Scale and memory effects are already present in case of diffusion processes with deterministic coefficients if the drift coefficients vary in space.

The density of the transition probability $g(\mathbf{x},t\mid\mathbf{x}_{0},t_{0})$ of a real diffusion process $\{X_{i}(t),t\geq 0,X_{i}\in\mathbb{R},i=1,2,3\}$ with drift coefficients $V_{i}$ and diffusion coefficients $D_{ij}$ is the solution of the Cauchy problem for the Fokker-Planck equation

with the initial condition $g(\mathbf{x},t_{0}\mid\mathbf{x}_{0},t_{0})=\delta(\mathbf{x}-\mathbf{x}_{0})$. Here and in the rest of the paper summation over repeated indices is implied.

The transition probability $g$ governs the evolution of the concentration,

where $c(\mathbf{x}_{0},t_{0})$ is the initial concentration. If $c(\mathbf{x}_{0},t_{0})$ is normalized to unity, so is $c(\mathbf{x},t)$, and both can be interpreted as one-point probability densities.

By virtue of (2.2), the concentration $c(\mathbf{x},t)$ verifies the equation (2.1) with the initial condition $c(\mathbf{x},t_{0})=c(\mathbf{x}_{0},t_{0})$. The equivalence with the concentration balance equation based on Fick’s law of diffusion,

is established by the relation $V^{*}_{i}=V_{i}-\partial D_{ij}/\partial x_{j}$ between velocity components $V^{*}_{i}$ and drift coefficients $V_{i}$ [43].

A diffusion process satisfies uniformly in $\mathbf{x}$ and $t$, for all $\epsilon>0$ [31, 59],

Condition (2.4) prevents instantaneous jumps and ensures the almost sure continuity of the sample paths $X(t)$, (2.5) defines the drift coefficients, and (2.6) the diffusion coefficients [59, 43].

In applications to transport in groundwater, the class of diffusion processes is restricted by imposing conditions for finite first and second spatial moments of $g$ at finite times [57, 104]:

With (2.7-2.8), the integrals in (2.5-2.6) extend over the entire $\mathbb{R}^{3}$. In fact, the local averages, over spheres of radius $\epsilon$, are used in (2.4-2.6) to avoid the hypothesis that the first two moments exist [31, p. 276]. However, the latter is always true when the drift coefficients are samples of random velocity fields with finite-range correlations, as well as for samples of fractional Gaussian noise velocity fields, the two situations considered in the following.

The conditions (2.7) and (2.8) are fulfilled, for instance, by the one-dimensional Gaussian diffusion with affine mean and linear variance,

constant drift and diffusion coefficients, $V=d\mu(t)/dt$, $D=\frac{1}{2}ds(t)/dt$, and transition probability density invariant to spatial translations,

For diffusion processes with variable coefficients, the linear time behavior of the first two moments no longer holds. General relations between moments and coefficients were recently derived in [104, Appendix A]. For a general diffusion precess satisfying (2.4-2.6) and the conditions for finite moments (2.7-2.8), the components of the first moment, $\mu_{i}$, and of the covariance, $s_{ij}$,  $i,j=1,2,3$, are given by:

where   $\overline{V_{i}}(t)=\int V_{i}(\mathbf{x},t)c(\mathbf{x},t)d\mathbf{x}$   and   $u_{i}(\mathbf{x},t)=V_{i}(\mathbf{x},t)-\overline{V}_{i}(t)$.

According to (2.11), the covariance $s_{ij}$ is decomposed in “dispersion terms” $s_{u,ij}$, positive definite, expressed through correlations of the drift coefficients, (2.12), and “memory terms”, consisting of correlations between drift coefficients and initial positions (2.13), which are no longer positive definite (see [104, Fig. 1]). The decomposition (2.11) can be viewed as an extension of an earlier result of Kitanidis [57]. Comparing [57, Eq. 25] and [104, Eq. A2] one can readily check that the cross correlations position-velocity occurring in the Kitanidis’ relation are precisely the sum between the dispersion terms (2.12) and the memory terms (2.13).

