Probability Distribution Of Displacement Research Articles

Diffusion of Anisotropic Colloids in Periodic Arrays of Obstacles

Colloidal suspensions in confined geometries exhibit rich diffusion dynamics governed by particle shapes and particle-confinement interactions. Here, we propose a colloidal system, consisting of ellipsoids in periodic array of obstacles, to investigate the confined diffusion of anisotropic colloids. From the obstacle density-dependent diffusion, we discover a decoupling of translational and rotational diffusion in which only rotational motion is localized while translational motion remains diffusive. Moreover, by evaluating the probability distributions of displacements, we found Brownian but non-Gaussian diffusion behaviors with increasing the obstacle densities, which originates from the shape anisotropy of the colloid and the multiplicity of the local configurations of the ellipsoids with respect to the obstacle. Our results suggest that the shape anisotropy and spatial confinements play a vital role in the diffusion dynamics. It is important for understanding the transportations of anisotropic objects in complex environments.

Probability Distributions of the Riemann Wave and an Integral of It

In this paper, we study the statistical characteristics of the Riemann wave and the integral of it. Such problems arise in the physics of fronts’ propagation, in particular, during the wave run-up on the shore, a flame front, and phase surfaces. Using the relation of the Lagrangian and Eulerian statistical descriptions, we obtained general expressions for the probability distributions of the front velocity and its displacement. We show that the joint probabilistic distribution of displacement, velocity, and acceleration at the input of a nonlinear medium are necessary to find the probability distribution of displacement. Based on the idea of probability as the time the signal spent in a certain interval, the characteristics of the waves after their breaking were calculated.

Temperature effect on the phase stability of hydrogen C2/c phase from first-principles molecular dynamics calculations

The structural stability of hydrogen C2/c phase from 0 K to 300 K is investigated by combining the first-principles molecular dynamics (MD) simulations and density functional perturbation theory. Without considering the temperature effect, the C2/c phase is stable from 150 GPa to 250 GPa based on the harmonic phonon dispersion relations. The hydrogen molecules at the solid lattice sites are sensitive to temperature. The structural stability to instability transition of the C2/c phase upon temperature is successfully captured by the radial distribution function and probability distribution of atomic displacements from first-principles MD simulations, confirmed by the phonon power spectrum analysis in the phase space. The existence of phonon quasiparticle for different normal modes is observed directly. The phonon power spectrum of specific normal modes corresponding to the Raman and infrared (IR) activations are depicted at different temperatures and pressures. The changes of frequency with temperature are in agreement with experimental results, supporting the C2/c as the hydrogen phase III. For the first time, the anharmonic phonon dispersion curves and density of states are predicted based on the phonon quasi-particle approach. Therefore, the temperature dependence of lattice vibrations can be observed directly, providing a more complete physical picture of phonon frequency distribution with respect to the Raman and IR spectra. It is found that the high-frequency regions adopt significant frequency shifts compared to the harmonic case.

Rouse model in crowded environment modeled by “diffusing diffusivity”

We derive exact results for the dynamics of Rouse model in crowded environment modeled by stochastically varying diffusivity, the concept of “diffusing diffusivity”. We specifically generalize the initial analytical formulation of this concept of diffusivity in single particle diffusion to many particle systems with localized interactions, particularly to chains of connected beads, the Rouse model. Independent Rouse modes are allowed to diffuse with stochastically varying time dependent diffusivities to model diffusion in the crowded rearranging environment, where zeroth mode represents the chain center of mass coordinate. We develop an analytical formalism to calculate the probability distribution of these individual mode displacements and subsequently average bead displacement distribution of the Rouse chain and its moments in terms of these modes. The resulting diffusion behavior of thus modified Rouse model is a function of mode resolved parameters which model local interactions and crowded environment collectively. Our analytically tractable implementation of the “diffusing diffusivity” concept to Rouse model is one of its kind that yields non-Gaussian diffusion with the scope of being anomalous and plausibly applied to dynamics in polymeric liquids.

