Predictability of anomalous transport on lattice networks with quenched disorder

We study stochastic transport through a lattice network with quenched disorder and evaluate the limits of predictability of the transport behavior across realizations of spatial heterogeneity. Within a Lagrangian framework, we perform coarse graining, noise averaging, and ensemble averaging, to obtain an effective transport model for the average particle density and its fluctuations between realizations. We show that the average particle density is described exactly by a continuous time random walk (CTRW), and the particle density variance is quantified by a novel two-particle CTRW.

Figure 1

(a) Schematic of the lattice network considered here, with two sets of links with orientation $\{-\alpha ,+\alpha \}$ with respect to the $x$ axis, and lattice spacing $l=1$. Each network realization exhibits quenched disorder in the particle velocity, with link velocities determined from an independent and identically distributed (i.i.d.) process $p_v(v)$. (b) Representation of transport through the network from particles released at the origin at $t=0$. Shown is the particle density (represented by circle size) at $t=30$ for a single realization with $\beta =1.5$. The random process generating velocity disorder induces large fluctuations in the particle density between realizations.

Figure 2

(a) Particle density at $t=30$ for four different realizations of quenched disorder in the velocity field. Each velocity field is generated using the one-sided truncated power law distribution with the same value of the exponent $\beta =1.5$. The disparity in the spatial distributions of particle density illustrates the need for quantifying fluctuations among realizations. (b) Mean particle density obtained by ensemble averaging, which displays a non-Gaussian shape even for long simulation times. We simulated ${10}^4$ realizations and ${10}^4$ particles per realization.

Figure 3

(a) Profile of the variance ${\sigma }_P^2$ along the $x$ axis at $t=30$ for a lattice with velocity disorder given by a power law exponent $\beta =1.5$. The predictions from the two-particle CTRW simulation are obtained with ${10}^5$ successful particle pairs. We compare these results with those obtained from Monte Carlo simulation with ${10}^4$ quenched disorder realizations and ${10}^4$ particles per realization. Inset: 2D spatial map of the logarithmic variance. (b) Time evolution of the variance at the point of maximum particle density, ${\sigma }_{P_m}^2$, showing a power law decay of the variance, ${\sigma }_{P_m}^2~t^{-\gamma }$. Inset: dependence of $\gamma$ on the power law exponent of the velocity distribution $\beta$. The results are obtained with the two-particle CTRW formulation, using ${10}^5$ successful particle pairs per simulation.