Modeling motions in random environments often requires relations between spatial moments of probability densities and the statistics of the process trajectories (e.g., [65, p. 287]). A systematical derivation of such relations can be achieved by means of suitably defined consistent finite-dimensional probability distributions (A.4) introduced in A.1. Using them, in A.2 it is proved that the decomposition (2.11) for the diagonal components of the variance of the process governed by the Fokker-Planck equation (2.1) with a constant diffusion coefficient $D$ can be recast in terms of trajectories as follows,

where $\langle\cdot\rangle_{{}_{DX_{{}_{0}}}}$ denotes the average over the realizations of the diffusion process and over the initial positions $\mathbf{X}_{{}_{0}}$ of the trajectories $\mathbf{X}(t)$.

For given coefficients $V_{i}$ and $D$ of the Fokker-Planck equation (2.1) it is possible to construct a process satisfying the Itô equation (e.g. [59, p. 144])

where $W_{i}$ is the standard Wiener process of mean $E(W_{i})=0$ and variance $E(W_{i}^{2})=2D(t-t_{0})$. Equation (2.15) describes the diffusion process in a weak sense, that is, only the coefficients are specified but not the Wiener process [59]. If sufficient conditions for the existence of weak solutions to (2.15) are fulfilled, the process has the same probability distribution as the diffusion process governed by the corresponding Fokker-Planck equation. Doob [31, Chap. VI, Sec. 3] also proved the equivalence of Itô and Fokker-Planck representations in a strong sense. That means, under more restrictive conditions, the path-wise unique solutions of the Itô equation are diffusion precesses satisfying (2.4-2.6) and, given a diffusion process with transition probability solving the Fokker-planck equation, the associated Itô equation admits path-wise unique solutions [59, Theorems 4.6.1, and 4.7.1].

When the variance of the process (2.15) is computed for a fixed Wiener process (e.g., by Itô formula, as in [99, p. 5]) one obtains, in addition to the terms of (2.14), a term consisting of correlations between the Wiener process and the velocity fluctuations [99, Eq. 27]. This term is canceled, for instance, when a weak solution to (2.15) is constructed by successive approximations with independent Wiener processes in each iteration [99, p. 9]. However, the representation (2.14) holds for either strong or weak solutions to Itô equation if the diffusion process satisfies the supplementary conditions (2.7) and (2.8) and, as proved in A.2, it is equivalent to the decomposition (2.11) derived within the Fokker-Planck description of the process. The GRW scheme presented in Section 6 below provides numerical approximations for such diffusion processes.

To emphasize the role of the initial conditions, it is useful to rewrite the variance (2.14) as

where the sum of the second and third term of (2.14) is expressed in terms of displacements $\widetilde{X}_{i}(t)=X_{i}(t)-X_{i}(t_{0})$,

The term $\widetilde{s}_{ii}$ describes an enhanced dispersion (with respect to the local dispersion $2D(t-t_{0})$ in (2.14)), which explains the scale effect in modeling subsurface transport processes (e.g. [42]). The last term in (2.16),

quantifies the memory effects.

Similarly to (2.16), for any three successive times, $t_{1}<t_{2}<t_{3}$, the variances of the increments of the process are related by

where

$\mathrm{var}\{\cdot\}$ denotes the variance, and $\mathrm{cov}\{\cdot\}$ the covariance. In fact, (2.18) is the general binomial rule saying that the variance of the sum of two random variables is the sum of variances plus two times their covariance (e.g., [77, p.213]) and as such it is valid for any stochastic process $X(t)$. The relation (2.16) above is retrieved for the increments of the process (2.15) with the expectation $E\{\cdot\}$ defined as in A.1 by $\langle\cdot\rangle_{{}_{DX_{{}_{0}}}}$.