Probing Intermittent Motion of Polymer Chains in Weakly Attractive Nanocomposites

In this study, we investigate the motion of polymer segments in polymer/nanoparticle composites by varying nanoparticle (NP) volume fractions. By studying the probability distribution of segment displacement, segment trajectory, and the square displacement of segment, we find the intermittent motion of segments, accompanied with the coexistence of slow and fast segments in polymer nanocomposites (PNCs). The displacement distribution of segments exhibits an exponential tail, rather than a Gaussian form. The intermittent dynamics of chain segments is comprised of a long-range jump motion and a short-range localized motion, which is mediated by the weakly attractive interaction between NP and chain segment and the strong confinement induced by NPs. Meanwhile, the intermittent motion of chain segments can be described by the adsorption-desorption transition at low particle loading and confinement effect at high particle loading. These findings may provide important information for understanding the anomalous motion of polymer chains in the presence of NPs.

A simplified reliability estimation method for pile-supported wharf on the residual displacement by earthquake

Pile-supported wharves undergo lateral displacement as well as structural damage due to earthquakes. Therefore, residual displacement and structural safety assessment are necessary for earthquake-resistant design of pile-supported wharves. This study introduces a simplified reliability estimation method for pile-supported wharves, involving the residual displacement caused by earthquakes and the variation of earthquake ground motions. As the residual displacement of wharves is strongly affected by the deformation of subsoil caused by earthquakes, a soil-structure system finite element earthquake response analysis was performed. To reduce the computational load for the evaluation of a probability distribution of residual displacement, a first-order second-moment method estimating the probabilistic response of structures with a small number of trials was applied. Thefinite element analyses demonstrate that the summed squared Fourier amplitudes (SSFAs) of the input seismic motion for a cumulative probability frequency band better estimated the residual displacement. A simplified reliability estimation method utilizing three results of finite element analyses is proposed following the demonstration of its applicability.

Phase stabilities of Cmcm and Pnma SnSe studied by phonon quasiparticle approach

We investigated the structural stability of SnSe from 0 to 800 K by the phonon quasiparticle approach combining first-principles molecular-dynamics (MD) simulations and lattice dynamics. At high temperature, we witness the dynamic stability of the $Cmcm$ phase and reveal the coupling of the polarization of phonon modes and the phase transition between the $Cmcm$ and $Pnma$ phases. Specifically, in real space, the probability distribution of atomic displacements from first-principles MD simulations successfully captures the structural instability at low temperature and the structural stability at high temperature for $Cmcm$ SnSe. An analysis of phonon power spectra of several modes also delivers the dynamic stabilization of $Cmcm$ at high temperature. Particularly, the soft modes of the ${Y}_{1}$ mode at the $\mathbf{q}=Y(\frac{1}{2},\frac{1}{2},0$) point and the ${\mathrm{\ensuremath{\Gamma}}}_{1}$ mode at the $\mathbf{q}=\mathrm{\ensuremath{\Gamma}}(0,0,0)$ point of the $Cmcm$ phase in the harmonic approximation become relatively rigid at elevated temperature, in agreement with experimental and previous theoretical results. The calculated anharmonic phonon dispersions and density of states are strongly temperature-dependent, and some phonon modes adopt giant frequency shifts. These aspects demonstrate the heavy anharmonicity in SnSe. At low temperature, the transition from the $Cmcm$ to the $Pnma$ phase induces a symmetry breaking of structure. Consequently, the degeneracy of associated electronic states (mainly $p$ states) is lifted, thus lowering the energy of the $Pnma$ phase.

Assembly Tolerance Design Based on Skin Model Shapes Considering Processing Feature Degradation

To increase the reliability and accuracy of tolerance design, more and more research works are considering not only orientation and position deviations; they are also forming errors in tolerance modeling. As a direct cause of form errors in industrial mass production, the processing features of the machining system degrade over time. Under the Industry 4.0 paradigm, an assembly tolerance design method based on Skin Model Shape is proposed to take the effect of degrading processing features into consideration. A continuous-time multi-dimensional Markov process is trained through maximum likelihood estimation based on the nodal sampling point set on the machined surface. Degradation of the machined surface is modeled based on the joint probability distribution of nodal displacements. Assembly force constraints and assembly entity constraints are applied to spatial assembly simulations. Tolerance synthesis takes the manufacturing cost and assembling probability as design objectives. A design example of the rotary feed component in a five-axis machine tool is proposed for explanation and verification.

