The Ehrenfest diffusion model in 1-dimensional lattice binomial distribution
Kostas Papamichalis PhD in Theoretical Physics (Athens University Greece)
Download the Theoretical Model: The Ehrenfest diffusion model in a 1-dimensional lattice
Synopsis
In this application we study and simulate the evolution of a system of N=630 particles which can jump along the cells of a 1-D finite lattice, according to the Ehrenfest model.
In a time-interval of length Dt, each particle can perform up to one jump between neighboring cells with a certain transition probability determined in the context of the model The Ehrenfest diffusion model
In a sequence of time-moments, the program of the simulation calculates the number of particles in every cell. The number of the particles in a cell is depicted by a certain cell-color. The correspondence of colors with the particles' numbers is explained in the bottom-window of the virtual environment. The initial state of the system and the number of the cells along the lattice, are selected by the user.
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 cell-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 shown in the same system of axes.
The first objective of the applet is to compare the data obtained in real-time in the running simulation, with the theoretical predictions of the model.
Furthermore, as a second objective, the user is getting able to confirm 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 a seperate window, the graph of a Lyapunov functional H corresponding to the dynamical system of the cells and particles, is created in real time. Each time-moment, the value of H is uniquely determined by the distribution of the particles in the cells of the lattice at this moment. In addition, by using the experimental graph of H over time, the user can estimate the relaxation time of the process towards the equilibrium-state.
The user is prompted to verify that irrespectively of the selected initial state, the system converges to the equilibrium-distribution.