Particles' transfer between three regions with a Markov process

Kostas Papamichalis Dr of Theoretical Physics

The theoretical model - Features of the simulation - Activities (pdf-file)

In this application we study the evolution of a system consisted of N=340 particles placed in three neighboring regions A, B and C. The particles can jump between A, B, C according to a Markov process:
In every time-intervals of the form: I_k=[kDt,(k+1)Dt], k=0,1,...k_max=3000, each particle performs just one jump to one of the neighboring regions with a certain transition-probability, or it remains in the region it was at time kDt (see the attached pdf-file).
In the virtual environment of the simulation, it is given that: Dt=0.1; the transition probabilities per unit time are symbolized: w01, w10, w12, w21, w20, w02, where the numbers correspond to the regions A, B and C: A to 0, B to 1, C to 2. The values of the transition probabilities are controled by the user. The range of the values is determined by the set {0, 0.1, 0.2, 0.3, 0.4}
In the left part of the first window, the user can watch the real evolution of the system, which is depicted by the variation of the regions' colors with time. The color of each region corresponds to a number-range of particles that are contained in this region at the time-moments t_k. The correspondence of colors with the particles' numbers is explained in the bottom-window of the simulation.
The initial distribution of the particles is selected by the user. It is depicted in the same -left- part of the first window at time t=0.
The equilibrium distribution of the particles, which is predicted by the theoretical model, is depicted in the right part of the first window. The form of the equilibrium state is determined by the selected values of the transition probabilities, only.
In the second window, we see two graphs. The left, showing the evolution of the particles' number in each region, and the right, showing the variation of the H-Lyapunov functional of the system, with time.
On the x-axis of the left graph we have placed the sequence of the three regions A, B, C. On the y-axis we count the number of particles. The program counts at each time moment the number of particles in A, B, C and construct the corresponding histogram. In this graph, you can see the initial distribution and the evolution of the particles distribution in real time, with the form of a histogram which is changing with time. In the same graph, you can also observe a series of three blue horizontal segments, which determine the numbers of particles in each region at the equilibrium state, according to the theoretical model. The user is invited to notice that irrespectively of the selected initial state, the system converges to the equilibrium-distribution. This is also confirmed in the right graph, by noticing that H converges to zero, irrespectively of the form of initial distribution (see the attached pdf-file).

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