Perpendicular diffusion of magnetic field lines in double curl Beltrami magnetic fields

The transport of magnetic field lines is investigated numerically by considering a static magnetic fluctuation with a uniform background field in which the fluctuating part is obtained as a particular solution of the double curl equation and is non force-free in nature. It is found that the transport of magnetic field lines is anomalous, in fact, subdiffusion for lower fluctuation level, δB∕B0, relative to the mean field. The situation corresponds to weak chaos and closed magnetic surfaces in the phase space. Stochasticity increases by increasing the fluctuation level. Higher fluctuation levels with δB∕B0>1 lead to global stochasticity and almost normal diffusion. These results for transport characteristics are supported by analyzing the kurtosis of the displacement distribution of the chaotic field lines. A mixed phase space leads to non-Gaussian structure of the displacement probability distribution for the chaotic trajectories, whereas a wholly chaotic phase space supports nearly Gaussian distribution.

Identifying the mechanism for superdiffusivity in mouse fibroblast motility.

We seek to characterize the motility of mouse fibroblasts on 2D substrates. Utilizing automated tracking techniques, we find that cell trajectories are super-diffusive, where displacements scale faster than t1/2 in all directions. Two mechanisms have been proposed to explain such statistics in other cell types: run and tumble behavior with Lévy-distributed run times, and ensembles of cells with heterogeneous speed and rotational noise. We develop an automated toolkit that directly compares cell trajectories to the predictions of each model and demonstrate that ensemble-averaged quantities such as the mean-squared displacements and velocity autocorrelation functions are equally well-fit by either model. However, neither model correctly captures the short-timescale behavior quantified by the displacement probability distribution or the turning angle distribution. We develop a hybrid model that includes both run and tumble behavior and heterogeneous noise during the runs, which correctly matches the short-timescale behaviors and indicates that the run times are not Lévy distributed. The analysis tools developed here should be broadly useful for distinguishing between mechanisms for superdiffusivity in other cells types and environments.

SMTracker: a tool for quantitative analysis, exploration and visualization of single-molecule tracking data reveals highly dynamic binding of B. subtilis global repressor AbrB throughout the genome

Single-particle (molecule) tracking (SPT/SMT) is a powerful method to study dynamic processes in living cells at high spatial and temporal resolution. Even though SMT is becoming a widely used method in bacterial cell biology, there is no program employing different analytical tools for the quantitative evaluation of tracking data. We developed SMTracker, a MATLAB-based graphical user interface (GUI) for automatically quantifying, visualizing and managing SMT data via five interactive panels, allowing the user to interactively explore tracking data from several conditions, movies and cells on a track-by-track basis. Diffusion constants are calculated a) by a Gaussian mixture model (GMM) panel, analyzing the distribution of positional displacements in x- and y-direction using a multi-state diffusion model (e.g. DNA-bound vs. freely diffusing molecules), and inferring the diffusion constants and relative fraction of molecules in each state, or b) by square displacement analysis (SQD), using the cumulative probability distribution of square displacements to estimate the diffusion constants and relative fractions of up to three diffusive states, or c) through mean-squared displacement (MSD) analyses, allowing the discrimination between Brownian, sub- or superdiffusive behavior. A spatial distribution analysis (SDA) panel analyzes the subcellular localization of molecules, summarizing the localization of trajectories in 2D- heat maps. Using SMTracker, we show that the global transcriptional repressor AbrB performs highly dynamic binding throughout the Bacillus subtilis genome, with short dwell times that indicate high on/off rates in vivo. While about a third of AbrB molecules are in a DNA-bound state, 40% diffuse through the chromosome, and the remaining molecules freely diffuse through the cells. AbrB also forms one or two regions of high intensity binding on the nucleoids, similar to the global gene silencer H-NS in Escherichia coli, indicating that AbrB may also confer a structural function in genome organization.

