We consider the Jeffreys-type equation as the foundation in three different models of mass transfer, namely, the Jeffreys-type and two-phase models and the $D_1$ approximation to the linear Boltzmann equation. We study two classic ($1+1$)-dimensional problems in the framework of each model. The first problem is the transfer of a substance initially confined at a point. The second problem is the transfer of a substance from a stationary point source. We calculate the mean-square displacement (MSD) for the solutions of the first problem. The temporal behavior of the MSD in the framework of the first and third models is found to be the same as that in the Brownian motion described by the standard Langevin equation. In addition, we find a remarkable phenomenon when a portion of the substance does not move.

• Received 26 June 2013

DOI:https://doi.org/10.1103/PhysRevE.88.062116