The condition of vanishing memory terms, $m_{ii}(t_{1},t_{2},t_{3})=0$, leads to

which expresses the uncorrelatedness of the increments of the process. As follows from (2.18), vanishing memory terms is equivalent with the additivity of the variance of the increments $\widetilde{\Sigma}_{ii}$ with respect to nonoverlapping time intervals [92, p. 5]. In the particular case of diffusion with constant coefficients the increments are uncorrelated, $m_{ii}(t_{1},t_{2},t_{3})=0$ , and (2.18) expresses the linearity of the variance $\widetilde{s}_{ii}(t_{m},t_{n})=2D(t_{n}-t_{m})$, $t_{n}>t_{m}$. It is also easy to see that the uncorrelatedness of the increments is a consequence of the translation invariance of the transition probability density (2.9). Moreover, it has been shown that the only Itô-diffusion process with space-homogeneous transition probabilities are Gaussian diffusion processes with constant drift and diffusion coefficients [1].

In general, according to Theorem II.3.2 of Doob [31, p. 74], for any real process with $E\{|X(t)|^{2}\}<\infty$ and uncorrelated increments there exists a wide sense version of $X(t)$ (i.e. with the same first two moments) which is a Gaussian process with independent increments [31, p. 100]. By the definition of the statistical independence, this process has space-homogeneous transition probabilities. Since the converse is clearly true, i.e. processes with homogeneous transition probabilities have uncorrelated increments, we have the following corollary [92, p. 5]:

If the memory terms of a real process with finite first and second moments vanish for arbitrary successive time increments, then, the transport process is a wide-sense version of a Gaussian processes with spatially homogeneous transition probabilities.

Thus, memory-free processes have homogeneous transition probabilities. Inhomogeneous transition probabilities as memory effects were also identified in case of rare and extreme events (where the memory-free limit corresponds to independent identically distributed variables) and for non-Markovian processes (generalized Langevin equation, diffusion equations with memory, or fractional diffusion) [92]. It is worth noting that the memory effects quantified by (2.17) are not necessarily an indication of non-Markovian behavior of the processes. They characterize all Itô processes such as (2.15), which are Markovian processes [59, Chap. 4], as far as their increments are correlated.

Figure 1 shows the long-time decay of the variance $R_{ii}$ of the center of mass process $X^{cm}_{i}(t)$. This implies, according to (3.4), that at large times the expectation of the second moment of the concentration may be approximated by the second moment of the mean concentration, $S_{ii}\approx\Sigma_{ii}$. Figure 1 also shows the decrease of $R_{ii}$ with increasing supports of the initial concentration. One expects therefore that, for sufficiently large contaminant sources, the approximation $S_{ii}\approx\Sigma_{ii}$ also holds at finite times. Following the terminology introduced by Dagan [26], large plumes, for which the expected second moment can be approximated by the second moment of the ensemble mean concentration, are usually called “ergodic plumes” in the hydrogeological literature. One hypothesizes also that the mean second moment $S_{ii}$ as well as the un-averaged moment $s_{ii}$ of an ergodic plume can be approximated, according to (3.6), by the one-particle dispersion $X_{ii}$ [98, 101].

Another ergodic property was formulated by Sposito et al. [89]. The transport in groundwater is called asymptotically ergodic if the solution to (3.1) for a given realization of the velocity field approaches that of the “macrodispersion” model, an upscaled advection-diffusion equation supposed to describe the transport in random velocity fields with finite correlation scales [25]. Various meanings of ergodicity in hydrogeological literature are particular cases of the general formulation proposed in [95]: an observable of the transport process is ergodic with respect to a stochastic model if the root mean square distance from the model prediction is smaller than a given threshold. The squared distance can be decomposed as sum between the squared deviation of the ensemble mean of the observable from the reference stochastic model and the variance of the observable about its mean [95, definition 5]. The usual statistical inference for ergodic estimators of the mean [118] is retrieved in this formulation when the observable is an average over the parameter range (time or space) and the stochastic model is the ensemble mean of the random function (see also [98]). The self-averaging property from statistical physics [11] corresponds to the particular case when the observable is the (un-averaged) process itself and the stochastic model is the ensemble mean of the process. The self-averaging property is thus ensured by a vanishing variance in the long time limit, and, obviously, it is implied by the asymptotic ergodicity.

