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Brownian motion; fluctuation dissipation theorem; ideal gas

We present an insightful ‘derivation’ of the Langevin equation and the fluctuation dissipation theorem in the specific context of a heavier particle moving through an ideal gas of much lighter particles. The Newton’s law of motion (mx = F) for the heavy particle reduces to a Langevin equation (valid on a coarser time-scale) with the assumption that the lighter gas particles follow a Boltzmann velocity distribution. Starting from the kinematics of the random collisions we show that (1) the average force 〈F〉 ∞ −x and (2) the correlation function of the fluctuating forceη = F — 〈F〉 is related to the strength of the average force.

Rangan Lahiri¹ Arvind^{2 3} Anirban Sain^{4 5}

1. Department of Physics, Indian Institute of Science, Bangalore - 560 012, India
2. Department of Physics, Carnegie Mellon University, Pittsburgh, PA - 15213, USA
3. Department of Physics, Guru Nanak Dev University, Amritsar - 143 005, India
4. Department of Physics, University of Waterloo, Ontario, Canada - N2L3G1, Canada
5. Department of Physics, Indian Institute of Technology, Powai, Mumbai - 400 076, India

Published

May 2004

Manuscript received

3 February 2003

Manuscript revised

12 January 2004

Accepted

17 January 2004