*
written by
Kostas Papamichalis *

This simulation models a Markov process that distributes particles between three three regions A, B, and C. In every time-intervals of the form: Ik=[tk, tk+1), tk =kDt k=0,1,...kmax, each particle performs just one jump to one of the neighboring regions with a given transition-probability, or it remains in the region it was at time tk. The evolution of the system from its initial state to the final state of equilibrium, is described by a master equation. The main objective of the simulation is the confirmation of the theoretical proposition that "irrespectively of the form of the initial distribution, the system converges to a certain equilibrium state which is determined by the transition probabilities".

In the time-moments tk=kDt, k=0,1,... the program of the simulation counts the real number of particles in every sector. The intermediate states of the system between the initial state and the final state of equilibrium are depicted by a varying histogram and a sequence of changing sector-colors. On the other hand, the distribution of the particles at the equilibrium state has been determined according to the theoretical model and it is depicted in the same graphs. The user compares the real-time data with the theoretical predictions.

Finally, a Lyapunov functional H is determined for the system. Each time-moment tk, the value of H is uniquely determined by the numbers of the particles that exist in each sector at tk. The corresponding graph is designed in real time. By using this graph, the user can estimate the relaxation time of the process toward the equilibrium-state.

Last Modified *January 4, 2023*

*
*