The decay with time of the variance $R_{ii}$ indicates the self-averaging of the center of mass process $X^{cm}_{i}(t)$. The self-averaging of the center of mass corresponds to a self-averaging property of the mean Lagrangian velocity $\langle u_{i}(\mathbf{X}(t^{\prime}))\rangle_{{}_{DX_{{}_{0}}}}$ [104, Fig. 2]. The variance of $(X^{cm}_{i})^{2}(t)$ also was found to decrease in time and with increasing source size [100, Fig. 1]. This implies the self-averaging of the quantity $(X^{cm}_{i})^{2}/(2t)$, which is the single-realization dispersion coefficient of the center of mass [99, Eq. 23]. Since, according to Slutsky’s theorem [118] a vanishing variance is a sufficient condition for ergodicity, the self-averaging implies the “usual” ergodicity, that is, the convergence of the time and space averages of the observables $X^{cm}_{i}(t)$), $\langle u_{i}(\mathbf{X}(t^{\prime}))\rangle_{{}_{DX_{{}_{0}}}}$, and $(X^{cm}_{i})^{2}/(2t)$.

The variance of the process $X^{ens}_{i}(t)$, i.e. the second moment $\Sigma_{ii}$ of the mean concentration, computed for different shapes and sizes of the source is shown in Fig. 2. Significant dependence on initial conditions of the ensemble dispersion corrected for the initial second moment, $\Sigma_{ii}-S_{ii}(0)$, manifests in case of asymmetric sources with large extension on the $i$-axis while the initial conditions have negligible influence for sources with direction of largest extension perpendicular to the $i$-axis. This behavior was attributed to the mean memory terms, which may be significantly large in the first case and negligible in the second case (according to relation (3.5), where the influence of the $Q_{ii}$ term was found to be negligible, see [104, 100, 98]).

From Fig. 2 we conclude that the second moments of the mean concentration depend on the size, geometry, and orientation of the source. They approximate the one-particle dispersion only in special cases of narrow sources with small extension on $i$ -th direction and small memory terms (2.17). Otherwise, $\Sigma_{ii}\neq S_{ii}(0)+X_{ii}$. This indicates that the Lagrangian homogeneity, which would imply (3.6), fails even though the velocity field considered in simulations is statistically homogeneous.

The variance (2.16) of the effective process $X^{eff}_{i}(t)$ computed for fixed realizations of the velocity field shows large sample to sample fluctuations in cases where $\Sigma_{ii}$ is also strongly influenced by the initial conditions (Fig. 3). The one-particle dispersion $X_{ii}$ shown in Fig. 3 was approximated by $\Sigma_{ii}-S_{ii}(0)$ in ergodic situations consisting of large slab sources perpendicular to the $i$-axis. The deviation $S_{ii}-X_{ii}$ of the mean of $s_{ii}$ from the one-particle dispersion is one to two orders of magnitude smaller than its standard deviation $SD(s_{ii})$. These are the two quantities which determine the deviation from ergodic behavior in the general formulation presented in Section 4.1. Thus, the results from Fig. 3 indicate that the single realization dispersion $s_{ii}$ is in general non-ergodic with respect to the one-particle dispersion $X_{ii}$ at finite times.

Ergodicity may be expected, within acceptable small root mean square distances, for longitudinal dispersion in case of large transverse slab sources and for transverse dispersion in case of longitudinal slab sources. However, the Monte Carlo results contradict a common belief on ergodic plumes: large transverse plumes do not necessarily imply the ergodicity of both longitudinal and transverse dispersion. On the contrary, increasing the plume dimensions might result in dramatic non-ergodic behavior, mainly for the transverse dispersion.