Diffusion in Nanoporous Materials: Novel Insights by Combining MAS and PFG NMR

Pulsed field gradient (PFG) nuclear magnetic resonance (NMR) allows recording of molecular diffusion paths (notably, the probability distribution of molecular displacements over typically micrometers, covered during an observation time of typically milliseconds) and has thus proven to serve as a most versatile means for the in-depth study of mass transfer in complex materials. This is particularly true with nanoporous host materials, where PFG NMR enabled the first direct measurement of intracrystalline diffusivities of guest molecules. Spatial resolution, i.e., the minimum diffusion path length experimentally observable, is limited by the time interval over which the pulsed field gradients may be applied. In “conventional” PFG NMR measurements, this time interval is determined by a characteristic quantity of the host-guest system under study, the so-called transverse nuclear magnetic relaxation time. This leads, notably when considering systems with low molecular mobilities, to severe restrictions in the applicability of PFG NMR. These restrictions may partially be released by performing PFG NMR measurements in combination with “magic-angle spinning” (MAS) of the NMR sample tube. The present review introduces the fundamentals of this technique and illustrates, via a number of recent cases, the gain in information thus attainable. Examples include diffusion measurements with nanoporous host-guest systems of low intrinsic mobility and selective diffusion measurement in multicomponent systems.

Non-Gaussianity of the van Hove function and dynamic-heterogeneity length scale.

Non-Gaussian nature of the probability distribution of particles' displacements in the supercooled temperature regime in glass-forming liquids are believed to be one of the major hallmarks of glass transition. It has already been established that this probability distribution, which is also known as the van Hove function, shows universal exponential tail. The origin of such an exponential tail in the distribution function is attributed to the hopping motion of particles observed in the supercooled regime. The non-Gaussian nature can also be explained if one assumes that the system has heterogeneous dynamics in space and time. Thus exponential tail is the manifestation of dynamic heterogeneity. In this work we directly show that non-Gaussianity of the distribution of particles' displacements occur over the dynamic heterogeneity length scale and the dynamical behavior course grained over this length scale becomes homogeneous. We study the non-Gaussianity of the van Hove function by systematically coarse graining at different length scales and extract the length scale of dynamic heterogeneity at which the shape of the van Hove function crosses over from non-Gaussian to Gaussian. The obtained dynamic heterogeneity scale is found to be in very good agreement with the scale obtained from other conventional methods.

Structural origin of dynamical relaxation in undercooled Cu50Zr50 metallic liquid

Molecular dynamics (MD) simulation has been performed to explore the correlation between dynamic characteristics and structural heterogeneity in undercooled Cu50Zr50 melt based on the embedded atom method. It is found that atoms in the undercooled liquid can be classified into three groups in terms of the probability distribution of atoms' displacement, named ‘fast atoms’ (FAs), ‘intermediate atoms’ (IAs), ‘slow atoms’ (SAs). Meanwhile, the undercooled liquid can be divided into three temperature regime: as the temperature is higher than 1120 K, the distribution of FAs become network connectivity, in which the SAs region are surrounded; as the temperature is in the range from 1120 K to 900 K, the network of FAs region decompose into several isolated region gradually; after that the single relaxation separates into α- and β-relaxation at temperature lower than 900 K, the FAs region become isolated islands, while the SAs have connected into networks where the FAs regions are surrounded, which cause the significant decreasing of atomic mobility and sharp increasing of the relaxation time. The FAs, located in loose packaging and isolated region, which corresponds to β-relaxation reflected by the time-scale of intermediate plateau in mean-squared displacement (MSD) and intermediate scattering function (ISF). The SAs which develop into the backbone of the system and surround clusters formed by FAs, are responsible for the glass formation and have a connection with α-relaxation. The compressive deformation in the Cu50Zr50 metallic glass shows that the ‘fast atoms’, anticipating in β-relaxation of undercooled liquid, mostly are located in large shear strain zones and the structural heterogeneity of undercooled liquid inherits into the glass. Since metallic glass (MG) can be regarded as frozen undercooled liquid, our study is helpful for understanding the structural origins of dynamical relaxation in MGs.

