We study the mechanics and statistical physics of dislocations interacting on cylinders, motivated by the elongation of rod-shaped bacterial cell walls and cylindrical assemblies of colloidal particles subject to external stresses. The interaction energy and forces between dislocations are solved analytically, and analyzed asymptotically. The results of continuum elastic theory agree well with numerical simulations on finite lattices even for relatively small systems. Isolated dislocations on a cylinder act like grain boundaries. With colloidal crystals in mind, we show that saddle points are created by a Peach-Koehler force on the dislocations in the circumferential direction, causing dislocation pairs to unbind. The thermal nucleation rate of dislocation unbinding is calculated, for an arbitrary mobility tensor and external stress, including the case of a twist-induced Peach-Koehler force along the cylinder axis. Surprisingly rich phenomena arise for dislocations on cylinders, despite their vanishing Gaussian curvature.

Small-angle neutron scattering (SANS) has been carried out to examine the block-copolymer-induced depletion interaction of charged silica nanoparticles in aqueous solution. The measurements have been performed on fixed concentrations (1 and 10 wt. $\%$) of anionic Ludox silica nanoparticles having sizes of 8 and 16 nm in the presence of 0.1$M$ NaCl and varying concentration of polyethylene oxide–polypropylene oxide–polyethylene oxide P85 [(EO)${}_{26}$(PO)${}_{39}$(EO)${}_{26}$] block copolymer. The presence of the block copolymer induces an attractive depletion interaction between charge-stabilized nanoparticles. The effective interaction of silica nanoparticles is modeled by a combination of two Yukawa potentials accounting for attractive depletion and repulsive electrostatic forces. The depletion interaction is found to be a long-range attraction whose magnitude and range increase with block-copolymer concentration. The depletion interaction is further enhanced by tuning the self-assembly of the block copolymer through the variation of temperature. The increase of the depletion interaction ultimately leads to clustering of nanoparticles and is confirmed by the presence of a Bragg peak in the SANS data. The positioning of the Bragg peak suggests simple-cubic-type packing of particles within the clusters. The scattering from the clusters in the low-$Q$ region is governed by the Porod scattering, indicating that clusters are quite large (order of microns). The depletion interaction is also found to be strongly dependent on the size of the nanoparticles.

Based on our formulation of the DNA electrophoresis near a pore [Rowghanian and Grosberg, Phys. Rev. E (to be published)], we address the electrophoretic DNA capture into a nanopore as a steady-state process of particle absorption to a sink placed on top of an energy barrier. Reproducing the previously observed diffusion-limited and barrier-limited regimes as two different limits of the particle absorption process and matching the data, our model suggests a slower growth of the capture rate with the DNA length for very large DNA molecules than the previous model, motivating more experiments beyond the current range of electric field and DNA length. At moderately weak electric fields, our model predicts a different effect, stating that the DNA length dependence of the capture rate first disappears as the field is reduced and eventually reverses to a decreasing trend with $N$.

Motivated by DNA electrophoresis near a nanopore, we consider the flow field around an “elongated jet,” a long thin source which injects momentum into a liquid. This solution qualitatively describes the electro-osmotic flow around a long rigid polymer, where due to electrohydrodynamic coupling, the solvent receives momentum from the electric field. Based on the qualitative behavior of the elongated jet solution, we develop a coarse-grained scheme which reproduces the known theoretical results regarding the electrophoretic behavior of a long rigid polymer and a polymer coil in a uniform field, which we then exploit to analyze the electrophoresis of a polymer coil in the nonuniform field near a nanopore.

We analyze the failure process of a two-component system with widely different fracture strength in the framework of a fiber bundle model with localized load sharing. A fraction $0\le \alpha \le 1$ of the bundle is strong and it is represented by unbreakable fibers, while fibers of the weak component have randomly distributed failure strength. Computer simulations revealed that there exists a critical composition ${\alpha }_c$ which separates two qualitatively different behaviors: Below the critical point, the failure of the bundle is brittle, characterized by an abrupt damage growth within the breakable part of the system. Above ${\alpha }_c$, however, the macroscopic response becomes ductile, providing stability during the entire breaking process. The transition occurs at an astonishingly low fraction of strong fibers which can have importance for applications. We show that in the ductile phase, the size distribution of breaking bursts has a power law functional form with an exponent $\mu =2$ followed by an exponential cutoff. In the brittle phase, the power law also prevails but with a higher exponent $\mu =\frac{9}{2}$. The transition between the two phases shows analogies to continuous phase transitions. Analyzing the microstructure of the damage, it was found that at the beginning of the fracture process cracks nucleate randomly, while later on growth and coalescence of cracks dominate, which give rise to power law distributed crack sizes.

