Chapter

Generalised Langevin Equation

Phil Attard

in Non-equilibrium Thermodynamics and Statistical Mechanics

Published in print October 2012 | ISBN: 9780199662760
Published online January 2013 | e-ISBN: 9780191745287 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199662760.003.0010
Generalised Langevin Equation

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The generalised Langevin equation is derived in three ways: perturbation theory, linear response theory, and projector operator formalism. The perturbation theory derives the trajectory entropy for a driven Brownian particle, with the contribution from the external time-dependent force appearing explicitly. Maximisation gives the most likely trajectory, and adding the random force gives the generalised Langevin equation. The memory function is related explicitly to the velocity autocorrelation function of the bare particle, and this determines both the dissipative force and the covariance of the random force, which is the generalised fluctuation dissipation theorem. Quantitative tests of the theory are made against computer simulation results for a driven Brownian particle. The theory is extended to the mixed parity case, and the results are discussed in comparison to the projector operator approach.

Keywords: Langevin equation; stochastic; memory function; perturbation; linear response; projector operator

Chapter.  33366 words.  Illustrated.

Subjects: Condensed Matter Physics

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