Lagrangian-Eulerian statistics and numerical modeling of transport in homogeneous random media

Physical Review E, 2010

Whenever one uses translation invariant mean Green's functions to describe the behavior in the mean and to estimate dispersion coefficients for diffusion in random velocity fields, the spatial homogeneity of the transition probability of the transport process is implicitly assumed. This property can be proved for deterministic initial conditions if, in addition to the statistical homogeneity of the space-random velocity field, the existence of unique classical solutions of the transport equations is ensured. When uniqueness condition fails and translation invariance of the mean Green's function cannot be assumed, as in the case of nonsmooth samples of random velocity fields with exponential correlations, asymptotic dispersion coefficients can still be estimated within an alternative approach using the Itô equation. Numerical simulations confirm the predicted asymptotic behavior of the coefficients, but they also show their dependence on initial conditions at early times, a signature of inhomogeneous transition probabilities. Such memory effects are even more relevant for random initial conditions, which are a result of the past evolution of the process of diffusion in correlated velocity fields, and they persist indefinitely in case of power law correlations. It was found that the transition probabilities for successive times can be spatially homogeneous only if a long-time normal diffusion limit exits. Moreover, when transition probabilities, for either deterministic or random initial states, are spatially homogeneous, they can be explicitly written as Gaussian distributions.

Applied Numerical Mathematics, 2009

In this note, we consider the random linear transport equation. We indicate that standard averaging approaches to obtain an equation for the evolution of the statistical mean of the solution may also be valid for all the statistical moments of the solution. With this result we can obtain more statistical information about the random solution, as illustrated in two particular examples.

MRS Proceedings, 1994

The transport of a scalar by jump-type random advection and diffusion is described by a mean field formulation in terms of a Langevin equation for the fluctuations of the scalar. It is shown how the distribution of the scalar fluctuations is sensistive to the relative strength of advection versus diffusion, and the details of the advection process.

Physical Review E, 2002

We consider the motion of a test particle in a one-dimensional system of equal-mass point particles. The test particle plays the role of a microscopic "piston" that separates two hard-point gases with different concentrations and arbitrary initial velocity distributions. In the homogeneous case when the gases on either side of the piston are in the same macroscopic state, we compute and analyze the stationary velocity autocorrelation function C(t). Explicit expressions are obtained for certain typical velocity distributions, serving to elucidate in particular the asymptotic behavior of C(t). It is shown that the occurrence of a non-vanishing probability mass at zero velocity is necessary for the occurrence of a long-time tail in C(t). The conditions under which this is a t −3 tail are determined. Turning to the inhomogeneous system with different macroscopic states on either side of the piston, we determine

Physical Review Letters, 2008

Single particle tracking of mRNA molecules and lipid granules in living cells shows that the time averaged mean squared displacement δ 2 of individual particles remains a random variable while indicating that the particle motion is subdiffusive. We investigate this type of ergodicity breaking within the continuous time random walk model and show that δ 2 differs from the corresponding ensemble average. In particular we derive the distribution for the fluctuations of the random variable δ 2 . Similarly we quantify the response to a constant external field, revealing a generalization of the Einstein relation. Consequences for the interpretation of single molecule tracking data are discussed. 05.40.Fb,87.10.Mn An ensemble of non interacting Brownian particles spreads according to Fick's law as a Gaussian packet. The ensemble averaged mean square displacement (MSD) is x 2 (t) = 2D 1 t where D 1 is the diffusion constant. By an Einstein relation D 1 is expressed in terms of statistical properties of the microscopic jumps according to D 1 = δx 2 /2 τ where τ is the average time between jumps and δx 2 is the variance of the jump lengths. Instead one can analyze the time series x(t) of the particle trajectory and determine the time averaged (TA) MSD

A mean-field equation for a passive scalar e.g., for a mean number density of particles in a random velocity field incompressible and compressible with a finite constant renewal time is derived. The finite renewal time of a random velocity field results in the appearance of high-order spatial derivatives in the mean-field equation for a passive scalar. We considered three models of a random velocity field: i a velocity field with a small renewal time; ii the Gaussian approximation for Lagrangian trajectories; and iii a small inhomogeneity of the velocity and mean passive scalar fields. For a small renewal time we recovered results obtained using the-function-correlated in time random velocity field. The finite renewal time and compressibility of the velocity field can cause a depletion of turbulent diffusion and a modification of an effective drift velocity. For a compressible velocity field the form of the mean-field equation for a passive scalar depends on the details of the velocity field, i.e., the universality is lost. For an incompressible velocity field the universality exists in spite of the finite renewal time. Results by Saffman J. Fluid Mech. 8, 273 1960 for the effect of molecular diffusivity in turbulent diffusion are generalized for the case of a compressible and anisotropic random velocity field. The obtained results may be of relevance in some atmospheric phenomena e.g., atmospheric aerosols and smog formation.

2011

Recent works call attention that basic concepts in statistical mechanics are still under discussion. In particular, we have shown that some of those concepts can be discussed in a direct and analytical way in diffusion.

Journal of Fluid Mechanics, 1982

Various theories seeking to relate the velocity statistics of Lagrangian particles to the statistics of the Eulerian flow in which they are embedded are examined. Mean particle drift, mean-square particle velocity and the frequency spectrum of velocity are examined for stationary, homogeneous and joint-normally distributed Eulerian fields. Predictions based on a third-order weak-interaction expansion, the successive approximation procedure of Phythian (1976), the quasi-normal approximation of Saffman (1969), the parametrized model of Saffman (1962), and a new procedure based on a statistical estimator of the kinematic equation are compared with simulations of particle motion in one-dimensional flow. Only the statistical estimator produces both acceptable mean-drift and frequency-spectrum predictions.

Physical Review E, 2009

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Monte Carlo and Quasi-Monte Carlo Methods 2008, 2009

Upscaled coefficients for diffusion in ergodic velocity fields are derived by summing up correlations of increments of the position process, or equivalently of the Lagrangian velocity. Ergodic estimations of the correlations are obtained from time averages over finite paths sampled on a single trajectory of the process and a space average with respect to the initial positions of the paths. The first term in this path decomposition of the diffusion coefficients corresponds to Markovian diffusive behavior and is the only contribution for processes with independent increments. The next terms describe memory effects on diffusion coefficients until they level off to the value of the upscaled coefficients. Since the convergence with respect to the path length is rather fast and no repeated Monte Carlo simulations are required, this method speeds up the computation of the upscaled coefficients over methods based on long-time limit and ensemble averages by four orders of magnitude.