A Decay Process Model

Kostas Papamichalis Dr of Theoretical Physics

The theoretical model - Activities (pdf-file)

In this application we study the evolution of a system consisted of N=800 identical, but discrete, cells placed at the vertices of a lattice. Each cell can exist in one of two states A and B. If at time t the j-cell is in state A, the probability p to get transitioned to the state B in the infinitesimal interval [t,t+Dt] equals wDt, where w is a real constant, named "transition probability per unit time". The value of w is chosen by the user. However, if at time t the j-cell is in state B, it remains in state B with probability 1.
In the environment of the simulation a cell is in the state A if its color is orange, and in B if its color is purple. By running the simulation, we see in real time the transition of each individual cell from state A to state B. The transition happens or not, in one of the time-intervals [t_k,t_k+1], where t_k=kDt, k=0,1,2,... and Dt=0.1. The decision of each transition is taken by using the random-number generator of JavaScript, with probability p=wDt.
In the adjacent, right-sided window, we see two graphs concerning the change with time of a certain Lyapunov functional corresponding to the master equation which describes the evolution of the system toward its equilibrium state (see the attached pdf-file: "A Model of Decay Process"):

The red one corresponds to the theoretical analytical expression of the Lyapunov functional.
The blue one depicts the variation of the Lyapunov functional in real time, according to the evolution of the system in the virtual environment of the simulation. This is achieved by counting the cells in state A each time moment t_k=kDt, k=0,1,2,...

It must be noticed that in the case of the decay process model, an appropriate Lyapunov functional, which is used in this application is the mean number of cells that are in state A at time t (The theoretical model - Activities).

Return to the initial page