In ergodic cases shown in Fig. 3 (slab sources perpendicular to $i$-axis) the relation $s_{ii}-S_{ii}(0)\approx X_{ii}$ is an acceptable approximation of (2.16). In non-ergodic cases, the memory terms (2.17) are no longer negligible and (2.16) is approximated by $s_{ii}-S_{ii}(0)\approx X_{ii}+m_{ii}$, which allows estimations of means and standard deviations of the memory terms $m_{ii}$ [104]. Since the deviation of the mean $S_{ii}-S_{ii}(0)-X_{ii}$ is negligible as compared with the standard deviation $SD(s_{ii})$ (see Fig. 3), $SD(m_{ii})\approx SD(s_{ii})$ also quantifies the non-ergodicity of $s_{ii}$ with respect to $X_{ii}$ (see definition in Section 4.1).

The Monte Carlo results presented in Fig. 4 show strong memory effects at finite times for asymmetric sources. The memory terms for Pe$=100$ are almost identical with those for pure advection (Pe$=\infty$). The mean-square convergence of $m_{ii}$ to zero indicates the asymptotic ergodicity of the actual dispersion $s_{ii}$.

Asymptotic ergodicity, and implicitly self-averaging behavior, is also indicated by Monte Carlo results on means and standard deviations of the cross-section space average concentration at the plume center of mass [95, Figs. 1 and 2].

The failure of Lagrangian homogeneity (Section 4.2) means that the mean $\langle\widetilde{X}_{i}\rangle_{{}_{DV}}$ and the variance $X_{ii}$ of the increment $\widetilde{X}_{i}(t)=X_{i}(t)-X_{i}(t_{0})$ depend on the deterministic initial position $X_{i}(t_{0})$. Since the transition density $g(\mathbf{x},t\mid\mathbf{x}_{0},t_{0})$ is the probability density of $\widetilde{X}_{i}$, the statistical homogeneity of $\widetilde{X}_{i}$ is equivalent to the invariance to space translations of the ensemble averaged transition density $\langle g\rangle_{{}_{V}}$.

The usual set-up for statistical homogeneity is as follows. Let $V(\omega,\mathbf{x})=\omega(\mathbf{x})$, $\omega\in\Omega$, be a homogeneous random function defined on the canonical probability space $(\Omega,\mathcal{A},P)$, where $\mathcal{A}$ is a $\sigma$ -algebra on $\Omega$. Measure-preserving shifts on $\Omega$ are defined through $(\tau_{\mathbf{x}_{{}_{0}}}\omega)(\mathbf{x})=\omega(\mathbf{x+x_{{}_{0}}})$, $P\circ\tau_{\mathbf{x}_{{}_{0}}}^{-1}=P$. A composed function $F(V)$ is also homogeneous if it depends on $\omega$ and $\mathbf{x}_{0}$ only through measure preserving shifts $F=F(\tau_{\mathbf{x}_{{}_{0}}}\omega)$ [119].

Let us consider the transition density solving (3.1) in a translated reference system, $\widetilde{\mathbf{x}}=\mathbf{x}-\mathbf{x}_{0}$,

As follows from (5.1), $g$ depends on velocity statistics only through $\tau_{\mathbf{x}_{{}_{0}}}\omega=\mathbf{V}(\widetilde{\mathbf{x}}+\mathbf{x}_{{}_{0}})$, hence, it is statistically homogeneous if the Eulerian velocity field $\mathbf{V}(\mathbf{x},\mathbf{\omega})$ is homogeneous. If, in addition, the solutions of (3.1) are unique in a classical sense, then $g(\mathbf{x},t\mid\mathbf{x}_{0},t_{0},\omega)=g(\mathbf{x}-\mathbf{x}_{0},t\mid\mathbf{0},t_{0},\tau_{\mathbf{x}_{0}}\omega)$ and the measure-preserving property of $\tau_{\mathbf{x}_{0}}$ implies the translation invariance of the mean transition density, that is, $\langle g\rangle_{{}_{V}}(\mathbf{x},t\mid\mathbf{x}_{0},t_{0})=\langle g\rangle_{{}_{V}}(\mathbf{x}-\mathbf{x}_{0},t\mid\mathbf{0},t_{0})$.

