Displacement Autocorrelation Function Research Articles

Extensive study of droplet dynamics with end-grafted polymer chains in Pickering emulsions: Molecular Dynamics simulation and a generic model derived from the generalized Langevin equation

This paper investigates Pickering emulsions (PEs), formed by suspending oil droplets as the dispersed phase in water as the continuous phase using uncharged particles at their interface. To prevent coalescence, the particles are grafted with polymer chains, as emulsifier, not only to provide steric repulsion forces through the excluded volume between monomers floating in water, compensating for the absence of Coulombic repulsive forces but also control emulsion stabilization/destabilization. We combined molecular dynamics (MD) simulations and a generalized Langevin equation (GLE) model to study the dynamics of hairy-droplets using two dynamic quantities: mean square displacement (MSD) and velocity autocorrelation function (VACF). The number of grafted-polymer-chains, f, was the main parameter of interest, while the other parameters of the system were kept constant. A statistical approach was used to estimate theoretical GLE-based model parameters along with their uncertainties, providing insight into the diffusion behavior of these hairy-droplets and specifically addressing the transition between different observed regimes. we capture three specific values of grafted-polymer-chains as, fb=47 and ft=185 describe the transitions of mushroom-brush conformation of grafted-polymer-chains and viscous-viscoelastic behavior of hairy-droplets, respectively, and fg=257 representing the gel-like state number.

Thermophysical properties and condensation of R514A through molecular dynamics simulation

Hydrofluoroolefins are the latest (4th) generation of working fluids (refrigerants) due to zero ozone depletion and very low global warming potential. The R514A is a mixture of R1336mzz(Z) (74.7 %) and R1130(E) (25.3 %) and is considered a class of Hydrofluoroolefins. In this paper, the art of molecular dynamics simulation has been executed to evaluate the dynamical and thermophysical properties of R514A during the condensation process. The effects of temperatures and pressures on dynamical, thermophysical properties and condensation time are computed, analyzed and discussed. Further, the effects of the simulation time step on the properties mentioned above are computed. The simulation outcomes explained that the condensation time significantly depends on condensation temperatures and pressure values. For example, little time is required to complete the condensation cycle of R514A at lower temperatures and higher pressure, and vice versa. The changes in the structure during the condensation process also examine the in-term bonding and nonbinding interaction of R514A in vapor and liquid phases. The mean squared displacement and velocity autocorrelation function explained the average diffusion for the molecules of R1336mzz(Z) and R1130(E), and increased with respect to simulation time with increasing temperature and pressure. The obtained simulation outcomes are reliable, accurate, and in excellent agreement with previously available simulation data. Current investigations indicated that the refrigerant R514A considering sustainable and eco-friendly in numerous applications in the replacement of R123.

The influence of force fields on the structure and dynamics of water confined in ZIF-8 from atomistic simulations.

The complexity of modeling flexible crystals, such as ZIF-8, mainly stems from the handling of intramolecular interactions. Numerous force fields have been proposed in the literature to describe the interactions between atoms in ZIF-8. We employ seven force fields to examine the structure and dynamic behavior of water molecules confined in ZIF-8, with the aim of investigating the impact of force fields on simulation results. Various structural characterization methods consistently indicate that the choice of different force fields has quantitative effects but no qualitative effects on the structural characteristics of confined water. Additionally, the force fields do not impact the qualitative description of the diffusion mechanism. Both mean-square displacement and van Hove autocorrelation function reveal two characteristic movements of water molecules diffusing in ZIF-8: a short-time intra-cavity hopping process and a long-time inter-cavity hopping process. However, the framework flexibility is found to play a crucial role in determining the order of spatial arrangement and local structure, self-diffusion coefficient and reorientational dynamics of confined water. Specifically, the DREIDING force field gives rise to an unrealistic stiff framework, enhancing the order of spatial arrangement and diminishing the local ordered structure of confined water. Meanwhile, it results in much slower translational and reorientational dynamics. Hence, the general DREIDING force field cannot be considered for providing a quantitative description of the water structure and dynamics.