The linear response of synchronized time-delayed chaotic systems to small external perturbations, i.e., the phenomenon of chaos pass filter, is investigated for iterated maps. The distribution of distances, i.e., the deviations between two synchronized chaotic units due to external perturbations on the transferred signal, is used as a measure of the linear response. It is calculated numerically and, for some special cases, analytically. Depending on the model parameters this distribution has power law tails in the region of synchronization leading to diverging moments of distances. This is a consequence of multiplicative and additive noise in the corresponding linear equations due to chaos and external perturbations. The linear response can also be quantified by the bit error rate of a transmitted binary message which perturbs the synchronized system. The bit error rate is given by an integral over the distribution of distances and is calculated analytically and numerically. It displays a complex nonmonotonic behavior in the region of synchronization. For special cases the distribution of distances has a fractal structure leading to a devil's staircase for the bit error rate as a function of coupling strength. The response to small harmonic perturbations shows resonances related to coupling and feedback delay times. A bidirectionally coupled chain of three units can completely filter out the perturbation. Thus the second moment and the bit error rate become zero.

We investigated a nonlinear advection-diffusion-reaction equation for a passive scalar field. The purpose is to understand how the compressibility can affect the front dynamics and the bulk burning rate. We study two classes of flows: periodic shear flow and cellular flow, analyzing the system by varying the extent of compressibility and the reaction rate. We find that the bulk burning rate $v_f$ in a shear flow increases with compressibility intensity $\epsilon$, following the relation $\Delta v_f\sim {\epsilon }^2$. Furthermore, the faster the reaction is, the more important the difference is with respect to the laminar case. The effect has been quantitatively measured, and it turns out to be generally small. For the cellular flow, two extreme cases have been investigated, with the whole perturbation situated either in the center of the vortex or in the periphery. The dependence in this case does not show a monotonic scaling with different behavior in the two cases. The enhancing remains modest and is always less than $20\%$.

Weakly conducting dielectric particles suspended in a dielectric liquid of higher conductivity can undergo a transition to spontaneous sustained rotation when placed in a sufficiently strong dc electric field. This phenomenon of Quincke rotation has interesting implications for the rheology of these suspensions, whose effective viscosity can be controlled and reduced by application of an external field. While previous models based on the rotation of isolated particles have provided accurate estimates for this viscosity reduction in dilute suspensions, discrepancies have been reported in more concentrated systems where particle-particle interactions are likely significant. Motivated by this observation, we extend the classic description of Quincke rotation based on the Taylor-Melcher leaky dielectric model to account for pair electrohydrodynamic interactions between two identical spheres using the method of reflections. A coupled system of evolution equations for the dipole moments and angular velocities of the spheres is derived that accounts for electric dipole-dipole interactions and hydrodynamic rotlet interactions up to order $O\left(R^{-5}\right)$, where $R$ is the separation distance between the spheres. A linear stability analysis of this system shows that interactions modify the value of the critical electric field for the onset of Quincke rotation: both electric and hydrodynamic interactions can either stabilize or destabilize the system depending on the orientation of the spheres, but the leading effect of interactions on the onset of rotation is hydrodynamic. We also analyze the dynamics in the nonlinear regime by performing numerical simulations of the governing equations. In the case of a pair of spheres that are fixed in space, we find that particle rotations always synchronize in magnitude at long times, though the directions of rotation of the spheres need not be the same. The steady-state angular velocity magnitude depends on the configuration of the spheres and electric field strength and agrees very well with an asymptotic estimate derived for corotating spheres. In the case of freely-suspended spheres, dipolar interactions are observed to lead to a number of distinct behaviors depending on the initial relative configuration of the spheres and on any infinitesimal initial perturbation introduced in the system: in some cases the spheres slowly separate in space while steadily rotating, while in other cases they pair up and either corotate or counterrotate depending on their orientation relative to the field.

A two-dimensional plasma crystal was melted by suddenly applying localized shear stress. A stripe of particles in the crystal was pushed by the radiation pressure force of a laser beam. We found that the response of the plasma crystal to stress and the eventual shear melting depended strongly on the crystal's angular orientation relative to the laser beam. Shear stress and strain rate were measured, from which the spatially resolved shear viscosity was calculated. The latter was shown to have minima in the regions with highest strain rate, thus demonstrating shear thinning. Shear-induced reordering was observed in the steady-state flow, where particles formed strings aligned in the flow direction.

