It is known that internal magnetic field gradients in porous materials, caused by susceptibility differences at the solid-fluid interfaces, alter the observed effective Nuclear Magnetic Resonance transverse relaxation times $T_{2,\text{eff}}$. The internal gradients scale with the strength of the static background magnetic field $B_0$. Here, we acquire data at various magnitudes of $B_0$ to observe the influence of internal gradients on $T_2\text{−}{T}_2$ exchange measurements; the theory discussed and observations made are applicable to any $T_2\text{−}{T}_2$ analysis of heterogeneous materials. At high magnetic field strengths, it is possible to observe diffusive exchange between regions of local internal gradient extrema within individual pores. Therefore, the observed exchange pathways are not associated with pore-to-pore exchange. Understanding the significance of internal gradients in transverse relaxation measurements is critical to interpreting these results. We present the example of water in porous sandstone rock and offer a guideline to determine whether an observed $T_{2,\text{eff}}$ relaxation time distribution reflects the pore size distribution for a given susceptibility contrast (magnetic field strength) and spin echo separation. More generally, we confirm that for porous materials $T_1$ provides a better indication of the pore size distribution than $T_{2,\text{eff}}$ at high magnetic field strengths $\left(B_0>1\text{T}\right)$, and demonstrate the data analysis necessary to validate pore size interpretations of $T_{2,\text{eff}}$ measurements.

A parallel [open multiprocessing (OpenMP)] implementation of the Wolff single-cluster algorithm has been developed and tested for the three-dimensional (3D) Ising model. The developed procedure is generalizable to other lattice spin models and its effectiveness depends on the specific application at hand. The applicability of the developed methodology is discussed in the context of the applications, where a sophisticated shuffling scheme is used to generate pseudorandom numbers of high quality, and an iterative method is applied to find the critical temperature of the 3D Ising model with a great accuracy. For the lattice with linear size $L=1024$, we have reached the speedup about 1.79 times on two processors and about 2.67 times on four processors, as compared to the serial code. According to our estimation, the speedup about three times on four processors is reachable for the $O\left(n\right)$ models with $n\ge 2$. Furthermore, the application of the developed OpenMP code allows us to simulate larger lattices due to greater operative (shared) memory available.

We consider various anomalous $d$-dimensional diffusion problems in the presence of an absorbing boundary with radial symmetry. The motion of particles is described by a fractional diffusion equation. Their mean-square displacement is given by $⟨r^2⟩\propto t^{\gamma }\left(0<\gamma \le 1\right)$, resulting in normal diffusive motion if $\gamma =1$ and subdiffusive motion otherwise. For the subdiffusive case in sufficiently high dimensions, divergent series appear when the concentration or survival probabilities are evaluated via the method of separation of variables. While the solution for normal diffusion problems is, at most, divergent as $t\to 0$, the emergence of such series in the long-time domain is a specific feature of subdiffusion problems. We present a method to regularize such series, and, in some cases, validate the procedure by using alternative techniques (Laplace transform method and numerical simulations). In the normal diffusion case, we find that the signature of the initial condition on the approach to the steady state rapidly fades away and the solution approaches a single (the main) decay mode in the long-time regime. In remarkable contrast, long-time memory of the initial condition is present in the subdiffusive case as the spatial part ${\Psi }_1\left(r\right)$ describing the long-time decay of the solution to the steady state is determined by a weighted superposition of all spatial modes characteristic of the normal diffusion problem, the weight being dependent on the initial condition. Interestingly, ${\Psi }_1\left(r\right)$ turns out to be independent of the anomalous diffusion exponent $\gamma$.

Molecular spiders are synthetic molecular motors featuring multiple legs that each can interact with a substrate through binding and cleavage. Experimental studies suggest the motion of the spider in a matrix is biased toward uncleaved substrates and that spider properties such as processivity can be altered by changing the binding strength of the legs to substrate [R. Pei, S. K. Taylor, D. Stefanovic, S. Rudchenko, T. E. Mitchell, and M. N. Stojanovic, J. Am. Chem. Soc. 128, 12693 (2006)]. We investigate the origin of biased motion and molecular motor properties of bipedal spiders using Monte Carlo simulations. Our simulations combine a realistic chemical kinetic model, hand-over-hand or inchworm modes of stepping, and the use of a one-dimensional track. We find that stronger binding to substrate, cleavage and spider detachment from the track are contributing mechanisms to population bias. We investigate the contributions of stepping mechanism to speed, randomness parameter, processivity, coupling, and efficiency, and comment on how these molecular motor properties can be altered by changing experimentally tunable kinetic parameters.