Scattering fingerprints of two-state dynamics

Particle transport in complex environments such as the interior of living cells is often (transiently) non-Fickian or anomalous, that is, it deviates from the laws of Brownian motion. Such anomalies may be the result of small-scale spatio-temporal heterogeneities in, or viscoelastic properties of, the medium, molecular crowding, etc. Often the observed dynamics displays multi-state characteristics, i.e. distinct modes of transport dynamically interconverting between each other in a stochastic manner. Reliably distinguishing between single- and multi-state dynamics is challenging and requires a combination of distinct approaches. To complement the existing methods relying on the analysis of the particle’s mean squared displacement, position- or displacement-autocorrelation function, and propagators, we here focus on ‘scattering fingerprints’ of multi-state dynamics. We develop a theoretical framework for two-state scattering signatures—the intermediate scattering function and dynamic structure factor—and apply it to the analysis of simple model systems as well as particle-tracking experiments in living cells. We consider inert tracer-particle motion as well as systems with an internal structure and dynamics. Our results may generally be relevant for the interpretation of state-of-the-art differential dynamic microscopy experiments on complex particulate systems, as well as inelastic or quasielastic neutron (incl. spin-echo) and x-ray scattering probing structural and dynamical properties of macromolecules, when the underlying dynamics displays two-state transport.

Non-Gaussian, transiently anomalous, and ergodic self-diffusion of flexible dumbbells in crowded two-dimensional environments: Coupled translational and rotational motions.

We employ Langevin-dynamics simulations to unveil non-Brownian and non-Gaussian center-of-mass self-diffusion of massive flexible dumbbell-shaped particles in crowded two-dimensional solutions. We study the intradumbbell dynamics of the relative motion of the two constituent elastically coupled disks. Our main focus is on effects of the crowding fraction ϕ and of the particle structure on the diffusion characteristics. We evaluate the time-averaged mean-squared displacement (TAMSD), the displacement probability-density function (PDF), and the displacement autocorrelation function (ACF) of the dimers. For the TAMSD at highly crowded conditions of dumbbells, e.g., we observe a transition from the short-time ballistic behavior, via an intermediate subdiffusive regime, to long-time Brownian-like spreading dynamics. The crowded system of dimers exhibits two distinct diffusion regimes distinguished by the scaling exponent of the TAMSD, the dependence of the diffusivity on ϕ, and the features of the displacement-ACF. We attribute these regimes to a crowding-induced transition from viscous to viscoelastic diffusion upon growing ϕ. We also analyze the relative motion in the dimers, finding that larger ϕ suppress their vibrations and yield strongly non-Gaussian PDFs of rotational displacements. For the diffusion coefficients D(ϕ) of translational and rotational motion of the dumbbells an exponential decay with ϕ for weak and a power-law variation D(ϕ)∝(ϕ-ϕ^{★})^{2.4} for strong crowding is found. A comparison of simulation results with theoretical predictions for D(ϕ) is discussed and some relevant experimental systems are overviewed.

Active Ornstein–Uhlenbeck model for self-propelled particles with inertia

Self-propelled particles, which convert energy into mechanical motion, exhibit inertia if they have a macroscopic size or move inside a gaseous medium, in contrast to micron-sized overdamped particles immersed in a viscous fluid. Here we study an extension of the active Ornstein–Uhlenbeck model, in which self-propulsion is described by colored noise, to access these inertial effects. We summarize and discuss analytical solutions of the particle’s mean-squared displacement and velocity autocorrelation function for several settings ranging from a free particle to various external influences, like a linear or harmonic potential and coupling to another particle via a harmonic spring. Taking into account the particular role of the initial particle velocity in a nonstationary setup, we observe all dynamical exponents between zero and four. After the typical inertial time, determined by the particle’s mass, the results inherently revert to the behavior of an overdamped particle with the exception of the harmonically confined systems, in which the overall displacement is enhanced by inertia. We further consider an underdamped model for an active particle with a time-dependent mass, which critically affects the displacement in the intermediate time-regime. Most strikingly, for a sufficiently large rate of mass accumulation, the particle’s motion is completely governed by inertial effects as it remains superdiffusive for all times.

