Basic CTRW Distributions

Compatibility

In the previous version of the CTRW software (as reported in Berkowitz et al., Ground Water, 2001), only FPTD functions were given and they were all named starting from "ctrw...". We list the relation of these old functions to the new FPTD functions. In the new functions, "C" denotes the "cumulative" distribution. Note that for β < 1 the new functions contain one additional variable.

Basic Distributions

The definitions below are based on Margolin and Berkowitz, (2004) Continuous time random walks revisited: First passage time and spatial distributions. Physica A, 334, 46-66.

Spatial and temporal distributions ((C)SCDs and (C)FPTDs) can be expressed in terms of functions f and f_c, defined below, as follows:

The first three arguments in these functions are parameters of a given distribution, while the last one is the variable; L is the travel distance from the origin and t is the travel time.

Spatial Moments

The mean and the standard deviation of the 1D propagator for 0 < β < 2, β ≠ 1 are calculated using

and

respectively, where γ º min(b,1), e º max(b,[ 1/(b)])-1 and m º [[ 1/(e)]]. The parameters R and κ are defined below.

Constants and Parameters

There are three constants of motion: β, C and C₁. The constant β represents the functional behavior of anomalous transport, while C and C₁ relate to absolute temporal and spatial scales of the process. They arise from the coefficients of the long-time Laplace inversion of the single transition probability density (e.g., Margolin and Berkowitz, submitted, 2003).

On the basis of these constants, four parameters are defined (two relevant for (C)SCDs and two others for (C)FPTDs). R and κ are functions of time at which the (C)SCDs are calculated, while T and r are functions of the travel distance at which the (C)FPTDs are calculated. R and T represent the absolute relevant orders of magnitudes of lengths and times, while κ and r are dimensionless and should be less then 1 by absolute value. As β approaches 1, they also generally grow towards 1. For β > 1 these parameters are strictly positive (because C₁ should be positive). For β ≤ 0.5, the rigorously correct choice is κ = r=C₁=0.

In the case 0 < β < 1 the following parameters are defined:

In the case 1 < β < 2 the following parameters are defined:

From these parameter definitions, it is seen how the transport parameters change with changing time t or distance L of measurement. If the parameters are known at a certain time (distance) they can be calculated for (C)SCDs ((C)FPTDs) at another time (distance).

Functions f and f_c

By definition,

and