Abstract
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.
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References
M A Bevan and D C Prieve,J. Chem. Phys. 113, 1228 (2000)
J Meiners and S R Quake,Phys. Rev. Lett. 84, 5014 (2000)
D S Sholl, M Fenwick, E S Atman and D C Prieve,J. Chem. Phys. 113, 9268 (2000)
L D Landau and E M Lifshitz,Fluid mechanics, 2nd edition (Pergamon Press, Oxford, 1987)
M Doi and S F Edwards,The theory of polymer dynamics (Oxford University Press, 1986) Ch. 3, Sec. 6
A Einstein,Ann. Phys. (Leipzig) 17, 549 (1905); appearing inThe collected papers of Albert Einstein, English translation by Anna Beck (Princeton University Press, 1989) vol. 2, p. 123
P Langevin,Comptes. Rendues 146, 530 (1908); for English translation, see Don S Lemons and Anthony Gythiel, Paul Langevin’s 1908 paper ‘On the Theory of Brownian Motion’,Am. J. Phys. 65, 1079 (1997)
S Chandrasekhar,Rev. Mod. Phys. 15, 1 (1943)
P MChaikin and TLubensky,Principles of condensed matter physics (Cambridge University Press, Cambridge, 1995)
P Hanngi, P Talkner and M Borokovek,Rev. Mod. Phys. 62, 251 (1990)
J Krug and H Spohn, Kinetic roughening of growing surfaces, inSolids far from equilibrium edited by C Godreche (Cambridge University Press, 1992) pp. 479–582
H Grabert,Projection operator techniques in nonequilibrium statistical mechanics (Springer, Berlin, 1982)
R Kubo,Rep. Prog. Phys. 29, 255 (1966)
B U Felderhof,J. Phys. A: Math. Gen. 11, 921 (1978)
D T Gillespie,Am. J. Phys. 61, 1077 (1993);64, 225 (1996)
D T Gillespie,Markov processes: an introduction for physical scientists (Academic Press, Boston, (1992)
N G van Kampen,Stochastic processes in physics and chemistry (North Holland, New York, 1992)
B G De Grooth,Am. J. Phys. 67, 1248 (1999)
K Huang,Statistical mechanics (Wiley, New York, 1963)
Carlo Cercignani,The Boltzmann equation and its applications (Springer-Verlag, New York, 1987).
J Zinn-Justin,Quantum field theory and critical phenomena (Clarendon Press, Oxford, 1989) Ch. 3
For a discussion, see [18], p. 43
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Lahiri, R., Arvind & Sain, A. Brownian motion in a classical ideal gas: A microscopic approach to Langevin’s equation. Pramana - J Phys 62, 1015–1028 (2004). https://doi.org/10.1007/BF02705249
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DOI: https://doi.org/10.1007/BF02705249