Analysis of phase transition, structural and dynamical properties of R290 using molecular dynamics simulation

Propane (R290), a hydrocarbon refrigerant, is an excellent choice of cooling fluids for use in refrigeration and air conditioning systems considering the environmental point of view and system performance. The phase transition phenomenon and structural and dynamic properties of R290 were analyzed through a molecular dynamics (MD) simulation. The densities, isobaric heat capacities and viscosities were computed and the variations of density, volume, potential energy and the nucleation process were examined to investigate the effects of condensation temperature on the phase transition rate. The mean square displacement and velocity autocorrelation function for different temperatures were simulated for dynamical analysis. Radial distribution functions were investigated to get insight into the structural analysis at the atomic level. Shear viscosity and isobaric heat capacity obtained by the present simulation showed a good agreement with the REFPROP data. The structural analysis revealed that the phase transition of R290 did not affect its intramolecular structure.

Analysis of Random Dynamics of Cell Segmented by a Modified Active Contour Method

To understand the dynamics of a living system, the analysis of particular and/or cellular dynamics has been performed based on shape-based center point detection. After collecting sequential time-lapse images of cellular dynamics, the trajectory of a moving object is determined from the set of center points of the cell analyzed from each image. The accuracy of trajectory is significant in understanding the stochastic nature of the dynamics of biological objects. In this study, to localize a cellular object in time-lapse images, three different localization methods, namely radial symmetry, circular Hough transform, and modified active contour, were considered. To analyze the accuracy of cellular dynamics, several statistical parameters such as mean square displacement and velocity autocorrelation function were employed, and localization error derived from these was reported for each localization method. In particular, through denoising using a Poisson noise filter, improved localization characteristics could be achieved. The modified active contour with denoising reduced localization error significantly, and thus allowed for accurate estimation of the statistical parameters of cellular dynamics.

Frenkel line crossover of confined supercritical fluids

We investigate the temperature evolution of dynamics and structure of partially confined Lennard Jones (LJ) fluids in supercritical phase along an isobaric line in the P-T phase diagram using molecular dynamics simulations. We compare the Frenkel line (FL) crossover features of partially confined LJ fluids to that of the bulk LJ fluids in supercritical phase. Five different spacings have been chosen in this study and the FL crossover characteristics have been monitored for each of these spacings for temperatures ranging from 240 K to 1500 K keeping the pressure fixed at 5000 bar. We characterize the FL crossover using density of states (DoS) function and find that partially confined supercritical fluids (SCF) exhibit a progressive shift of FL crossover point to higher temperatures for smaller spacings. While the DoS perpendicular to the walls shows persistent oscillatory modes, the parallel component exhibits a smooth crossover from an oscillatory to non-oscillatory characteristics representative of FL crossover. We find that the vanishing of peaks in DoS parallel to the walls indicates that the SCF no longer supports shear mode excitations and could serve as an identifier of the FL crossover for confined systems just as is done for the bulk. Layer heights of density profiles, self-diffusivity and the peak heights of radial distribution function parallel to the walls also feature the FL crossover consistent with the DoS criteria. Surprisingly, self-diffusivity undergoes an Arrhenius to super-Arrhenius crossover at low temperatures for smaller spacings as a result of enhanced structural order evidenced via pair-excess entropy. This feature, typical of glass-forming liquids and binary supercooled liquids, is found to develop from the glass-like characteristic slowdown and strong caging in confined supercritical fluid, evidenced via mean squared displacement and velocity autocorrelation function respectively, over intermediate timescales.

Transient superdiffusion of polydisperse vacuoles in highly motile amoeboid cells.

We perform a detailed statistical analysis of diffusive trajectories of membrane-enclosed vesicles (vacuoles) in the supercrowded cytoplasm of living Acanthamoeba castellanii cells. From the vacuole traces recorded in the center-of-area frame of moving amoebae, we examine the statistics of the time-averaged mean-squared displacements of vacuoles, their generalized diffusion coefficients and anomalous scaling exponents, the ergodicity breaking parameter, the non-Gaussian features of displacement distributions of vacuoles, the displacement autocorrelation function, as well as the distributions of speeds and positions of vacuoles inside the amoeba cells. Our findings deliver novel insights into the internal dynamics of cellular structures in these infectious pathogens.

