This is a simulation of the simple exclusion process with open boundaries. Intuitively, this process is described as follows: We have a segment of size N where each site of the segment is occupied by at most one particle at a time. The particles move according to the following rule: Each particle is equipped with an Exponential-1-clock. When the clock rings, the particle tries to move to the right with probability p and to the left with probability 1-p. However, if the target already occupied, this move is suppressed. In addition, we have two reservoirs of particles at the endpoints of the segment and place Exponential clocks with rates α and γ at left-hand side as well as with rates β and δ on the right-hand side. Whenever a clock of rate α or δ (respectively β or γ) rings, we try to place (respectively remove) a particle from the endpoint .

The transition rates of the simple exclusion process with open boundaries are visualized as follows:

In the simulation, you are free to choose the size of the segment N, the drift parameter p, and the boundary parameters. Moreover, there is the option of running two simple exclusion processes (with different boundary parameters) within the canonical coupling. This means that the same vertex is chosen to be updated in both dynamics. In this case, the corresponding disagreement process is shown in the third row. All first class particles are shown in red, while second class particles are drawn in blue. Depending on the choice of parameters, there are various different regimes and there are examples for some of the regimes below. In order to keep track of the particle moves individually, there is the option to slow down the simulation. Also, you can show and hide the height function representation and the boxes in the segment.

Enough for the description - Have fun with the simulation :)