In general nonlinear lattices, the existence of renormalized phonons due to the nonlinear interactions has been independently discovered by many research groups. Regarding these renormalized phonons as the energy carriers responsible for the heat transport, the scaling laws of temperature-dependent thermal conductivities of one-dimensional nonlinear lattices can be derived from the phenomenological effective phonon approach. For the paradigmatic nonlinear ${\varphi }^4$ lattice, $\kappa \left(T\right)\propto T^{-1.35}$, which was numerically obtained more than a decade ago, can be well explained by the current approach. Most importantly, this approach is able to predict the scaling laws of temperature-dependent thermal conductivities of generalized nonlinear Klein-Gordon lattices. These theoretical predictions are compared by numerical simulations, and perfect agreements have been found.

In order to more accurately describe the fracture process of extensive biological fibers, a fiber-bundle model with stick-slip dynamics and a variable Young modulus is constructed. In this model, the Young modulus of a fiber is assumed to increase or decrease by multiplying with a changing ratio after local sliding events. So, the maximum number of stick-slip events of a single fiber and the changing ratio of the Young modulus are the two key parameters of the model. By means of analytical theory and numerical simulation, the constitutive law, the critical stress, the average size of the largest avalanche, and the avalanche size distribution are shown against the two parameters of the model. From a macroscopic viewpoint, the constitutive curves show different morphologies varying from a local plastic state to a unimodal parabola, while from a microscopic viewpoint, the avalanche size distributions can be well fitted into a power law relationship, which is in accord with the classical fiber-bundle model.

Shear-induced self-diffusion in a dilute suspension of non-Brownian spheres under a simple shear flow is studied in the limit of zero Reynolds number, for different volume fractions $\varphi$. Particles are simulated considering the first term in the multipolar expansion to take into account the long-range hydrodynamic interactions, and a repulsive force is added to avoid interpenetration. The final diffusive regime is established after a long time ($t_{\mathrm{diffusion}}\sim {\varphi }^{-1.5}$), and the diffusion coefficient is proportional to ${\varphi }^2$, as expected in the presence of the short-range repulsive force. Before the diffusive regime is established, there is a rich subdiffusive behavior, particularly in the gradient direction. Each pairwise hydrodynamic interaction is reversible, not leading to streamline migration. Due to the incoherence of the different pair interactions, a plateau in the mean square displacement is first observed, lasting for a period that increases as $\varphi$ is decreased. Then, a first diffusive regime is established due to three-particle interactions, with a diffusion coefficient of $D^{\left(1\right)}\sim {\varphi }^{2.4}$. At longer times, a phenomenon similar to caging is observed. Particles diffuse for long times in the vicinity of some positions, and eventually large displacements are produced, moving the particle to a new position and resulting in transient large values of the kurtosis of the displacement distribution. This migration is produced by collisions through the repulsive potential. After several of those large displacements, the final diffusive regime is established.

We have measured the hydration-level dependence of the single-particle dynamics of water confined in the ordered mesoporous silica MCM-41. The dynamic crossover observed at full hydration is absent at monolayer hydration. The monolayer dynamics are significantly slower than those of water in a fully hydrated pore at ambient temperatures. At low temperatures, the opposite is found to be true. These results underscore the importance of water's tetrahedral hydrogen-bond network in accounting for its low temperature dynamic properties.

We study the evolution of crystallization in dense mono- and polydisperse hard sphere fluids, initially quenched to an amorphous configuration. We use as signatures of crystallization both the decay of the reduced pressure $Z$ and the increase in the local and global orientational order parameters ${\overline{Q}}_6$. For a given realization of the crystallization process these parameters show sudden changes, both large and small, separated by long periods of quiescence. However, averaging over a large number of realizations, a well-defined scenario for their evolution appears. We find an initial fast relaxation to a disordered state, followed by a period of slow variation, associated to the presence of nucleation events, followed by a fast change, composed of the growth of a few crystals with different orientations, and a final and slow coarsening in a domain-growth process. No clear scaling for this whole process was found. We also find that the transition to an stable glassy fluid is quite sharp as the polydispersity is increased, showing a probable first-order phase transition behavior. A well-defined boundary between crystallizing and permanently amorphous fluids should exist, at least for a region in packing fractions. We looked for segregation at large values of polydispersity, but no effects of this type were found.

In simulations of the two-dimensional Ising model, we examine heterogeneous nucleation induced by a small impurity consisting of a line of $l$ fixed spins. As $l$ increases, we identify a limit of stability beyond which the metastable phase is not defined. We evaluate the free energy barrier for nucleation of the stable phase and show that, contrary to expectation, the barrier does not vanish on approach to the limit of stability. We also demonstrate that our values for the height of the barrier yield predictions for the nucleation time (from transition state theory) and the size of the critical cluster (from the nucleation theorem) that are in excellent agreement with direct measurements, even near the limit of stability.