The order formation dynamics of self-assembled monolayers (SAM) of long-chain molecules were studied using coarse-grained molecular dynamics simulations. The primary kinetic processes of surface order formation from solution are adsorption to the surface and surface diffusion. For long-chain molecules, the degrees of freedom of the chain structure and motion add various complexities to the order formation dynamics. Specifically, the strength of the chain interaction, the chain flexibility and the chain length play a significant role, and this work focused on the effects of these chain properties on the order formation dynamics. The adsorption dynamics of SAM molecules can be explained by the same theoretical framework as the polymer brush. On the other hand, the evolution of highly ordered structure is specific to SAM systems. Simulation results revealed that the development of oriented domains can be grouped into three types, isolated island growth, packing growth, and growth suppression, which depend on temperature and chain flexibility. In packing growth, oriented domains are formed gradually due to the decrease in free volume as the surface density becomes high, while the tilt of the adsorbed chain molecules does not become upright gradually as a whole. Rather, inside the oriented domains, the adsorbed chains adopt “standing” states with tilt angles almost equal to the final values, which contributes to the gradual increase in the total tilt order. The effect of chain length was also studied. In the case of semirigid chain molecules, longer-chain systems showed slightly slower growth in adsorption but faster growth in oriented domains. These simulation results reveal how chain properties influence the dynamics of oriented structure formation on surfaces.

An assumption commonly used in cable theory is revised by taking into account electrical amplification due to intracellular capacitive effects in passive dendritic cables. A generalized cable equation for a cylindrical volume representation of a dendritic segment is derived from Maxwell’s equations under assumptions: (i) the electric-field polarization is restricted longitudinally along the cable length; (ii) extracellular isopotentiality; (iii) quasielectrostatic conditions; and (iv) homogeneous medium with constant conductivity and permittivity. The generalized cable equation is identical to Barenblatt’s equation arising in the theory of infiltration in fissured strata with a known analytical solution expressed in terms of a definite integral involving a modified Bessel function and the solution to a linear one-dimensional classical cable equation. Its solution is used to determine the impact of thermal noise on voltage attenuation with distance at any particular time. A regular perturbation expansion for the membrane potential about the linear one-dimensional classical cable equation solution is derived in terms of a Green’s function in order to describe the dynamics of free charge within the Debye layer of endogenous structures in passive dendritic cables. The asymptotic value of the first perturbative term is explicitly evaluated for small values of time to predict how the slowly fluctuating (in submillisecond range) electric field attributed to intracellular capacitive effects alters the amplitude of the membrane potential. It was found that capacitive effects are almost negligible for cables with electrotonic lengths $L>0.5$, contributes up to 10% of the signal for cables with electrotonic lengths in the range between $0.25<L<0.5$, and dominates the membrane potential for electrotonically short cables $\left(L<0.2\right)$. These results show that electrotonically short dendritic cables with both ends sealed are prone to significant neurobiological thermal noise due to intracellular capacitive effects. The presence of significant thermal noise weakens the assumption of intracellular isopotentiality when approximating dendrites with compartments.

The energy distributions of secondary electrons produced by energetic carbon ions (in the energy range used, e.g., in hadron therapy), incident on liquid water, are discussed. For low-energy ions, a parametrization of the singly differential ionization cross sections is introduced, based on tuning the position of the Bragg peak. The resulting parametrization allows a fast calculation of the energy spectra of secondary electrons at different depths along the ion’s trajectory, especially near the Bragg peak. At the same time, this parametrization provides penetration depths for a broad range of initial-ion energies within the therapeutically accepted error. For high-energy ions, the energy distribution is obtained with a use of the dielectric-response function approach. Different models are compared and discussed.

We consider a long-range Ising antiferromagnet put in a transverse field (LRTIAF) with disorder. We have obtained the phase diagrams for both the classical and quantum cases. For the pure case applying quantum Monte Carlo method, we study the variation in order parameter (spin correlation in the Trotter direction), susceptibility, and average energy of the system for various values of the transverse field at different temperatures. The antiferromagnetic order is seen to get immediately broken as soon as the thermal or quantum fluctuations are added. We discuss generally the phase diagram for the same LRTIAF model with perturbative Sherrington-Kirkpatrick-type disorder. We find that while the antiferromagnetic order is immediately broken as one adds an infinitesimal transverse field or thermal fluctuation to the pure LRTIAF system, an infinitesimal SK spin-glass disorder is enough to induce a stable glass order in the LRTIAF. This glass order eventually gets destroyed as the thermal or quantum fluctuations are increased beyond their threshold values and the transition to paramagnetic phase occurs. Analytical studies for the phase transitions are discussed in details in each case. These transitions have been confirmed by applying classical and quantum Monte Carlo methods. We show here that the disordered LRTIAF has a surrogate incubation property of the SK spin glass phase.

