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Diffusion of particles along an 1-d lattice
written by Kostas Papamichalis
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.
1 supplemental document is available
1 source code document is available
Subjects Levels Resource Types
Thermo & Stat Mech
- Kinetic and Diffusive Processes
= Diffusion
- Instructional Material
= Curriculum
= Interactive Simulation
Intended Users Formats Ratings
- Learners
- Educators
- text/html
- application/pdf
- application/javascript
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This material is released under a Creative Commons Attribution-Share Alike 4.0 license.
Rights Holder:
Kostas Papamichalis
Record Creator:
Metadata instance created November 2, 2022 by kostas papamichalis
Record Updated:
November 9, 2022 by Wolfgang Christian
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K. Papamichalis, (2022), WWW Document, (https://www.compadre.org/Repository/document/ServeFile.cfm?ID=16356&DocID=5664).
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K. Papamichalis, Diffusion of particles along an 1-d lattice (2022), <https://www.compadre.org/Repository/document/ServeFile.cfm?ID=16356&DocID=5664>.
APA Format
Papamichalis, K. (2022). Diffusion of particles along an 1-d lattice. Retrieved April 23, 2024, from https://www.compadre.org/Repository/document/ServeFile.cfm?ID=16356&DocID=5664
Chicago Format
Papamichalis, Kostas. Diffusion of particles along an 1-d lattice. 2022. https://www.compadre.org/Repository/document/ServeFile.cfm?ID=16356&DocID=5664 (accessed 23 April 2024).
MLA Format
Papamichalis, Kostas. Diffusion of particles along an 1-d lattice. 2022. 23 Apr. 2024 <https://www.compadre.org/Repository/document/ServeFile.cfm?ID=16356&DocID=5664>.
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@misc{ Author = "Kostas Papamichalis", Title = {Diffusion of particles along an 1-d lattice}, Volume = {2024}, Number = {23 April 2024}, Year = {2022} }
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%A Kostas Papamichalis %T Diffusion of particles along an 1-d lattice %D 2022 %U https://www.compadre.org/Repository/document/ServeFile.cfm?ID=16356&DocID=5664 %O text/html

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%0 Electronic Source %A Papamichalis, Kostas %D 2022 %T Diffusion of particles along an 1-d lattice %V 2024 %N 23 April 2024 %9 text/html %U https://www.compadre.org/Repository/document/ServeFile.cfm?ID=16356&DocID=5664

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