We have investigated the electric conductivity, dielectric relaxation behavior, and viscosity for the aqueous solution of cyanobacterial megamolecules, molecular weight $=1.6\times {10}^7$ g/mol, named sacran. Sacran is an anionic polyelectrolyte which has carboxylate and sulfate groups on the saccharide chain. The electric conductivity and the zero shear viscosity demonstrated three crossover concentrations at 0.004, 0.02, and 0.1 wt$\%$. The viscosity was found to be scaled as ∼$c^{1.5}$, ∼$c^{0.5}$, ∼$c^{1.5}$, and ∼$c^{3.0}$ with increasing the sacran concentration. At 0.1 wt$\%$, the sacran chain formed a weak gel which exhibits macroscopic liquid crystal domains including Schlieren texture. Therefore, these crossover concentrations are considered to be the overlap concentration, entanglement concentration, and gelation concentration (or critical polyelectrolyte concentration), respectively. Dielectric relaxation analysis exhibited the fact that sacran has two types of counterions with different counterion-polyion interaction, i.e., strongly bound and loosely bound counterions. The dielectric parameters such as relaxation time or relaxation strength are sensitive to both the entanglement concentration and the gelation concentration, but not the overlap concentration. The number density of bound counterions calculated from the relaxation strength revealed that the counterion is condensed on the sacran chain with raising the sacran concentrations. The decrease in the charge density of the sacran chain reduces the repulsive force between the chains and this would cause the helix transformation or gelation behavior. The chain conformation of sacran in pure water and the gelation mechanism are discussed in relation with the behavior of polyelectrolytes and liquid crystals.

Stochasticity in gene expression gives rise to fluctuations in protein levels across a population of genetically identical cells. Such fluctuations can lead to phenotypic variation in clonal populations; hence, there is considerable interest in quantifying noise in gene expression using stochastic models. However, obtaining exact analytical results for protein distributions has been an intractable task for all but the simplest models. Here, we invoke the partitioning property of Poisson processes to develop a mapping that significantly simplifies the analysis of stochastic models of gene expression. The mapping leads to exact protein distributions using results for mRNA distributions in models with promoter-based regulation. Using this approach, we derive exact analytical results for steady-state and time-dependent distributions for the basic two-stage model of gene expression. Furthermore, we show how the mapping leads to exact protein distributions for extensions of the basic model that include the effects of posttranscriptional and posttranslational regulation. The approach developed in this work is widely applicable and can contribute to a quantitative understanding of stochasticity in gene expression and its regulation.

Cross-linked semiflexible polymer networks are omnipresent in living cells. Typical examples are actin networks in the cytoplasm of eukaryotic cells, which play an essential role in cell motility, and the spectrin network, a key element in maintaining the integrity of erythrocytes in the blood circulatory system. We introduce a simple mechanical network model at the length scale of the typical mesh size and derive a continuous constitutive law relating the stress to deformation. The continuous constitutive law is found to be generically nonlinear even if the microscopic law at the scale of the mesh size is linear. The nonlinear bulk mechanical properties are in good agreement with the experimental data for semiflexible polymer networks, i.e., the network stiffens and exhibits a negative normal stress in response to a volume-conserving shear deformation, whereby the normal stress is of the same order as the shear stress. Furthermore, it shows a strain localization behavior in response to an uniaxial compression. Within the same model we find a hierarchy of constitutive laws depending on the degree of nonlinearities retained in the final equation. The presented theory provides a basis for the continuum description of polymer networks such as actin or spectrin in complex geometries and it can be easily coupled to growth problems, as they occur, for example, in modeling actin-driven motility.

A methodology is formulated for studying the effects of network structure on the dynamics and spread of an epidemic. We study the well known SIS (susceptible-infected-susceptible) epidemic model and introduce exact moment equations relevant for any network connections. The equations take into account all possible 2${}^{\mathrm{N}}$ states of the network and the specific topology of the network at hand. For the star topology we are able to calculate the threshold value analytically, and also the proportion of infected individuals at steady state for any parameters. Our analytical results are verified with computer simulations. This study of epidemic dynamics on the star network evaluates both the threshold as well as the stability properties of the model's steady state.

We model the expansion of an interacting atomic Bose-Einstein condensate in a disordered lattice with a nonlinear diffusion equation normally used for a variety of classical systems. We find approximate solutions of the diffusion equation that well reproduce the experimental observations for both short and asymptotic expansion times. Our study establishes a connection between the peculiar shape of the expanding density profiles and the microscopic nonlinear diffusion coefficients.