According to (2.15), to Fokker-Plank equation (5.1) one associates an Itô equation solving for displacements $\widetilde{X}_{i}(t)=X_{i}(t)-x_{0i}$ from the deterministic initial position $X_{i}(t_{0})=x_{0i}$,

If the solutions to (5.2) are pathwise unique, then the displacement field $\widetilde{\mathbf{X}}(t;\mathbf{x}_{0},\mathbf{\omega})=\widetilde{\mathbf{X}}(t;\mathbf{0},\tau_{\mathbf{x}_{0}}\mathbf{\omega})$ is homogeneous [119, Remark 6.7 and Prposition 6.1]. Homogeneity of $\widetilde{\mathbf{X}}(t)$ implies homogeneity of the Lagrangian velocity field, $V^{L}_{i}(\mathbf{x}_{0},t)=V_{i}(\mathbf{X}(t))=V_{i}(\widetilde{\mathbf{X}}(t)+\mathbf{x}_{0})$, which depends on statistics of $\widetilde{\mathbf{X}}$ through measure-preserving shifts. Conversely, assuming the homogeneity of $V_{i}(\mathbf{x}_{0},t)$, (5.2) implies the homogeneity of $\widetilde{\mathbf{X}}$ (see also [92, p. 2]).

We have thus the following result which summarizes the homogeneity properties of the process of diffusion in random velocity fields:

If (1) the Eulerian velocity field $\mathbf{V}(\mathbf{x},\mathbf{\omega})$ is statistically homogeneous and (2) the Fokker-Plank equation (3.1) admits unique classical solutions for (3) deterministic initial conditions, then
(a) The mean transition density $\langle g\rangle_{{}_{V}}$ is invariant to spatial translations, $\langle g\rangle_{{}_{V}}(\mathbf{x},t\mid\mathbf{x}_{0},t_{0})=\langle g\rangle_{{}_{V}}(\mathbf{x}-\mathbf{x}_{0},t\mid\mathbf{0},t_{0})$.
   If, in addition, (4) the associated Itô equation (2.15) admits pathwise unique solutions, then
(b) the following statements are equivalent:
(b1) The displacement field $\widetilde{\mathbf{X}}=X_{i}(t)-x_{0i}$ is statistically homogeneous.
(b2) The Lagrangian velocity field $V^{L}_{i}(\mathbf{x}_{0},t)$ is statistically homogeneous.
(b3) The ensemble mean transition density $\langle g\rangle_{{}_{V}}$ is translation-invariant.

The statements (b1) and (b2) in can be proved independently, without requiring the existence of a density $g$ for the transition probability. The first proof of homogeneity property (b2) and of the equality between Lagrangian and Eulerian means was given by Lumley [64] for purely advective transport, under the implicit assumption of analytical velocity realizations. Port and Stone [80] extended this result by considering diffusion in random advection fields and provided a rigorous proof for the equality of the Lagrangian and Eulerian one-point probability distributions under milder conditions, i.e continuity of the first-order spatial derivatives of the velocity samples. Zirbel [119] extended previous homogeneity results to statistical stationarity in case of space-time velocity fields and generalized the results of Port and Stone by replacing the Wiener process by a family of martingales which allow including the diffusion in the random environment.

