This simulation uses a Markov chain to simulate the diffusion of N particles along an one dimension, finite lattice, toward the state of equilibrium. The particles are distributed along the 1D lattice cells. In a time interval of length Dt, each particle can perform just one jump between neighboring cells with a certain transition probability. The evolution of the system from its initial state to the final state of equilibrium, is described by a master equation. The initial state of the system and the transition probabilities are selected by the user. 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 a sequence of time-moments the program of the simulation counts the real number of particles in every cell. 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 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, the value of H is uniquely determined by the corresponding distribution of the particles in the cells of the lattice. The variation of H with time is depicted in real time. By using this graph, the user can estimate the relaxation time of the process toward the equilibrium-state.

Diffusion of Partilces Along a 1D Lattice Theory
A description of the Diffusion of Partilces Along a 1D Lattice and the theory of the Markov Process used in this simulation. download 217kb .pdf
Last Modified: November 9, 2022

Diffusion of Particles on 1D Lattice Source Code
EJS source code for Diffusion of Particles on 1D Lattice Model. download 69kb .zip
Last Modified: November 9, 2022

Papamichalis, K. (2022). Diffusion of particles along an 1-d lattice. Retrieved September 8, 2024, from https://www.compadre.org/Repository/document/ServeFile.cfm?ID=16356&DocID=5664

Papamichalis, Kostas. Diffusion of particles along an 1-d lattice. 2022. https://www.compadre.org/Repository/document/ServeFile.cfm?ID=16356&DocID=5664 (accessed 8 September 2024).

Papamichalis, Kostas. Diffusion of particles along an 1-d lattice. 2022. 8 Sep. 2024 <https://www.compadre.org/Repository/document/ServeFile.cfm?ID=16356&DocID=5664>.

@misc{
Author = "Kostas Papamichalis",
Title = {Diffusion of particles along an 1-d lattice},
Volume = {2024},
Number = {8 September 2024},
Year = {2022}
}

%0 Electronic Source %A Papamichalis, Kostas %D 2022 %T Diffusion of particles along an 1-d lattice %V 2024 %N 8 September 2024 %9 text/html %U https://www.compadre.org/Repository/document/ServeFile.cfm?ID=16356&DocID=5664

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