum neck radius $R_n$ goes to zero like a power law, but with a new scaling exponent of $1∕2$. However, for a significant part of the neck’s macroscopic evolution the scaling exponent is found to be much closer to $1∕3$. This observation may be consistent with simulations and theoretical results showing a very slow approach to the asymptotic pinch solution, making the “critical region” very small, both in time and space. After pinch-off, we observe a similar $1∕3$-scaling for the recoil of a crystal tip, both in simulation and experiment. For very early times our experiments are consistent with an approximate theory predicting an asymptotic regime with exponent $1∕2$. Future experiments must show whether the transient $1∕3$ scaling is a universal feature of crystal melting, or perhaps an artifact of our experimental setup.

We show that a simple lattice model can provide a unified description of the thermal denaturation of DNA oligomers and polymers. The model quantitatively reproduces experimental melting curves and reduces in limiting cases to the nearest-neighbor model and a suitable modified Poland-Sheraga model. Our results support the interpretation of the cooperativity parameter $\sigma$ for bubble opening in terms of an interfacial (forking) free energy which also affects chain opening from the ends. The lattice model treats long-ranged excluded volume interactions between all parts of the molecule explicitly, provides access to an ensemble of three dimensional structures (and hence the response to external mechanical forces), and includes fluctuations in situations without pre-determined secondary structure.

We investigate the wetting condition for diffuse-interface methods in the simulation of two-fluid flows with moving contact lines. A current method, which uses a surface-energy approach, is shown not to result in a slope of the interface that is consistent with the prescribed value of the contact angle. A geometric formulation is proposed that does result in the prescribed contact angle. Test results are presented for the axisymmetric droplet spreading due to the capillary force and the motion of a droplet on a solid substrate in a shear flow.

A simple multiscale approach to the diffusion-driven adsorption from a solution to a solid surface is presented. The model combines two important features of the adsorption process: (i) The kinetics of the chemical reaction between adsorbing molecules and the surface and (ii) geometrical constraints on the surface made by molecules which are already adsorbed. The process (i) is modeled in a diffusion-driven context, i.e., the conditional probability of adsorbing a molecule provided that the molecule hits the surface is related to the macroscopic surface reaction rate. The geometrical constraint (ii) is modeled using random sequential adsorption (RSA), which is the sequential addition of molecules at random positions on a surface; one attempt to attach a molecule is made per one RSA simulation time step. By coupling RSA with the diffusion of molecules in the solution above the surface the RSA simulation time step is related to the real physical time. The method is illustrated on a model of chemisorption of reactive polymers to a virus surface.

The quantum Brownian motion of a particle in a cosine periodic potential $V\left(x\right)=-V_0\cos (x∕x_0)$ is treated using the master equation for the time evolution of the Wigner distribution function $W(x,p,t)$ in phase space $(x,p)$. The dynamic structure factor, escape rate, and jump-length probabilities are evaluated via matrix continued fractions in the manner customarily used for the classical Fokker-Planck equation. The escape rate so yielded is compared with that given analytically by the quantum-mechanical reaction rate solution of the Kramers turnover problem. The matrix continued fraction solution substantially agrees with the analytic solution.

We investigate spinodal decomposition and structuring effects in binary immiscible and ternary amphiphilic fluid mixtures under shear by means of three-dimensional lattice Boltzmann simulations. We show that the growth of individual fluid domains can be arrested by adding surfactant to the system, thus forming a bicontinuous microemulsion. We demonstrate that the maximum domain size and the time of arrest depend linearly on the concentration of amphiphile molecules. In addition, we find that for a well-defined threshold value of amphiphile concentration, the maximum domain size and time of complete arrest do not change. For systems under constant and oscillatory shear we analyze domain growth rates in directions parallel and perpendicular to the applied shear. We find a structural transition from a sponge to a lamellar phase by applying a constant shear and the occurrence of tubular structures under oscillatory shear. The size of the resulting lamellae and tubes depends strongly on the amphiphile concentration, shear rate, and shear frequency.

We derive a set of equations for the dynamics of evolving fluid membranes, such as cell membranes, in the presence of bulk fluids. We model the membrane as a surface endowed with a director field, which describes the local average orientation of the molecules on the membrane. A model for the elastic energy of a surface endowed with a director field is derived using liquid crystal theory. This elastic energy reduces to the well-known Helfrich energy in the limit when the directors are constrained to be normal to the surface. We then derive the full dynamic equations for the membrane that incorporate both the elastic and viscous effects, with and without the presence of bulk fluids. We also consider the effect of local spontaneous curvature, arising from the presence of membrane proteins. Overall, the systems of equations allow us to carry out stable, accurate, and robust numerical modeling for the dynamics of the membranes.

The morphology evolution of immiscible polymer-liquid crystal systems is quite different from flexible polymer-polymer mixtures due to the anisotropic properties of liquid crystals. The deformation and retraction of a single low molar mass liquid crystal $4^{\prime }$-pentyl-4-biphenylcarbonitrile (5CB) droplet and $4^{\prime }$-octyl-4-biphenylcarbonitrile (8CB) dispersed in polydimethyl-siloxane under two-dimensional linear flow was investigated by a computer-controlled four-roll mill, which is equipped with an optical microscope and a digital camera. The deformation parameter and orientation angle during deformation versus capillary number was obtained and compared with calculations using the Maffettone-Minale (MM) model and the Yu-Zhou liquid-crystal (YZ-LC) model. The MM model can describe the behavior of a Newtonian droplet in another Newtonian matrix whereas the YZ-LC model can describe the behavior of a LC droplet in a Newtonian matrix. The results showed that the deformation and rotation of a LC droplet is more difficult than viscoelastic droplets, possibly because of the resistance of the nematic elastic energy induced by the nematic mesogens deformation and orientation under flow field. Furthermore, the different behavior between flow-aligning 5CB and flow-tumbling 8CB droplets and the influence of droplet size of LC on deformation and retraction were discussed by experiment and calculation; the results reveal that the different size LC droplets show different evolution curves.

The mechanisms of photon propagation in random media in the diffusive multiple scattering regime have been previously studied using diffusion approximation. However, similar understanding in the low-order (subdiffusion) scattering regime is not complete due to difficulties in tracking photons that undergo very few scatterings events. Recent developments in low-coherence enhanced backscattering (LEBS) overcome these difficulties and enable probing photons that travel very short distances and undergo only a few scattering events. In LEBS, enhanced backscattering is observed under illumination with spatial coherence length $L_{sc}$ less than the scattering mean free path $l_s$. In order to understand the mechanisms of photon propagation in LEBS in the subdiffusion regime, it is imperative to develop analytical and numerical models that describe the statistical properties of photon trajectories. Here we derive the probability distribution of penetration depth of LEBS photons and report Monte Carlo numerical simulations to support our analytical results. Our results demonstrate that, surprisingly, the transport of photons that undergo low-order scattering events has only weak dependence on the optical properties of the medium ($l_s$ and anisotropy factor $g$) and strong dependence on the spatial coherence length of illumination, $L_{sc}$ relative to those in the diffusion regime. More importantly, these low-order scattering photons typically penetrate less than $l_s$ into the medium due to the low spatial coherence length of illumination and their penetration depth is proportional to the one-third power of the coherence volume (i.e., $\left[l_s\pi L_s^2{]}^{1∕3}\right)$.

We present a model for the diffusive trafficking of protein receptors along the surface of a neuron’s dendrite. Distributed along the dendrite are spatially localized trapping regions that represent submicrometer mushroomlike protrusions known as dendritic spines. Within these trapping regions receptors can be internalized via endocytosis and either reinserted into the surface via exocytosis or degraded. We calculate the steady-state distribution of receptors along the dendrite assuming a constant flux of receptors inserted at one end, adjacent to the soma where receptors are synthesized, and use this to investigate the effectiveness of membrane diffusion as a transport mechanism. We also calculate the mean first passage time of a receptor to travel a certain distance along the cable and use this to derive an effective surface diffusivity.

Actuating periodically an elastic filament in a viscous liquid generally breaks the constraints of Purcell’s scallop theorem, resulting in the generation of a net propulsive force. This observation suggests a method to design simple swimming devices—which we call “elastic swimmers”—where the actuation mechanism is embedded in a solid body and the resulting swimmer is free to move. In this paper, we study theoretically the kinematics of elastic swimming. After discussing the basic physical picture of the phenomenon and the expected scaling relationships, we derive analytically the elastic swimming velocities in the limit of small actuation amplitude. The emphasis is on the coupling between the two unknowns of the problems—namely the shape of the elastic filament and the swimming kinematics—which have to be solved simultaneously. We then compute the performance of the resulting swimming device and its dependence on geometry. The optimal actuation frequency and body shapes are derived and a discussion of filament shapes and internal torques is presented. Swimming using multiple elastic filaments is discussed, and simple strategies are presented which result in straight swimming trajectories. Finally, we compare the performance of elastic swimming with that of swimming micro-organisms.

Despite their topological complexity almost all functional properties of metabolic networks can be derived from steady-state dynamics. Indeed, many theoretical investigations (like flux-balance analysis) rely on extracting function from steady states. This leads to the interesting question as to how metabolic networks avoid complex dynamics and maintain a steady-state behavior. Here, we expose metabolic network topologies to binary dynamics generated by simple local rules. We find that the networks’ response is highly specific: Complex dynamics are systematically reduced on metabolic networks compared to randomized networks with identical degree sequences. Already small topological modifications substantially enhance the capacity of a network to host complex dynamic behavior and thus reduce its regularizing potential. This exceptionally pronounced regularization of dynamics encoded in the topology may explain why steady-state behavior is ubiquitous in metabolism.

Using large-scale numerical calculations we explore spatiotemporal chaos in Rayleigh-Bénard convection for experimentally relevant conditions. We calculate the spectrum of Lyapunov exponents and the Lyapunov dimension describing the chaotic dynamics of the convective fluid layer at constant thermal driving over a range of finite system sizes. Our results reveal that the dynamics of fluid convection is truly chaotic for experimental conditions as illustrated by a positive leading-order Lyapunov exponent. We also find the chaos to be extensive over the range of finite-sized systems investigated as indicated by a linear scaling between the Lyapunov dimension of the chaotic attractor and the system size.