Dynamic Characterization of a Bistable Energy Harvester Under Gaussian White Noise for Larger Time Constant

In this paper, the system parameters of a nonlinear bistable energy harvester excited by Gaussian white noise was investigated. Using Fokker–Planck–Kolmogorov equation, the probability distribution functions of displacement and velocity of the oscillator are obtained. The effect of various system parameters on the probability distributions of displacement and velocity of the oscillator and the mean square of the output voltage are investigated when the time constant of the piezoelectric circuit takes a larger value to achieve maximum voltage gain. The maximum peak values of the joint probability distribution of displacement and velocity of the oscillator decrease with the larger values of noise strength. The effect of parameters of bistable potential function on mean square of output voltage was also investigated. The system equations are numerically solved and mean square value of output voltage is numerically estimated and is seen to be increased as noise intensity increases and decreased as viscous damping increases. The result also shows that better power output can be achieved when the time constant takes larger value.

Transport of Internetwork Magnetic Flux Elements in the Solar Photosphere

Abstract The motions of small-scale magnetic flux elements in the solar photosphere can provide some measure of the Lagrangian properties of the convective flow. Measurements of these motions have been critical in estimating the turbulent diffusion coefficient in flux-transport dynamo models and in determining the Alfvén wave excitation spectrum for coronal heating models. We examine the motions of internetwork flux elements in Hinode/Narrowband Filter Imager magnetograms and study the scaling of their mean squared displacement and the shape of their displacement probability distribution as a function of time. We find that the mean squared displacement scales super-diffusively with a slope of about 1.48. Super-diffusive scaling has been observed in other studies for temporal increments as small as 5 s, increments over which ballistic scaling would be expected. Using high-cadence MURaM simulations, we show that the observed super-diffusive scaling at short increments is a consequence of random changes in barycenter positions due to flux evolution. We also find that for long temporal increments, beyond granular lifetimes, the observed displacement distribution deviates from that expected for a diffusive process, evolving from Rayleigh to Gaussian. This change in distribution can be modeled analytically by accounting for supergranular advection along with granular motions. These results complicate the interpretation of magnetic element motions as strictly advective or diffusive on short and long timescales and suggest that measurements of magnetic element motions must be used with caution in turbulent diffusion or wave excitation models. We propose that passive tracer motions in measured photospheric flows may yield more robust transport statistics.

Nonstationary Markovian replication process causing diverse diffusions.

We introduce a single generative mechanism that can be used to describe diverse nonstationary diffusions. A nonstationary Markovian replication process for steps is considered for which we derive analytically the time evolution of the probability distribution of the walker's displacement and the generalized telegrapher equation with time-varying coefficients, and we find that diffusivity can be determined by temporal changes of replication of an immediate step. By controlling the replications, we realize diverse diffusions such as alternating diffusion, superdiffusion, subdiffusion, and marginal diffusion, which originate from oscillating, increasing, decreasing, and slowly increasing or decreasing replications with time, respectively.

