Diffusion of particles along an 1-D lattice

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=840 particles which can jump along the cells of an 1-D lattice, according to a Markov process.
In every time-intervals of the form: I_k=[kDt,(k+1)Dt], k=0,1,...t_max=1000, each particle performs just one jump to the right-neighboring cell with transition-probability w⁺Dt, or to the left-neighboring cell with transition-probability w^-Dt, or it remains in the cell it was at time kDt, with probability 1-w⁺Dt-w^-Dt (see the attached pdf-file). It is given that Dt=0.1; the quantities w⁺ and w^-, are controled by the user.
In the first window, the user can watch the real evolution of the system, which is depicted by the variation of the cells' colors with time. Each color corresponds to a range of numbers of particles that are contained in the cell 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 first window of the simulation, with a series of colors.
In the third window, we see two graphs. On the x-axis of the left graph we have placed the sequence of the 12 cells of the lattice. On the y-axis the program counts the number of particles in each cell. 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.
The equilibrium distribution of the particles in the lattice, which is predicted by the theoretical model, is depicted in the second window, with a series of colors. The system's equilibrium distribution is also shown in the left graph, with the series of the blue horizontal segments, which determine the numbers of particles at each cell.
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, where the evolution of the Lyapunov functional H[p(j,t_k)] of the system with time is depicted (see the attached pdf-file).

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