We consider the statistical properties of the gravitational field $F$ in an infinite one-dimensional homogeneous Poisson distribution of particles using an exponential cutoff of the pair interaction to control and study the divergences which arise. Deriving an exact analytic expression for the probability density function (PDF) $P\left(F\right)$, we show that it is badly defined in the limit in which the well-known Holtzmark distribution is obtained in the analogous three-dimensional case. A well-defined $P\left(F\right)$ may, however, be obtained in the infinite range limit by an appropriate renormalization of the coupling strength giving a Gaussian form. Calculating the spatial correlation properties we show that this latter procedure has a trivial physical meaning. Finally we calculate the PDF and correlation properties of differences of forces (at separate spatial points), which are well defined without any renormalization. We explain that the convergence of these quantities is in fact sufficient to allow a physically meaningful infinite system limit to be defined for the clustering dynamics from Poissonian initial conditions.

Motivated by subdiffusive motion of biomolecules observed in living cells, we study the stochastic properties of a non-Brownian particle whose motion is governed by either fractional Brownian motion or the fractional Langevin equation and restricted to a finite domain. We investigate by analytic calculations and simulations how time-averaged observables (e.g., the time-averaged mean-squared displacement and displacement correlation) are affected by spatial confinement and dimensionality. In particular, we study the degree of weak ergodicity breaking and scatter between different single trajectories for this confined motion in the subdiffusive domain. The general trend is that deviations from ergodicity are decreased with decreasing size of the movement volume and with increasing dimensionality. We define the displacement correlation function and find that this quantity shows distinct features for fractional Brownian motion, fractional Langevin equation, and continuous time subdiffusion, such that it appears an efficient measure to distinguish these different processes based on single-particle trajectory data.

An error correcting code using a treelike multilayer perceptron is proposed. An original message ${\mathit{s}}^0$ is encoded into a codeword ${\mathit{y}}_0$ using a treelike committee machine (committee tree) or a treelike parity machine (parity tree). Based on these architectures, several schemes featuring monotonic or nonmonotonic units are introduced. The codeword ${\mathit{y}}_0$ is then transmitted via a binary asymmetric channel where it is corrupted by noise. The analytical performance of these schemes is investigated using the replica method of statistical mechanics. Under some specific conditions, some of the proposed schemes are shown to saturate the Shannon bound at the infinite codeword length limit. The influence of the monotonicity of the units on the performance is also discussed.

Seeing through the dynamics of turbid media is often demanded in both science and technology. Dynamic light scattering is one of the most powerful means to study the dynamics of various condensed matter systems, in particular, soft matter systems, in a nondestructive manner. However, its applicability is often limited by multiple scattering of light, which severely distorts the signal. To overcome this problem, cross correlation spectroscopy has been developed. Here, we propose another physical principle by which we can avoid the effects of multiple scattering. We demonstrate that this method allows not only Rayleigh but also Brillouin scattering measurements even in very turbid colloidal suspensions. This method opens a possibility of optically characterizing the dynamics (translational and rotational diffusion, thermal diffusion, and acoustic phonon dynamics) of turbid materials with mode selectivity.

Self-terminating electrochemical fabrication was used within a microfluidic channel to create a junction between two Au electrodes separated by a gap of $75\mu \text{m}$. During the electrochemical process of etching from the anode to deposition at the cathode, flow could be applied in the anode-to-cathode direction. Without applied flow, dendritic growth and dense branching morphologies were typically observed at the cathode. The addition of applied flow resulted in a densely packed gold structure that filled the channel. A computer simulation was developed to explore regimes where the diffusion, flow, and electric field between the electrodes individually dominated growth. The model provided good qualitative agreement relating flow to the experimental results. The model was also used to contrast the effects of open and closed boundaries and electric field strength, as factors related to tapering.

Spike-triggered analysis is a statistical method used to elucidate encoding properties in neural systems by estimating the statistical structure of input stimulus preceding spikes. A recent numerical study suggested that the profile of the spike-triggered average (STA) changes depending on whether the mean input stimuli are subthreshold or suprathreshold. Here we analytically verify the difference between subthreshold STA and suprathreshold STA by using the spike response model (SRM). We show by moment expansion that the suprathreshold STA is proportional to the first derivative of the response kernel, and that the subthreshold STA is expressed by a linear combination of the response kernel and its first derivative. We verify whether the analytical results obtained from the SRM can be applied to a multicompartment model with Hodgkin-Huxley type dynamics.

We predict the complex polarizability of a realistic model of a red blood cell (RBC), with an inhomogeneous dispersive and anisotropic membrane. In this model, the frequency-dependent complex electrical parameters of the individual cell layers are described by the Debye equation while the dielectric anisotropy of the cell membrane is taken into account by the different permittivities along directions normal and tangential to the membrane surface. The realistic shape of the RBC is described in terms of the Jacobi elliptic functions. To calculate the polarizability, we evoke the effective dipole moment method to determine the cell internal electric field distribution, employing an adaptive finite-element numerical approach. We have furthermore investigated the influence of the anisotropic membrane and dispersive electrical parameters of each individual cell layer on the total complex polarizability. Our findings suggest that the individual layer contribution depends on two factors: the volume of the layer and the associated induced electric field, which in turn is influenced by other layers of the cell. These results further show that the average polarizability spectra of the cell are significantly impacted by the anisotropy and associated dispersion of the cellular compartments.

