It is known that the sphere-packing problem and the number-variance problem (closely related to an optimization problem in number theory) can be posed as energy minimizations associated with an infinite number of point particles in $d$-dimensional Euclidean space ${\mathbb{R}}^d$ interacting via certain repulsive pair potentials. We reformulate the covering and quantizer problems as the determination of the ground states of interacting particles in ${\mathbb{R}}^d$ that generally involve single-body, two-body, three-body, and higher-body interactions. This is done by linking the covering and quantizer problems to certain optimization problems involving the “void” nearest-neighbor functions that arise in the theory of random media and statistical mechanics. These reformulations, which again exemplify the deep interplay between geometry and physics, allow one now to employ theoretical and numerical optimization techniques to analyze and solve these energy minimization problems. The covering and quantizer problems have relevance in numerous applications, including wireless communication network layouts, the search of high-dimensional data parameter spaces, stereotactic radiation therapy, data compression, digital communications, meshing of space for numerical analysis, and coding and cryptography, among other examples. In the first three space dimensions, the best known solutions of the sphere-packing and number-variance problems (or their “dual” solutions) are directly related to those of the covering and quantizer problems, but such relationships may or may not exist for $d\ge 4$, depending on the peculiarities of the dimensions involved. Our reformulation sheds light on the reasons for these similarities and differences. We also show that disordered saturated sphere packings provide relatively thin (economical) coverings and may yield thinner coverings than the best known lattice coverings in sufficiently large dimensions. In the case of the quantizer problem, we derive improved upper bounds on the quantizer error using sphere-packing solutions, which are generally substantially sharper than an existing upper bound in low to moderately large dimensions. We also demonstrate that disordered saturated sphere packings yield relatively good quantizers. Finally, we remark on possible applications of our results for the detection of gravitational waves.

Injection-driven immiscible flow in radial Hele-Shaw cells results in highly ramified patterns if the injection rate is constant in time. Likewise, time-dependent gap immiscible flow in lifting Hele-Shaw cells leads to intricate morphologies if the cell’s gap width grows exponentially with time. Recent studies show that the rising of these complex fingered structures can be controlled by properly adjusting the injection rate, and the time-dependent gap width. We investigate the effectiveness of these control strategies assuming that the fluids involved are miscible. Despite the absence of surface tension effects, intensive numerical simulations support the stabilizing role of these controlling protocols. Splitting, merging and competition of fingers are all inhibited. The sensitivity of the system to changes in the initial conditions and Péclet numbers is also discussed.

The near-surface motility of bacteria is important in the initial formation of biofilms and in many biomedical applications. The swimming motion of Escherichia coli near a solid surface is investigated both numerically and experimentally. A boundary element method is used to predict the hydrodynamic entrapment of E. coli bacteria, their trajectories, and the minimum separation of the cell from the surface. The numerical results show the existence of a stable swimming distance from the boundary that depends only on the shape of the cell body and the flagellum. The experimental validation of the numerical approach allows one to use the numerical method as a predictive tool to estimate with reasonable accuracy the near-wall motility of swimming bacteria of known geometry. The analysis of the numerical database demonstrated the existence of a correlation between the radius of curvature of the near-wall circular trajectory and the separation gap. Such correlation allows an indirect estimation of either of the two quantities by a direct measure of the other without prior knowledge of the cell geometry. This result may prove extremely important in those biomedical and technical applications in which the near-wall behavior of bacteria is of fundamental importance.

Surface nanobubbles emerging at solid-liquid interfaces of submerged hydrophobic surfaces show extreme stability and very small (gas-side) contact angles. In a recent paper Ducker [W. A. Ducker, Langmuir 25, 8907 (2009)]. conjectured that these effects may arise from the presence of impurities at the air-water interface of the nanobubbles. In this paper we present a quantitative analysis of this hypothesis by estimating the dependence of the contact angle and the Laplace pressure on the fraction of impurity coverage at the liquid-gas interface. We first develop a general analytical framework to estimate the effect of impurities (ionic or nonionic) in lowering the surface tension of a given air-water interface. We then employ this model to show that the (gas-side) contact angle and the Laplace pressure across the nanobubbles indeed decrease considerably with an increase in the fractional coverage of the impurities, though still not sufficiently small to account for the observed surface nanobubble stability. The proposed model also suggests the dependencies of the Laplace pressure and the contact angle on the type of impurity.

We measured the splay $\left(K_{11}\right)$ and bend $\left(K_{33}\right)$ elastic constants in the nematic phase of a liquid crystal with T-shaped molecules. We find that the ratio, $K_{33}/K_{11}\simeq 1$ in the entire nematic range except very close to the nematic to $\text{Sm-}A$ (SN) transition. Both $K_{33}$ and $K_{11}$ show pretransitional divergence as the SN transition is approached from higher temperature. The ratio, $K_{33}/K_{11}$ suggests that the length $\left(L\right)$ to effective width $\left(D\right)$ ratio (i.e., $L/D$) is significantly smaller due to the presence of long and flexible lateral group, compared to that of rigid rodlike molecules. It is argued that apart from the extra contribution to the elasticity the long and flexible lateral group also has a significant contribution to the suppression of the splay fluctuations in the onset of smectic short-range fluctuation. The structure of the $\text{Sm-}A$ phase is investigated by using small angle x-ray diffraction, and a possible arrangement of the molecules in the $\text{Sm-}A$ layer is proposed.

