By in situ monitoring structural changes with the reflection spectrometer during the colloidal crystallization, we present direct experimental evidence of liquid-bcc-fcc phase transition in crystallization of charged colloidal particles, as a manifestation of the Ostwald’s step rule. In addition, the lifetime of the bcc metastable structure in this system decreases significantly with increasing particle volume fraction, offering a possible explanation for “exceptions” to the step rule.

A Langevin equation whose deterministic part undergoes a saddle-node bifurcation is investigated theoretically. It is found that statistical properties of relaxation trajectories in this system exhibit divergent behaviors near a saddle-node bifurcation point in the weak-noise limit, while the final value of the deterministic solution changes discontinuously at the point. A systematic formulation for analyzing a path probability measure is constructed on the basis of a singular perturbation method. In this formulation, the critical nature turns out to originate from the neutrality of exiting time from a saddle point. The theoretical calculation explains results of numerical simulations.

In most nucleation theories, the dynamics of nucleation is characterized by the evolution in time of the mass of droplets, and this time evolution is described as a combination of drift and diffusion. This assumes that the mass fluctations are described by a Markovian, i.e., memoryless, stochastic process. This paper presents a method to assess in how far this assumption of Markovianity is valid. The method is employed in nucleation studies in a two-dimensional Ising model at temperature $T=0.88T_c$, both with spin flip dynamics and with local spin exchange dynamics. In the first case, it shows that the evolution of droplet masses might be effectively described by a Markov process on large time scales. In the latter case, however, the dynamics are far from Markovian. We argue that this is due to the presence of a locally conserved quantity.

In this paper, we propose a method to distinguish between mechanisms leading to single molecule subdiffusion in confinement. We show that the method of $p$-variation, introduced in the recent paper [M. Magdziarz et al., Phys. Rev. Lett. 103, 180602 (2009)], can be successfully applied also for confined systems. We propose a test which allows distinguishing between heavy-tailed continuous-time random walk and fractional Brownian motion in the presence of binding potentials and reflecting boundaries. We apply our test to the experimental data describing motion of mRNA molecules inside E. coli cells. The results of the test show that it is more likely that fractional Brownian motion is the underlying process.

A collection of spherical particles can be packed tightly together into an amorphous packing known as “random close packing” (RCP). This structure is of interest as a model for the arrangement of molecules in simple liquids and glasses, as well as the arrangement of particles in sand piles. We use confocal microscopy to study the arrangement of colloidal particles in an experimentally realized RCP state. We image a large volume containing more than $450000$ particles with a resolution of each particle position to better than 0.02 particle diameters. While the arrangement of the particles satisfies multiple criteria for being random, we also observe a small fraction (less than 3%) of tiny crystallites (4 particles or fewer). These regions pack slightly better and are thus associated with locally higher densities. The structure factor of our sample at long length scales is nonzero, $S\left(0\right)=0.049\pm 0.008$, suggesting that there are long wavelength density fluctuations in our sample. These may be due to polydispersity or tiny crystallites. Our results suggest that experimentally realizable RCP systems may be different from simulated RCP systems, in particular, with the presence of these long wavelength density fluctuations.

Although the time scale governing diffusion-limited transport of soluble species from solution onto a planar interface is well understood, the time scale governing transport onto a spherical interface is not. The time scales that have been proposed in the literature for spherical interfaces do not capture the correct asymptotic behavior for increasing bubble radius and do not capture previously reported experimental observations of the effect of concentration. This paper develops a diffusion-limited time scale that is dependent on an intrinsic length scale termed the spherical depletion depth. The time scale is determined by considering a specific example of diffusion-limited transport of surfactant species to a water-air interface and is verified using numerical simulations and experiments. This newly derived diffusion time scale will have a significant impact on our understanding of fundamental phenomena at spherical fluid-fluid and fluid-solid interfaces, especially those involving micrometer and nanometer length scales.

In real communication and transportation networks, the geographical positions of nodes are very important for the efficiency and the tolerance of connectivity. Considering spatially inhomogeneous positions of nodes according to a population, we introduce a multiscale quartered (MSQ) network that is stochastically constructed by recursive subdivision of polygonal faces as a self-similar tiling. It has several advantages: the robustness of connectivity, the bounded short path lengths, and the shortest distance routing algorithm in a distributive manner. Furthermore, we show that the MSQ network is more efficient with shorter link lengths and more suitable with lower load for avoiding traffic congestion than other geographical networks which have various topologies ranging from river to scale-free networks. These results will be useful for providing an insight into the future design of ad hoc network infrastructures.

We introduce a measure of greedy connectivity for geographical networks (graphs embedded in space) and where the search for connecting paths relies only on local information, such as a node’s location and that of its neighbors. Constraints of this type are common in everyday life applications. Greedy connectivity accounts also for imperfect transmission across established links and is larger the higher the proportion of nodes that can be reached from other nodes with a high probability. Greedy connectivity can be used as a criterion for optimal network design.