Molecular dynamics study of diffusion of xenon in water at different temperatures

Molecular Dynamics simulation was performed using 2 xenon atoms as solute and 300 water molecules as solvent. We have studied the structural properties as well as transport property. As structural properties, we have determined the radial distribution function (RDF) of xenon-xenon, xenon-water, and water-water interactions. Study of RDF of xenon-xenon and oxygen-oxygen interactions of water shows that there is hydrophobic behavior of xenon in the presence of water. We have studied the self diffusion coefficient of xenon, water, and mutual diffusion coefficients of xenon in water. The self diffusion coefficient of xenon was estimated using both mean-squared displacement (MSD) and velocity autocorrelation function (VACF), while only MSD was used for water. The temperature dependence of the diffusion coefficient of xenon and water were found to follow the Arrhenius behavior. The activation energies obtained are 12.156 KJ/mole with MSD and 14.617 KJ/mole with VACF in the temperature range taken in this study.

Construction of non-Markovian coarse-grained models employing the Mori-Zwanzig formalism and iterative Boltzmann inversion.

We propose a new coarse-grained (CG) molecular simulation technique based on the Mori-Zwanzig (MZ) formalism along with the iterative Boltzmann inversion (IBI). Non-Markovian dissipative particle dynamics (NMDPD) taking into account memory effects is derived in a pairwise interaction form from the MZ-guided generalized Langevin equation. It is based on the introduction of auxiliary variables that allow for the replacement of a non-Markovian equation with a Markovian one in a higher dimensional space. We demonstrate that the NMDPD model exploiting MZ-guided memory kernels can successfully reproduce the dynamic properties such as the mean square displacement and velocity autocorrelation function of a Lennard-Jones system, as long as the memory kernels are appropriately evaluated based on the Volterra integral equation using the force-velocity and velocity-velocity correlations. Furthermore, we find that the IBI correction of a pair CG potential significantly improves the representation of static properties characterized by a radial distribution function and pressure, while it has little influence on the dynamic processes. Our findings suggest that combining the advantages of both the MZ formalism and IBI leads to an accurate representation of both the static and dynamic properties of microscopic systems that exhibit non-Markovian behavior.

A self-avoiding walk with neural delays as a model of fixational eye movements

Fixational eye movements show scaling behaviour of the positional mean-squared displacement with a characteristic transition from persistence to antipersistence for increasing time-lag. These statistical patterns were found to be mainly shaped by microsaccades (fast, small-amplitude movements). However, our re-analysis of fixational eye-movement data provides evidence that the slow component (physiological drift) of the eyes exhibits scaling behaviour of the mean-squared displacement that varies across human participants. These results suggest that drift is a correlated movement that interacts with microsaccades. Moreover, on the long time scale, the mean-squared displacement of the drift shows oscillations, which is also present in the displacement auto-correlation function. This finding lends support to the presence of time-delayed feedback in the control of drift movements. Based on an earlier non-linear delayed feedback model of fixational eye movements, we propose and discuss different versions of a new model that combines a self-avoiding walk with time delay. As a result, we identify a model that reproduces oscillatory correlation functions, the transition from persistence to antipersistence, and microsaccades.

Atomistic insights into the thermodynamics, structure, and dynamics of ionic liquid 1-hexyl-3-methylimidazolium hexafluorophosphate via molecular dynamics study

Molecular dynamics (MD) simulations were performed to compute the properties of the ionic liquid 1-hexyl-3-methylimidazolium hexafluorophosphate, [hmim][PF6], at three different temperatures (298, 323, and 348K) and atmospheric pressure. Thermodynamic, structural, and dynamical properties, such as average density, isobaric thermal expansion coefficient, isothermal compressibility factor, radial distribution function (RDF), spatial distribution function (SDF), mean square displacement (MSD) and velocity autocorrelation function (VACF) were computed and interpreted. The ionic self-diffusion coefficients were calculated using both the Einstein and Green–Kubo formulas and it was found that the cation diffuses faster than the anion and contributes more in the electric current. The ionic self-diffusion coefficients obtained from the Green–Kubo method agreed reasonably well with the experimental data in compared to the Einstein method. Also, the self-diffusion coefficients obtained from the two methods were used to calculate the ionic conductivity from the Nernst–Einstein equation. The effects of the system size and simulation run-time on determining diffusivity using the Einstein method were also investigated. The RDFs show that anions interact preferentially with the cation ring especially in direction of the acidic hydrogen atom of the imidazolium ring. This interaction was identified as hydrogen bonding interaction and observed using SDF.

A network of discrete events for the representation and analysis of diffusion dynamics.

We developed a coarse-grained description of the phenomenology of diffusive processes, in terms of a space of discrete events and its representation as a network. Once a proper classification of the discrete events underlying the diffusive process is carried out, their transition matrix is calculated on the basis of molecular dynamics data. This matrix can be represented as a directed, weighted network where nodes represent discrete events, and the weight of edges is given by the probability that one follows the other. The structure of this network reflects dynamical properties of the process of interest in such features as its modularity and the entropy rate of nodes. As an example of the applicability of this conceptual framework, we discuss here the physics of diffusion of small non-polar molecules in a microporous material, in terms of the structure of the corresponding network of events, and explain on this basis the diffusivity trends observed. A quantitative account of these trends is obtained by considering the contribution of the various events to the displacement autocorrelation function.

Calibration of optical tweezers with non-spherical probes via high-resolution detection of Brownian motion

Optical tweezers are commonly used and powerful tools to perform force measurements on the piconewton scale and to detect nanometer-scaled displacements. However, the precision of these instruments relies to a great extent on the accuracy of the calibration method. A well-known calibration procedure is to record the stochastic motion of the trapped particle and compare its statistical behavior with the theory of the Brownian motion in a harmonic potential. Here we present an interactive calibration software which allows for the simultaneous fitting of three different statistical observables (power spectral density, mean square displacement and velocity autocorrelation function) calculated from the trajectory of the probe to enhance fitting accuracy. The fitted theory involves the hydrodynamic interactions experimentally observable at high sampling rates. Furthermore, a qualitative extension is included in our model to handle the thermal fluctuations in the orientation of optically trapped asymmetric objects. The presented calibration methodology requires no prior knowledge of the bead size and can be applied to non-spherical probes as well. The software was validated on synthetic and experimental data. Program summaryProgram title: PFMCalCatalogue identifier: AEXH_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEXH_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 206,399No. of bytes in distributed program, including test data, etc.: 10,319,465Distribution format: tar.gzProgramming language: MatLab 2011a (MathWorks Inc.).Computer: General computer running MatLab (MathWorks Inc.), using Statistics Toolbox.Operating system: Any which supports Matlab using Statistics Toolbox.RAM: 10 MBClassification: 3, 4.9, 18, 23.Nature of problem: Calibration of optical tweezers by measuring the Brownian motion of the trapped object. The voltage-to-displacement ratio of the detection system (the inverse of the sensitivity), the stiffness of the trap and the size of the bead are obtained via the simultaneous fitting of the power spectral density (PSD), mean square displacement (MSD) and velocity autocorrelation (VAF) functions calculated from the trajectory. The calibration can be performed for non-spherical probes as well.Solution method: Initialization points for all parameters are inferred from characteristic features of the statistical observables (PSD, MSD and VAF) based on the method developed by Grimm et al. in [1]. Theoretical functions for the PSD, the MSD and the VAF are calculated from the model of Brownian motion confined by a harmonic potential taking hydrodynamic interactions into consideration [2–4]. This calibration methodology has been successfully used in actual experiments with micro-spheres [5, 6]. Calculated functions are fitted to the measurement data via the Levenberg–Marquardt least square fitting routine available in the MatLab Optimalization Toolbox, using the nlinfit function. If the error to the measured data has been estimated, the corresponding data values can be weighted by the inverse of the standard error squared in order to eliminate bias introduced by heteroscedasticity. In order to increase robustness and avoid convergence to local minima, minimum search from multiple initial values in the vicinity of the first guess is possible.Additional comments: Input of the program is the experimental PSDexp, MSDexp, and VAFexp data points calculated from the measured x,y and z projections of the trajectory of the particle. The data may best be blocked and may optionally contain an error for each data point for better results. The data should be formatted into three columns: 1. independent variables (time or frequency array), 2. function values, 3. optionally, error values (e.g. standard error calculated from the binning) to improve fitting efficiency. In case error values were not estimated, the third column should be filled with zeros. The first row is reserved for a header, data is read from the second row. Data size should not be larger than a few hundreds of rows, otherwise, using larger blocks for binning is advised. In order to observe the short time behavior of the Brownian motion, influenced by hydrodynamic effects, the sampling rate should be typically higher than 100 kHz. Total sampling time is typically tens of seconds, to achieve a good resolution in the frequency range. The optical axis of the laser, the microscope and the detector system should be co-aligned to exclude artificial crosstalk between the x,y and z channels of the position detector.Running time: Seconds

Diffusion and surface excess of a confined nanoswimmer dispersion

The diffusivity and surface excess of nanoswimmers which are confined in two plates with the separation H are explored by dissipative particle dynamics. Both mean squared displacement and velocity autocorrelation function methods are used to study the diffusive behavior of nanoswimmers with the Brownian diffusivity D0 and the results obtained from both methods are consistent. The active diffusivity of confined nanoswimmers (D - D0) depends on the wall separation, swimming speed v(a), and run time τ. Our simulation results show that (D-D0)/v(a)(2)τ is a function of v(a)τ/H. The reduction in the diffusivity of active colloids is more significant than that of passive particles. The distribution of nanoswimmers between two parallel walls is acquired and two regions can be identified. The accumulation of nanoswimmers near walls is quantitatively described by the surface excess Γ. It is found that Γ grows as the nanoswimmer concentration c(b), swimming speed v(a), and run time τ are increased. The coupling between the ballistic trajectory of nanoswimmers and the walls results in nanoswimmer accumulation. The simulation outcomes indicate that Γ/Hc(b) is a function of H/v(a)τ.

Temperature dependence of diffusion coefficient of nitrogen gas in water: A molecular dynamics study

We have carried out the molecular dynamics (MD) simulation to study the structural properties and to estimate the diffusivity of molecular nitrogen ( N 2) gas (solute) in extended simple point charge model (SPC/E) water (solvent) with N 2 mole fraction of 0.018 at different temperatures. For the structural properties of the system, we have determined radial distribution function (RDF). The solute–solute, solute–solvent and solvent–solvent RDF have been evaluated. Self-diffusion coefficient of N 2 was estimated by evaluating mean-squared displacement (MSD) and velocity autocorrelation function (VACF) separately. The diffusion coefficients obtained from the two methods agree within 3%. The results are in agreement with the experimentally determined values within 10%. The self-diffusion coefficient of water ( H 2 O ) was also estimated by evaluating MSD. Mutual diffusion coefficient of the system have also been estimated invoking Darken's relation. The temperature dependance of the diffusion coefficients were found to follow Arrhenius relation.

Temperature dependence of diffusion coefficient of carbon monoxide in water: A molecular dynamics study

In the present work, molecular dynamics simulations of a mixture of carbon monoxide (CO) gas in SPC/E water (H2O) with CO as a solute and water as a solvent have been performed to understand the self-diffusion coefficient of the components, i.e. CO and water along with the mutual diffusion coefficients and structural properties of the system by studying the radial distribution function (RDF) of the components present in the system at different temperatures of 293K, 303K, 313K, 323K, and 333K. The mole fraction of CO in the system is 0.018 and that of water is 0.982. The solvent–solvent, solute–solute and solute–solvent radial distribution functions have been estimated to study the structural properties of the system. The self-diffusion coefficient of CO is calculated using mean square displacement (MSD) and velocity autocorrelation function (VACF) and that of water is estimated using MSD method only. The self-diffusion coefficient of CO agrees within around 20% of the experimental results and that of water agrees within around 10% of the experimental results. The mutual diffusion coefficient of solute–solvent is calculated using Darken's relation. The temperature dependence of self-diffusion coefficients of CO and water and mutual diffusion coefficients of CO in water all follow the Arrhenius behavior from where we estimate the activation energy of the diffusion process. Thus estimated activation energies also agree to experimental values.

Analytical solution of the generalized Langevin equation with hydrodynamic interactions: Subdiffusion of heavy tracers

We consider a generalized Langevin equation that can be used to describe thermal motion of a tracer in a viscoelastic medium by accounting for inertial and hydrodynamic effects at short times, subdiffusive scaling at intermediate times, and eventual optical trapping at long times. We derive a Laplace-type integral representation for the linear response function that governs the diffusive dynamics. This representation is particularly well suited for rapid numerical computation and theoretical analysis. In particular, we deduce explicit formulas for the mean and variance of the time averaged (TA) mean square displacement (MSD) and velocity autocorrelation function (VACF). The asymptotic behavior of the TA MSD and TA VACF is investigated at different time scales. Some biophysical and microrheological applications are discussed, with an emphasis on the statistical analysis of optical tweezers' single-particle tracking experiments in polymer networks and living cells.