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Non-equilibrium Thermodynamics and Statistical MechanicsFoundations and Applications$
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Phil Attard

Print publication date: 2012

Print ISBN-13: 9780199662760

Published to Oxford Scholarship Online: January 2013

DOI: 10.1093/acprof:oso/9780199662760.001.0001

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Brownian Motion

Brownian Motion

Chapter:
(p.61) Chapter 3 Brownian Motion
Source:
Non-equilibrium Thermodynamics and Statistical Mechanics
Author(s):

Phil Attard

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199662760.003.0003

Brownian motion is presented as a stochastic process, and as a bridge between thermodynamics and statistical mechanics. Einstein’s result for the linear growth in time of the mean square displacement and the consequent diffusion equation are given. Langevin’s equation and the molecular statistical basis of the fluctuation dissipation theorem are established from the relation between a stochastic process and the second entropy. The non-equilibrium probability distribution for a Brownian particle in a moving trap is derived in two complementary ways: from fluctuation theory and from fundamental statistical and physical considerations. Computer simulation results are used to illustrate the veracity of the fluctuation formulation. The time evolution of the entropy and of the probability is established in general from the second entropy. The Fokker-Planck equation and Liouville’s theorem are derived and critically analysed. A generalised non-equilibrium equipartition theorem is given.

Keywords:   Brownian motion, stochastic process, Langevin, fluctuation dissipation, Fokker-Planck, Liouville, equipartition, non-equilibrium probability

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