We have derived an exact analytical expression for the average forward rate of a reversible electron transfer reaction, modeled through a reaction coordinate undergoing diffusive motion in arbitrary potential wells of the reactant and the product in presence of a localized sink of arbitrary location and strength. The dynamics of diffusive motion is described by employing two coupled generalized diffusion reaction (Smoluchowski) equations with coordinate dependent diffusivity and delta sink. The average forward electron transfer rate constant obtained here for the system, with equilibrium or nonequilibrium distributions as initial condition, is determined by the forward and backward rate constants calculated based on the transition state theory and the weighted average rate for the well dynamics. We also discuss various limiting cases for the rate of electron transfer reactions corresponding to the different experimental situations. As an illustrative example, we have considered back electron transfer (ET) reaction and shown that the present theory can explain the non-Marcus free energy gap dependence of the rate of ET reactions. More importantly, the approach presented here can easily be extended to systems describing the dynamics of diffusive motion in coupled multipotential surfaces associated with electron transfer reactions.

In this paper, the multiband transmission resonant properties of single-layer Minkowski fractal planar frequency-selective surfaces (FSSs) at normal-incidence case were numerically investigated in detail. The self-similar iterative transformational method was proposed to construct the fractal FSSs. Both the finite integration technology and finite element method were used to calculate the magnitudes and phases of transmittances, and then the induced surface current and magnetic energy density distributions were demonstrated to analyze the insight physical picture of the multiband resonant feature. The results indicate that the individual resonant frequencies of our fractal FSSs present a periodical linear-resonant phenomenon and they could be designed easily and precisely just according to the geometrical characteristic of each iteration. With the intriguing resonant properties, the fractal FSS has a potential application in the fields of dual-band or multiband resonance.

Ratchet dynamics of topological solitons of the forced and damped discrete double sine-Gordon system are studied. Directed transport occurring both in regular and in chaotic regions of the phase space and its dependence on damping, amplitude, and frequency of the driving, asymmetry parameter, and coupling constant, has been extensively investigated. We show that the passage from ratchet phase-locked regime to chaotic ratchets occurs via a period doubling route to chaos and that, quite surprisingly, pinned states can exist inside phase locking and chaotic transport regions for intermediate values of the coupling constant. The possibility to control chaotic discrete soliton ratchets by means of both small subharmonic signals and more general periodic drivings, has also been investigated.

On the basis of the stationary Schrödinger equation, the virial theorem in an inhomogeneous external field for the canonical ensemble is proved. It is shown that the difference in the form of virial theorem is conditioned by the value of the wave-function derivative on the surface of the volume, surrounding the system under consideration. The stress tensor in such a system is determined by the average values of the wave-function space derivatives.

Under certain circumstances, the time behavior of a random walk is modulated by logarithmic-periodic oscillations. Using heuristic arguments, we give a simple explanation of the origin of this modulation for diffusion on a substrate with two properties: self-similarity and finite ramification order. On these media, the time dependence of the mean-square displacement shows log-periodic modulations around a leading power law, which can be understood on the basis of a hierarchical set of diffusion constants. Both the random walk exponent and the period of oscillations are analytically obtained for a pair of examples, one is fractal and the other is nonfractal, and confirmed by Monte Carlo simulations. The last example shows that the anomalous diffusion can arise from substrates without holes of all sizes.

We analyze the exact formulas for the resistance between two arbitrary notes in a rectangular network of resistors under free, periodic and cylindrical boundary conditions obtained by Wu [J. Phys. A 37, 6653 (2004)]. Based on such expression, we then apply the algorithm of Ivashkevich, Izmailian, and Hu [J. Phys. A 35, 5543 (2002)] to derive the exact asymptotic expansions of the resistance between two maximally separated nodes on an $M\times N$ rectangular network of resistors with resistors $r$ and $s$ in the two spatial directions. Our results is $\frac{1}{s}{R}_{M\times N}\left(r,s\right)=c\left(\rho \right)\text{ln}S+c_0\left(\rho ,\xi \right)+{\sum }_{p=1}^{\infty }\frac{c_{2p}\left(\rho ,\xi \right)}{S^p}$ with $S=MN$, $\rho =r/s$ and $\xi =M/N$. The all coefficients in this expansion are expressed through analytical functions. We have introduced the effective aspect ratio ${\xi }_{eff}=\sqrt{\rho }\xi$ for free and periodic boundary conditions and ${\xi }_{eff}=\sqrt{\rho }\xi /2$ for cylindrical boundary condition and show that all finite-size correction terms are invariant under transformation ${\xi }_{eff}\to 1/{\xi }_{eff}$.