We show a microscopic derivation of a quantum master equation with counting terms which describes the electron statistics. A localized spin behaves as a probe whose precession angle monitors the net electron current by the magnetic moment interaction. The probe Hamiltonian is proportional to the current and is determined self-consistently for a model of a quantum dot. Then it turns out that the quantum master equation for the spin precession contains the counting terms. As an application, we show the fluctuation theorem for the electron current.

We have measured first and second order statistics of the velocity of phase singularities $v$ in evolving speckle patterns of microwave radiation transmitted through random quasi-1D samples as the frequency is swept and relate these to global statistics of speckle evolution. When $v$ is normalized by the standard deviation of the fractional intensity change, the probability distribution and correlation function of $v$ approach those for random Gaussian fields even for localized waves. Analogous results are found for transmitted intensity normalized by the total transmission. This provides a unified framework for the statistics of speckle evolution and intensity.

The molecular mechanism of slip at the interface between polymer melts and weakly attractive smooth surfaces is investigated using molecular dynamics simulations. In agreement with our previous studies on slip flow of shear-thinning fluids, it is shown that the slip length passes through a local minimum at low shear rates and then increases rapidly at higher shear rates. We found that at sufficiently high shear rates, the slip flow over atomically flat crystalline surfaces is anisotropic. It is demonstrated numerically that the friction coefficient at the liquid-solid interface (the ratio of viscosity and slip length) undergoes a transition from a constant value to the power-law decay as a function of the slip velocity. The characteristic velocity of the transition correlates well with the diffusion velocity of fluid monomers in the first fluid layer near the solid wall at equilibrium. We also show that in the linear regime, the friction coefficient is well described by a function of a single variable, which is a product of the magnitude of surface-induced peak in the structure factor and the contact density of the adjacent fluid layer. The universal relationship between the friction coefficient and induced fluid structure holds for a number of material parameters of the interface: fluid density, chain length, wall-fluid interaction energy, wall density, lattice type and orientation, thermal or solid walls.

We investigate the shock wave produced by an energetic ion in liquid water. This wave is initiated by a rapid energy loss when the ion moves through the Bragg peak. The energy is transferred from the ion to secondary electrons, which then transfer it to the water molecules. The pressure in the overheated water increases by several orders of magnitude and drives a cylindrical shock wave on a nanometer scale. This wave eventually weakens as the front expands further; but before that, it may contribute to DNA damage due to large pressure gradients developed within a few nanometers from the ion’s trajectory. This mechanism of DNA damage may be a very important contribution to the direct chemical effects of low-energy electrons and holes.

The emergence of scaling in transport through interconnected systems is a consequence of the topological structure of the network and the physical mechanisms underlying the transport dynamics. We study transport by advection and diffusion in scale-free and Erdős-Rényi networks. Velocity distributions derived from a flow potential exhibit power-law scaling with exponent $\nu \approx \gamma +1$, where $\gamma$ is the exponent of network connectivity. Using stochastic particle simulations, we find anomalous (nonlinear) scaling of the mean-square displacement with time. We show the connection with existing descriptions of anomalous transport in disordered systems, and explain the mean transport behavior from the coupled nature of particle jump lengths and transition times.

We provide some general theoretical results to guide the optimization of transverse hydrodynamic phenomena in superhydrophobic channels. Our focus is on the canonical micro- and nanofluidic geometry of a parallel-plate channel with an arbitrary two-component (low-slip and high-slip) coarse texture, varying on scales larger than the channel thickness. By analyzing rigorous bounds on the permeability, over all possible patterns, we optimize the area fractions, slip lengths, geometry, and orientation of the surface texture to maximize transverse flow. In the case of two aligned striped surfaces, very strong transverse flows are possible. Optimized superhydrophobic surfaces may find applications in passive microfluidic mixing and amplification of transverse electrokinetic phenomena.

Thermal convection experiments in a liquid gallium layer were carried out with various intensities of uniform horizontal magnetic fields. The gallium layer was in a rectangular vessel with a 4:1:1 length ratio (1 is the height), where the magnetic field is applied in the direction normal to the longest vertical wall. An ultrasonic velocity profiling method was used to visualize the spatiotemporal variations in the flow pattern, and the temperature fluctuations in the gallium layer were also monitored. The observed flow pattern without a magnetic field shows oscillating rolls with axes normal to the longest vertical wall of the vessel. The oscillatory motion of the flow pattern was suppressed when increasing the applied magnetic field. The flow behavior was characterized by the fluctuation amplitude of the oscillation and the frequency in the range of Rayleigh numbers from $9.3\times {10}^3$ to $3.5\times {10}^5$ and Chandrasekhar numbers 0–1900. The effect of the horizontal magnetic field on the flow pattern may be summarized into three regimes with increases in the magnetic intensity: (1) no effect of the magnetic field, (2) a decrease in the oscillation of the roll structure, and (3) a steady two-dimensional roll structure with no oscillation. These regimes may be explained as a result of an increase in the dominance of Lorentz forces over inertial forces. The power spectrum from the temperature time series showed the presence of a convective-inertial subrange above Rayleigh numbers of $7\times {10}^4$, which suggests that turbulence has developed, and such a subrange was commonly observed above this Rayleigh number even with applied magnetic fields when the rolls oscillate.

