Second Law of
Thermodynamics and Boltzmann's statistics
This application performs successive simulations of an isolated assembly of 1000 vibrating molecules modelled as quantum harmonic oscillators. The objective is to reach, by means of molecular random collisions, the thermodynamic equilibrium detected through the entropy evolution and the approach to Boltzmann's distribution.
Each successive
simulation, executed by pressing the Play button, consists of 2000
random collisions (the simulation number is indicated in the middle
yellow field) after which static plots are displayed. Dynamic plots
can be produced by ticking the DynPlot
check box .
The
reduced entropy (S/k) evolution is shown at the top plotting panel,
after every 50 collisions, as a function of the total number of
collisions.The numerical values are given at the top yellow field.
The bottom plotting panel shows the number of particles at each energy level after the simulations, being the red curve the theoretical Boltzmann's distribution. The total number of collisions is recorded at the bottom yellow field.
As the entropy of the system tends, on average, to a maximum value (the equilibrium "ceiling"), the molecular distributions approach Boltzmann's distribution despite ever present spontaneous fluctuations.
Author: Fernando M.S. Silva Fernandes
Faculty of Sciences, University of Lisboa, Portugal
Email: fsilva@fc.ul.pt
Acknowledgement
This application was inspired by:
- "A Computer Problem in Statistical Mechanics", P.
Schettler, J. Chem. Education, 51 (1974) 250.
- "Thermodynamic
statistique chimique", O. Dessaux, P. Goudmand, F. Langrand, Dunod
Université, 2nd edition, Bordas, Paris, 1982.
to which my colleague,
Isabel Lampreia, called my attention many years ago. Since then, it
was included in the course contents on programming, computational
chemistry and statistical thermodynamics that I have taught at the
Department of Chemistry and Biochemistry, Faculty of Sciences,
University of Lisbon.