The asymptotic behavior of the survival or reunion probability of vicious walks with short-range interactions is generally well studied. In many realistic processes, however, walks interact with a long-ranged potential that decays in $d$ dimensions with distance $r$ as $r^{-d-\sigma }$. We employ methods of renormalized field theory to study the effect of such long-range interactions. We calculate the exponents describing the decay of the survival probability for all values of parameters $\sigma$ and $d$ to first order in the double expansion in $\epsilon =2-d$ and $\delta =2-d-\sigma$. We show that there are several regions in the $\sigma \text{−}d$ plane corresponding to different scalings for survival and reunion probabilities. Furthermore, we calculate the leading logarithmic corrections.

The plant microtubule cortical array is a striking feature of all growing plant cells. It consists of a more or less homogeneously distributed array of highly aligned microtubules connected to the inner side of the plasma membrane and oriented transversely to the cell growth axis. Here, we formulate a continuum model to describe the origin of orientational order in such confined arrays of dynamical microtubules. The model is based on recent experimental observations that show that a growing cortical microtubule can interact through angle dependent collisions with pre-existing microtubules that can lead either to co-alignment of the growth, retraction through catastrophe induction or crossing over the encountered microtubule. We identify a single control parameter, which is fully determined by the nucleation rate and intrinsic dynamics of individual microtubules. We solve the model analytically in the stationary isotropic phase, discuss the limits of stability of this isotropic phase, and explicitly solve for the ordered stationary states in a simplified version of the model.

Alpha rhythm is a major component of spontaneous electroencephalographic (EEG) data. We develop a novel method that can be used to estimate the instantaneous phases and amplitudes of the alpha rhythm with high accuracy by modeling the alpha rhythm phase and amplitude as Markov random field (MRF) models. By using a belief propagation technique, we construct an exact-inference algorithm that can be used to estimate instantaneous phases and amplitudes and calculate the marginal likelihood. Maximizing the marginal likelihood enables us to estimate the hyperparameters on the basis of type-II maximum likelihood estimation. We prove that the instantaneous phase and amplitude estimation by our method is consistent with that by the Hilbert transform, which has been commonly used to estimate instantaneous phases and amplitudes, of a signal filtered from observed data in the limited case that the observed data consist of only one frequency signal whose amplitude is constant and a Gaussian noise. Comparison of the performances of observation noise reduction by our method and by a Gaussian MRF model of alpha rhythm signal indicates that our method reduces observation noise more efficiently. Moreover, the instantaneous phase and amplitude estimates obtained using our method are more accurate than those obtained by the Hilbert transform. Application of our method to experimental EEG data also demonstrates that the relationship between the alpha rhythm phase and the reaction time emerges more clearly by using our method than the Hilbert transform. This indicates our method’s practical usefulness. Therefore, applying our method to experimental EEG data will enable us to estimate the instantaneous phases and amplitudes of the alpha rhythm more precisely.

We use Brownian dynamics simulations and analytical theory to investigate the physical principles underlying subdiffusive motion of a polymer. Specifically, we examine the consequences of confinement, self-interaction, viscoelasticity, and random waiting on monomer motion, as these physical phenomena may be relevant to the behavior of biological macromolecules in vivo. We find that neither confinement nor self-interaction alter the fundamental Rouse mode relaxations of a polymer. However, viscoelasticity, modeled by fractional Langevin motion, and random waiting, modeled with a continuous time random walk, lead to significant and distinct deviations from the classic polymer-dynamics model. Our results provide diagnostic tools—the monomer mean square displacement scaling and the velocity autocorrelation function—that can be applied to experimental data to determine the underlying mechanism for subdiffusive motion of a polymer.

With the goal of understanding the intricate behavior and dynamics of collections of neurons, we present superconducting circuits containing Josephson junctions that model biologically realistic neurons. These “Josephson junction neurons” reproduce many characteristic behaviors of biological neurons such as action potentials, refractory periods, and firing thresholds. They can be coupled together in ways that mimic electrical and chemical synapses. Using existing fabrication technologies, large interconnected networks of Josephson junction neurons would operate fully in parallel. They would be orders of magnitude faster than both traditional computer simulations and biological neural networks. Josephson junction neurons provide a new tool for exploring long-term large-scale dynamics for networks of neurons.

We consider decoupled continuous time random walk with finite characteristic waiting time and jump length variance. We take approximate jump length probability distribution and waiting time probability distribution given by a product of power-law and exponential function. Using this waiting time probability distribution we study diffusion behaviors for all the time. Due to the finite characteristic waiting time and jump length variance the model presents normal diffusive behavior in the long-time limit. However, the model can describe anomalous behavior at the short and intermediate times. In particular, the model can describe subdiffusive, normal, and superdiffusive behaviors at the short times. Moreover, exact solution for probability distribution of the system is also investigated.