The phenomenon of liquid film retraction, occurring in bursting bubbles, is investigated in the simplest case of a semi-infinite sheet of inviscid liquid. Discretized equations of motion are solved on a staggered grid, with the boundary condition adapted to handle movement and breakup of liquid domain. Two independent approaches show a perfect agreement. The calculation shows a periodic pinchoff of small droplets, in contrast to the published results on viscid fluids where the liquid accumulates at the rim. A quantitative analysis of simulated liquid profiles shows that the characteristic parameters are very close to the theoretical predictions obtained from energy conservation and laws of motion.

We study the effect of quenched randomness in the arc-length dependent spontaneous curvature of a wormlike chain under tension. In the weakly bending approximation in two dimensions, we obtain analytic results for the force-elongation curve and the width of transverse fluctuations. We compare quenched and annealed disorder and conclude that the former cannot always be reduced to a simple change in the stiffness of the pure system. We also discuss the effect of a random transverse force on the stretching response of a wormlike chain without spontaneous curvature.

Experiments have shown that DNA molecules in capillary electrophoresis migrate across field lines if a pressure gradient is applied simultaneously. We suggest that this migration results from an electrically driven flow field around the polyelectrolyte, which generates additional contributions to the center-of-mass velocity if the overall polymer conformation is asymmetric. This hypothesis leads to a coarse-grained polymer model, without explicit charges, that quantitatively explains the experimentally observed migration. The simulations contradict the widely held notion that charge neutrality eliminates the effects of hydrodynamic interactions in electrically driven flows of polyelectrolytes. We predict a measurable increase in the electrophoretic velocity of a sheared polyelectrolyte that depends on chain length.

The mass transport in an inhomogeneous medium is modeled as the limiting case of a two-component lattice gas with excluded volume constraint and one of the components fixed. In the long-wavelength approximation, the density relaxation of mobile particles is governed by diffusion and interaction with a medium inhomogeneity represented by the static component distribution. It is shown that the density relaxation can be locally accompanied by density distribution compression, i.e., the local mass transport directed from low-to high-density regions. The origin of such a “negative” mass transport is shown to be associated with the presence of a stationary drift flow defined by the medium inhomogeneity. In the quasi-one-dimensional case, the compression dynamics manifests itself in the hoppinglike motion of packet front position of diffusing substance due to staged passing through inhomogeneity barriers, and it leads to fragmentation of the packet and retardation of its spreading. The root-mean-square displacement reflects only the averaged packet front dynamics and becomes inappropriate as the transport characteristic in this regime. In the stationary case, the mass transport throughout the whole system may be directed from the boundary with lower concentration towards the boundary with higher concentration. Implications of the excluded volume constraint and particle distinguishability for these effects are discussed.

Non-linear and non-Gaussian signal inference problems are difficult to tackle. Renormalization techniques permit us to construct good estimators for the posterior signal mean within information field theory (IFT), but the approximations and assumptions made are not very obvious. Here we introduce the simple concept of minimal Gibbs free energy to IFT, and show that previous renormalization results emerge naturally. They can be understood as being the Gaussian approximation to the full posterior probability, which has maximal cross information with it. We derive optimized estimators for three applications, to illustrate the usage of the framework: (i) reconstruction of a log-normal signal from Poissonian data with background counts and point spread function, as it is needed for gamma ray astronomy and for cosmography using photometric galaxy redshifts, (ii) inference of a Gaussian signal with unknown spectrum, and (iii) inference of a Poissonian log-normal signal with unknown spectrum, the combination of (i) and (ii). Finally we explain how Gaussian knowledge states constructed by the minimal Gibbs free energy principle at different temperatures can be combined into a more accurate surrogate of the non-Gaussian posterior.

The phase rule states that a fluid element is determined locally by $c+1$ independent intensive variables when there are $c$ components. In solving the Boltzmann equation for gas transport coefficients, the standard procedures involve only $c$ independent gradients of intensive variables for the local properties (exclusive of the convective motion). This paper addresses this disparity and proposes an alternative set of gradients when solving the Boltzmann equation. The consequence is added terms in the expressions for the thermal diffusion coefficients. As well, the standard expression for the binary diffusion constant is identified as being valid only at constant total density. Irreversible thermodynamics is used as a reference base for comparing the alternate formulations.