Relative dispersion in generalized two-dimensional turbulence

The statistical properties of turbulent fluids depend on how local the energy transfers among scales are, i.e. whether the energy transfer at some given scale is due to the eddies at that particular scale, or to eddies at larger (non-local) scale. This locality in the energy transfers may have consequences for the relative dispersion of passive particles. In this paper, we consider a class of generalized two-dimensional flows (produced by the so-called $\unicode[STIX]{x1D6FC}$-turbulence models), theoretically possessing different properties in terms of locality of energy transfers. It encompasses the standard barotropic quasi-geostrophic (QG) and the surface quasi-geostrophic (SQG) models as limiting cases. The relative dispersion statistics are examined, both as a function of time and as a function of scale, and compared to predictions based on phenomenological arguments assuming the locality of the cascade. We find that the dispersion statistics follow the predicted values from local theories, as long as the parameter $\unicode[STIX]{x1D6FC}$ is small enough (dynamics close to that of the SQG model), for sufficiently small initial pair separations. In contrast, non-local dispersion is observed for the QG model, a robust result when looking at relative displacement probability distributions. However, we point out that spectral energy transfers do have a non-local contribution for models with different values of $\unicode[STIX]{x1D6FC}$, including the SQG case. This indicates that locality/non-locality of the turbulent cascade may not always imply locality/non-locality in the relative dispersion of particles and that the self-similar nature of the turbulent cascade is more appropriate for determining the relative dispersion locality.

Lévy flight with absorption: A model for diffusing diffusivity with long tails.

We consider diffusion of a particle in rearranging environment, so that the diffusivity of the particle is a stochastic function of time. In our previous model of "diffusing diffusivity" [Jain and Sebastian, J. Phys. Chem. B 120, 3988 (2016)JPCBFK1520-610610.1021/acs.jpcb.6b01527], it was shown that the mean square displacement of particle remains Fickian, i.e., 〈x^{2}(T)〉∝T at all times, but the probability distribution of particle displacement is not Gaussian at all times. It is exponential at short times and crosses over to become Gaussian only in a large time limit in the case where the distribution of D in that model has a steady state limit which is exponential, i.e., π_{e}(D)∼e^{-D/D_{0}}. In the present study, we model the diffusivity of a particle as a Lévy flight process so that D has a power-law tailed distribution, viz., π_{e}(D)∼D^{-1-α} with 0<α<1. We find that in the short time limit, the width of displacement distribution is proportional to sqrt[T], implying that the diffusion is Fickian. But for long times, the width is proportional to T^{1/2α} which is a characteristic of anomalous diffusion. The distribution function for the displacement of the particle is found to be a symmetric stable distribution with a stability index 2α which preserves its shape at all times.

The mechanisms of the self-propelled spherical Janus micromotor: Self-diffusiophoresis and microbubble propulsion

Micro/nano-motor is a new and interdisciplinary field. The Janus micromotor that utilizes the heterogeneity between its two hemispheres to generate self-propulsion has attracted great research interests. This paper focuses on the self- propulsion of platinum-silica (Pt-SiO2) Janus microsphere suspended in hydrogen peroxide solution (H2O2). We systematically review the experimental and numerical results of the two instinct mechanism: Self-diffusiophoresis and microbubble propulsion. The motion of Janus microsphere whose size is smaller than 5 μ m is dominated by self- diffusiophoresis. In this regime, the Janus microsphere experiences simple Brownian motion at short times, self- diffusiophoresis at intermedia times and Brownian like motion at long times as indicated by the time variation of dimensionless mean square displacement. The non-Gaussianity due to propulsion is revealed by the probability distribution of displacement. The Janus microsphere’s rotation is found to be governed by Brownian torque, and the influences of solid wall and shear flow are discussed. When the Janus microsphere’s size is about 20–50 μ m, the microbubble propulsion is observed. Three typical stages, self-diffusiophoresis, microbubble growth and bubble collapse, are identified based on the Janus microsphere’s movement. Three different scaling laws are also established to describe that the microbubble growth in one period is respectively governed by viscous force, capillary force, and ambient fluid pressure, as bubble size increass. It is the first time to observe that a microjet occurs after the microbubble’s asymmetric collapse, and it will prople the Janus microsphere with a speed up to 0.1 m/s. In view of application, this paper introduces a new microshuttle technique which uses dieletrophoresis under AC electric field to manipulate Janus microsphere’s motion, and disscuses some important issues like the difference between self-electrophoresis and self-diffusiophoresis and improving the efficiency of Janus micromotor. A thorough understanding of the physics of Janus microsphere’s self-propulsion will shed light on designing better micromotors.