The Parking Lot Model

This model can describe the (reversible) adsorption of proteins, as well as the compaction of a granular material subject to a sequence of taps. The PLM can be defined in any dimension. Its realization in one-dimension is a system of hard rods adsorbing and desorbing on a line of length L. The rods adsorb at a rate k+ and desorb at a rate k-.  A rod can only adsorb if  it does not overlap  with any other rod. If the process starts from an empty line, the density ρ = N/L increases until a steady state is reached.


The following graph shows the evolution of the density for different values of K (shown as different colors) obtained from a numerical simulation of the model. In all cases, the density evolves towards a steady state value (indicated by the dashed lines). The steady state density increases as K increases (and approaches one in the limit of infinite K). Note that, for large K, the density increases very slowly (it's a log scale).


At large values of K, there are three distinct regimes as shown:


The first regime, where the density varies as 1/t, is characteristic of irreversible adsorption. At much larger times, rods start to desorb and large-scale reorganizations take place that create additional space, allowing the insertion of extra rods. The kinetics is now characterized by 1/ln(t) and it is this behavior that has been observed in some granular compaction experiments. Finally at still longer times, there is an exponential approach to the steady state.

To run a Java simulation of the two-dimensional PLM, written by John Parry, Click here

For more information, see: Talbot, J., G. Tarjus, P. R. Van Tassel and P. Viot, “From Car Parking to Protein Adsorption: A Review of Sequential Addition Processes”, Colloids and Surfaces A. 165 287-324, 2000.