The results obtained so far do not go beyond the equality of the one-dimensional probability distributions of the Lagrangian and Eulerian velocity fields. Since the higher-order distributions do not coincide, the probability laws of the Lagrangian and Eulerian fields are in general different [119]. The invariance to spatial translations of the mean density $\langle g\rangle_{{}_{V}}$ (statement (a)) is also a one-point statistical property, which implies the homogeneity of $\langle\widetilde{X}_{i}\rangle_{{}_{DV}}$ and $X_{ii}$, cancels the mean of the memory terms (2.17), and ensures the validity of the expression (3.6) for the second moment.

These homogeneity properties hold true for unique solutions of the transport equations. The irregularity of the velocity samples for the exponentially correlated $\ln K$ field used in Monte Carlo simulations (see [92, Fig. 1]), which do not ensure the uniqueness of the solutions, may explain the non-vanishing mean memory terms indicated by Fig. 2 and Fig. 4.

Also essential is the assumption of deterministic initial conditions. In case of random initial conditions, the velocity with the spatial argument translated by $\mathbf{x}_{0}(\omega)$ in equations (5.1) and (5.2) is not a measure-preserving shift and we are no longer in the frame of the usual homogeneity setup. Since the mean memory terms are time integrals of the Lagrangian velocity covariance [92, Eq. 15], they are non-vanishing as long as the velocity is correlated. This implies that translation-invariance of the mean transition probabilities associated to successive increments of the process may be expected only in case of un-correlated velocity fields or asymptotically in the long time limit, for velocity fields with finite correlation scales. In such cases, by the Corollary formulated in Section 2.3, the transport process is a wide-sense version of a Gaussian diffusion.

Theoretical investigations in subsurface hydrology are often based on first-order approximations for the variance of the effective, ensemble, and center of mass processes defined in (3.3) [45, 23, 2, 28, 34]. Such approximations are essentially asymptotic expansions truncated at the first order in the variance of the velocity field. Approximations obtained by Eulerian approaches, based on Fourier representations of solutions to partial differential equations similar to (3.1) [45, 2, 28, 15, 34], are in very good agrement with those derived from trajectory equations of type (2.15) [12, 34, 96, 35]. Moreover, if the same conventions in defining Fourier transform are used, the mathematical expressions of the first order approximations obtained by Eulerian and Lagrangian approaches are identical [105, Ramark 4.2]. This is just as we would expect from the Itô - Fokker-Planck equivalence (e.g. A.2). Itô representation is to be preferred here since it renders the computations easier and leads to simpler physical interpretations [92].

In the following, the first-order approximation approach is illustrated for the ensemble process $X_{i}^{ens}$. Approximations for the effective, $X_{i}^{eff}$, and center of mass, $X_{i}^{cm}$, processes can be obtained similarly ([99, Sec. 4]). The Itô process starting from $\mathbf{x}_{0}=\mathbf{0}$, in non-dimensional form, reads

where $Pe=U\lambda/D$ is the Péclet number with respect to the correlation length in the mean flow direction $\lambda=\lambda_{11}$, $U=\langle V_{1}\rangle_{{}_{V}}$, $u_{i}=V_{i}-\delta_{i,1}U$, $\langle u_{i}\rangle_{{}_{V}}=0$, $\langle u_{i}^{2}\rangle_{{}_{V}}=\sigma^{2}$, and $\epsilon=\sigma/U$ are velocity fluctuations. Within the order of magnitude hypothesis $Pe^{-1/2}=\mathcal{O}(\epsilon^{\alpha})$, $\alpha>0$, the ensemble process $X_{i}^{ens}(t)=X_{i}(t)-\langle X_{i}(t)\rangle_{{}_{DV}}$ is described, according to (5.3), by

The half time derivative of the variance of $X_{i}^{ens}$ (see (3.3)), which defines ensemble dispersion coefficients $D^{ens}_{ii}(t)=\frac{1}{2}d\Sigma_{ii}(t)/dt$ [2, 28], yields a Taylor-Green-Kubo formula [92]: