Velocity Autocorrelation Function Of Particles Research Articles

Comment on "Brownian motion with time-dependent friction and single-particle dynamics in liquids".

Recently, Lad, Patel, and Pratap [Phys. Rev. E 105, 064107 (2022)10.1103/PhysRevE.105.064107] revisited a microscopic theory of molecular motion in liquids, proposed by Glass and Rice [Phys. Rev. 176, 239 (1968)10.1103/PhysRev.176.239]. They argued that the friction coefficient for a Brownian particle in a liquid should exponentially depend on time and derived an equation of motion for the particle's velocity autocorrelation function (VAF). The equation was solved numerically and fitted to the results of molecular dynamics simulations on different liquids. We show that this solution, obtained under the condition of zero derivative of the VAF at time t=0, is physically incorrect at long times. This is evidenced by our exact analytical solution for the VAF, not found by Lad etal., and numerically, by using the same method as in the commented work.

The elements and richness of particle diffusion during sediment transport at small timescales

AbstractThe ideas of advection and diffusion of sediment particles are probabilistic constructs that emerge when the Master equation, a precise, probabilistic description of particle conservation, is approximated as a Fokker–Planck equation. The diffusive term approximates nonlocal transport. It ‘looks’ upstream and downstream for variations in particle activity and velocities, whose effects modify the advective term. High‐resolution measurements of bedload particle motions indicate that the mean squared displacement of tracer particles, when treated as a virtual plume, primarily reflects a nonlinear increase in the variance in hop distances with increasing travel time, manifest as apparent anomalous diffusion. In contrast, an ensemble calculation of the mean squared displacement involving paired coordinate positions independent of starting time indicates a transition from correlated random walks to normal (Fickian) diffusion. This normal behavior also is reflected in the particle velocity autocorrelation function. Spatial variations in particle entrainment produce a flux from sites of high entrainment toward sites of low entrainment. In the case of rain splash transport, this leads to topographic roughening, where differential rain splash beneath the canopy of a desert shrub contributes to the growth of a soil mound beneath the shrub. With uniform entrainment, rain splash transport, often described as a diffusive process, actually represents an advective particle flux that is proportional to the land surface slope. Particle diffusion during both bedload and rain splash transport involves motions that mostly are patchy, intermittent and rarefied. The probabilistic framework of the Master equation reveals that continuous formulations of the flux and its divergence (the Exner equation) represent statistically expected behavior, analogous to Reynolds‐averaged conditions. Key topics meriting clarification include the mechanical basis of particle diffusion, effects of rarefied conditions involving patchy, intermittent motions, and effects of rest times on diffusion of tracer particles and particle‐borne substances. Copyright © 2016 John Wiley & Sons, Ltd.

Composite generalized Langevin equation for Brownian motion in different hydrodynamic and adhesion regimes.

We present a composite generalized Langevin equation as a unified framework for bridging the hydrodynamic, Brownian, and adhesive spring forces associated with a nanoparticle at different positions from a wall, namely, a bulklike regime, a near-wall regime, and a lubrication regime. The particle velocity autocorrelation function dictates the dynamical interplay between the aforementioned forces, and our proposed methodology successfully captures the well-known hydrodynamic long-time tail with context-dependent scaling exponents and oscillatory behavior due to the binding interaction. Employing the reactive flux formalism, we analyze the effect of hydrodynamic variables on the particle trajectory and characterize the transient kinetics of a particle crossing a predefined milestone. The results suggest that both wall-hydrodynamic interactions and adhesion strength impact the particle kinetics.

A classical long-time tail in a driven granular fluid

I derive a mode-coupling theory for the tagged particle velocity autocorrelation function, ψ(t), in a fluid of randomly driven inelastic hard spheres far from equilibrium. With this, I confirm a conjecture from simulations that the velocity autocorrelation function decays algebraically, ψ(t) ∝ t−3/2, if momentum is conserved. I show that the slow decay is due to the coupling to transverse currents.

Long-time behavior of the velocity autocorrelation function at low densities and near the critical point of simple fluids

Numerous theoretical and numerical works have been devoted to the study of the algebraic decrease at large times of the velocity autocorrelation function of particles in a fluid. The derivation of this behavior, the so-called long-time tail, generally based on linearized hydrodynamics, makes no reference to any specific characteristic of the particle interactions. However, in the literature doubts have been expressed about the possibility that by numerical simulations the long-time tail can be observed in the whole fluid phase domain of systems in which the particles interact by soft-core and attractive pair potentials. In this work, extensive and accurate molecular-dynamics simulations establish that the predicted long-time tail of the velocity autocorrelation function exists in a low-density fluid of particles interacting by a soft-repulsive potential and near the liquid-gas critical point of a Lennard-Jones system. These results contribute to the confirmation that the algebraic decay of the velocity autocorrelation function is universal in these fluid systems.

Velocity autocorrelation functions of particles and clusters in liquids. A possible criterion for correlation length of incipient glass formation

We have carried out molecular-dynamics simulations over a range of densities in two and three dimensions for particles that interact through soft repulsive potentials. We have also carried out calculations of the corresponding systems in which all particles except a tagged particle and its neighbors within a certain distance are frozen. Velocity autocorrelation functions for a single particle, for clusters containing the particle, and for the velocity of the particle relative to an embedding cluster were obtained. The single-particle velocity autocorrelation function can be resolved into correlation functions describing the local rattling in a cage or a cluster, the motion of the cluster itself, and a small cross-correlation term; the function for the single particle is sensitive to the structure of the fluid over a much shorter time scale than are those of clusters, and the shape of the single-particle velocity autocorrelation function comes primarily from rattling motion within a cage. We show that the velocity autocorrelation functions of clusters are probably better probes than that for the single particle for investigating incipient glass formation since they can be used to establish a correlation length which increases when a liquid is cooled. The dynamics of clusters at a given state point depend upon their sizes, and the nature of their motions changes qualitatively from ‘‘rattling’’ for small to ‘‘diffusional’’ for large clusters, the ‘‘critical’’ size at which the change occurs increasing with decreasing temperature. A simple model for this cluster behavior is presented.

Molecular motions in a liquid‐crystalline lipid bilayer. Molecular dynamics simulation

AbstractThe results of a molecular dynamics study of a lipid bilayer are discussed. As a model we consider an ensemble of 2 × 10 flexible chains consisting of 16 groups in a box with lateral (x, y) periodicity conditions. The chains interact with each other and with the aqueous medium. Terminal groups are bound near to the bilayer surfaces by a harmonic potential. All numerical results are compared with those obtained for an isotropic polymer liquid with the same density as for the bilayer. From a study of the order parameters we conclude that the bilayer is in the liquid‐crystalline state with the chains expanding in the z‐direction normal to the surfaces. Segmental order parameters are in reasonable agreement with experimental NMR data. Dynamic properties discussed include the frequency spectrum of the velocity autocorrelation function of particles, the incoherent dynamic structure factor, dielectric absorption, and luminescence polarization. It is found that compared with the bulk polymer liquid, the molecular motions in the bilayer are quite different. Such features arise mainly from existing strong correlations in the bilayer and from a different molecular organization of the chains in the bilayer and in the bulk liquid.

Interparticle velocity correlations in simple liquids

The momentum transfer of a test particle to a cluster of atoms initially in the first shell of neighbors is investigated at liquid densities by molecular dynamics methods. The comparison of the results obtained for Lennard-Jones and soft spheres models is discussed. In the short time regime the interpretation of the data is shown to parallel that of the single particle velocity autocorrelation function. At longer times, a simple physical model can account for the decay of the cross correlation in terms of the increasing number of atoms involved in the process.

Langevin approach to the dynamics of interacting Brownian particles

Describes an approach to the dynamics of particles in liquid suspension, based on Langevin equations, which allows rather direct calculation of power series expansions in time tau of such quantities as the particle velocity autocorrelation function, mean-square displacement and dynamic structure factors. The expansions are evaluated to order tau 3 if hydrodynamic interactions are neglected, but only to order tau in their presence. The longer-time diffusion coefficients are also considered, and the importance of the structural relaxation time tau I to the theoretical development is emphasised. Further similarities between the dynamics of particle suspensions and atoms in simple fluids are pointed out.

Observation of a long-time tail in Brownian motion

By photon correlation dynamic laser light scattering the authors have measured the time dependence of the mean-square displacement ( Delta x2(t)) of spherical particles (radius approximately 1.7 mu m) in Brownian motion. Clear evidence was found for the existence of a t1/2 term in ( Delta x2(t)) which corresponds to the expected t-3/2 'long-time tail' in the particle velocity autocorrelation function. The experimentally determined amplitude of the t1/2 term was about 74+or-3% of the value predicted theoretically. Despite detailed consideration of possible systematic errors the authors were unable to explain the magnitude of this disagreement.

Dynamics of a Brownian particle in a plasma in the long-time limit

Abstract The velocity autocorrelation function (VAF) of a Brownian particle in a plasma is calculated in the long-time limit. The Brownian particle VAF exhibits the same qualitative behavior as the electron VAF in a one-component plasma: oscillations at the plasma frequency and decay ∼t- 3 2 .

Brownian motion in a critical fluid; A critical long time tail

The frictioncoefficient of a Brownian particle in a critical fluid was evaluated, taking into account the finite correlation length of density fluctuations. A t −3 2 tail of the Brownian particle velocity autocorrelation function exists close to the critical point whose amplitude is 10 2–10 3 times the amplitude of the usual long time tail found far from the critical point.

Computer study of the collective modes of a one dimensional disordered chain

It is found, on the basis of molecular dynamics experiments, that the density fluctuation of wavenumber k, in a one dimensional disordered chain of Lennard-Jones rods, decays like a damped cosine with frequency ωk and lifetime τk such that the dispersion relation ω = ωk is very similar to that of an harmonic chain and that τk goes as k−1/3 at small k. This latter observation is shown to be consistent with the conjecture that the single particle velocity autocorrelation function behaves asymptotically as t−3. It is concluded that the Lennard-Jones chain does not display a hydrodynamic decay.