This Demonstration shows the evolution of a partially asymmetric simple exclusion process (PASEP) on the integer lattice . In PASEP, a particle waits an exponential amount of time, then moves to the right with probability or to the left with probability , unless the site is occupied. In the case , the dynamic is totally asymmetric, and the process is known as a totally asymmetric simple exclusion process (TASEP).

Here, we let the system evolve on a one-dimensional lattice of length for a time range and simulate the presence of other particles outside the bulk by adding some boundary conditions: particles may enter or leave the bulk with a probability depending on the initial configuration of the system.

Three particle configurations can be chosen as initial conditions:

"flat": particles are located on even sites of ;

"step": particles are located on negative integers;

"stationary": particles are distributed according to a Bernoulli product measure with density .

The evolution of the asymmetric simple exclusion process (ASEP) is represented on the -plane using a graphical construction: in each column the arrows and are generated as independent Poisson processes of density , and the direction of the arrows is given by a random variable of Bernoulli distribution with parameter .

Once the initial condition is given, the trajectories of the process are uniquely determined by arrow configurations. The trajectory of a particle is represented by a continuous line. Every time a particle meets an arrow, it changes its position according to the direction of the arrow, if the site indicated by the arrow is empty. For this reason, lines never cross, and the order of the particles is preserved by the evolution.

The trajectory of a particle exiting and then reentering the bulk is represented in the same color.

Snapshot 1: flat initial conditions

Snapshot 2: step initial conditions

Snapshot 3: stationary initial conditions

Reference

[1] T. M. Liggett, Interacting Particle Systems, New York: Springer-Verlag, 1985.