In this paper, to study the interaction between network structure and dynamical property in the context of synchronization, a previously proposed adaptive coupling method is generalized where the coupling strength of a node from its neighbors not only develops adaptively according to the local synchronization property between the node and its neighbors (dynamical part) but also is modulated by its local structure, degree of the node with the form $1/k_i^{\alpha }$ (topological part). We can show both numerically and analytically that the input coupling strength of the network after adaptation displays a power-law dependence on the degree, $k^{-\theta }$, where the exponent $\theta$ is controlled by $\alpha$ as $\theta =\left(1+\alpha \right)/2$. Compared to the original adaptive coupling method, after the addition of modulation, the distribution of the node’s intensity is tunable and can be more homogenous with $\alpha \approx 1$, which results in better synchronizability. It is also found that the synchronization time can shrink greatly. Our theoretical work in the context of synchronization provides not only a deeper understanding of the interplay between structure and dynamics in real world systems, such as opinion formation and concensus, but also potential approaches to manipulate the global collective dynamics through local adaptive control.

Nonlinear (non-Boussinesq) variations in fluid’s density, viscosity, and thermal conductivity caused by a large temperature gradient in a flow domain lead to a wide variety of instability phenomena in mixed convection channel flow of a simple gas such as air. It is known that in strongly nonisothermal flows, the instabilities and the resulting flow patterns are caused by competing buoyancy and shear effects [see S. A. Suslov and S. Paolucci, J. Fluid Mech. 302, 91 (1995)]. However, as is the case in the Boussinesq limit of small temperature gradients, in moderately non-Boussinesq regimes, only a shear instability mechanism is active. Yet in contrast to Boussinesq flows, multiple instability modes are still detected. By reducing the system of full governing Navier-Stokes equations to a dynamical system of coupled Landau-type disturbance amplitude equations we compute a comprehensive parametric map of various shear-driven instabilities observed in a representative moderately non-Boussinesq regime. Subsequently, we analyze nonlinear interaction of unstable modes and reveal physical reasons for their appearance.

The escape process from the native valley for proteins subjected to a constant stretching force is examined using a model for a $\beta$ barrel. For a wide range of forces, the unfolding dynamics can be treated as one-dimensional diffusion, parametrized in terms of the end-to-end distance. In particular, the escape times can be evaluated as first passage times for a Brownian particle moving on the protein free-energy landscape, using the Smoluchowski equation. At strong forces, the unfolding process can be viewed as a diffusive drift away from the native state, while at weak forces thermal activation is the relevant mechanism. An escape-time analysis within this approach reveals a crossover from an exponential to an inverse Gaussian escape-time distribution upon passing from weak to strong forces. Moreover, a single expression valid at weak and strong forces can be devised both for the average unfolding time as well as for the corresponding variance. The analysis offers a possible explanation of recent experimental findings for the proteins ddFLN4 and ubiquitin.

We investigate how the largest synchronized connected component (LSCC) is formed and evolves to achieve a global synchronization on complex networks using Kuramoto model. In this study we use two different networks, Erdösi-Rényi network and Barabási-Albert network. From the finite-size scaling analysis, we find that the scaling exponents for the percolation order parameter and mean cluster size on both networks agree with the mean-field percolation theory, $\beta =\gamma =1$. We also find that the finite-size scaling exponent, $\overline{\nu }$, also agrees with the mean-field percolation result, $\overline{\nu }=3$. Moreover, we also show that the cluster size distributions are identical with the mean-field percolation distribution on both networks. Combining with the analysis for the merging clusters, we directly show that the LSCC on both networks evolves by merging clusters of various sizes.

We study the influence of the symmetries of competing microstructures on the emergence of different mesoscopic morphologies in the growth by vapor deposition of thin solid films. We show the results of numerical simulations in $\left(1+1\right)$- and $\left(2+1\right)$-dimensional systems including different microstructures, as well as thermally activated surface diffusion in combination with a ballistic algorithm to model the deposition process. We focus on the characterization of the transitional structures that appear in the empirical structure zone model (SZM) through the evaluation of the mean packing density and the mean coordination number. We show that the maximum coordination number of the underlying microstructure classifies the statistics of the transitional morphologies at the border between zone I in the SZM, characterized by the formation of fractal-like patterns, and zone II, where pronounced faceting develops. We analyze the appearance of lattice frustration and texture competition effects in complex microstructures having mutually exclusive symmetries.