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Introduction to Stochastic Processes Read chapter Read first chapter

2014 | OriginalPaper | Chapter

8. Derivation of the Langevin Equation

Author : Grigorios A. Pavliotis

Published in: Stochastic Processes and Applications

Publisher: Springer New York

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Abstract

In this chapter, we derive the Langevin equation from a simple mechanical model for a small system (which we will refer to as a Brownian particle) that is in contact with a thermal reservoir that is at thermodynamic equilibrium at time t = 0. The full dynamics, Brownian particle plus thermal reservoir, are assumed to be Hamiltonian. The derivation proceeds in three steps. First, we derive a closed stochastic integrodifferential equation for the dynamics of the Brownian particle, the generalized Langevin equation (GLE). In the second step, we approximate the GLE by a finite-dimensional Markovian equation in an extended phase space. Finally, we use singular perturbation theory for Markov processes to derive the Langevin equation, under the assumption of rapidly decorrelating noise. This derivation provides a partial justification for the use of stochastic differential equations, in particular the Langevin equation, in the modeling of physical systems.

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previous chapter Exit Problems for Diffusion Processes and Applications
next chapter Linear Response Theory for Diffusion Processes
Footnotes
1
A few basic facts about Gaussian measures in Hilbert spaces can be found in Sect. B.5 in the appendix.
 
2
For a functional of the form \(\mathcal{H}(\phi ) =\int _{\mathbb{R}}H(\phi,\partial _{x}\phi )\,dx,\) the functional derivative is given \(\textrm{by }\frac{\delta \mathcal{H}} {\delta \phi } = \frac{\partial H} {\partial \phi } - \frac{\partial } {\partial x} \frac{\partial H} {\partial (\partial _{x}\rho )}.\) We apply this definition to the functional (8.14) to obtain \(\frac{\delta \mathcal{H}} {\delta \pi } =\pi,\quad \frac{\delta \mathcal{H}} {\delta \varphi } = -\partial _{x}^{2}\varphi - q\partial _{x}\rho\).
 
3
Assuming, of course, that the heat bath is described by a wave equation, i.e., assuming that \(\mathcal{A}\) is the wave operator.
 
4
In fact, the autocorrelation function depends also on the operator \(\mathcal{A}\) in (8.17).
 
5
In fact, the last term in (8.38) should read \(\beta ^{-1}A_{s}: D_{z}, where A_{s} = \frac{1}{2}\left(A + A^{T}\right)\) denotes the symmetric part of A. However, since D_{z} is symmetric, we can write it in the form \(\beta^{-1} A : D_{z}\).
 
Literature
[3]
[4]
[38]
E. Cortes, B. J. West, and K. Lindenberg. On the generalized Langevin equation—classical and quantum-mechanical. Journal of Chemical Physics, 82(6):2708–2717, 1985.CrossRefMathSciNet
[47]
D. Dürr, S. Goldstein, and J. L. Lebowitz. A mechanical model of Brownian motion. Comm. Math. Phys., 78(4):507–530, 1980/81.
[48]
J.-P. Eckmann and M. Hairer. Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators. Comm. Math. Phys., 212(1): 105–164, 2000.CrossRefMATHMathSciNet
[50]
J-P. Eckmann, C-A. Pillet, and L. Rey-Bellet. Entropy production in nonlinear, thermally driven Hamiltonian systems. J. Statist. Phys., 95(1–2): 305–331, 1999.
[51]
J.-P. Eckmann, C.-A. Pillet, and L. Rey-Bellet. Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures. Comm. Math. Phys., 201(3):657–697, 1999.CrossRefMATHMathSciNet
[59]
[60]
G. W. Ford, M. Kac, and P. Mazur. Statistical mechanics of assemblies of coupled oscillators. J. Mathematical Phys., 6:504–515, 1965.CrossRefMATHMathSciNet
[71]
D. Givon, R. Kupferman, and A. M. Stuart. Extracting macroscopic dynamics: model problems and algorithms. Nonlinearity, 17(6):R55–R127, 2004.CrossRefMATHMathSciNet
[82]
P. Hanggi, P. Talkner, and M. Borkovec. Reaction-rate theory: fifty years after Kramers. Rev. Modern Phys., 62(2):251–341, 1990.CrossRefMathSciNet
[103]
[105]
V. Jakšić and C.-A. Pillet. Ergodic properties of classical dissipative systems. I. Acta Math., 181(2):245–282, 1998.
[123]
[124]
P. Krée and C. Soize. Markovianization of nonlinear oscillators with colored input. Rend. Sem. Mat. Univ. Politec. Torino, pages 135–150, 1982.
[127]
R. Kubo, M. Toda, and N. Hashitsume. Statistical physics. II, volume 31 of Springer Series in Solid-State Sciences. Springer-Verlag, Berlin, second edition, 1991. Nonequilibrium statistical mechanics.
[129]
[131]
[139]
J. T. Lewis and L. C. Thomas. How to make a heat bath. In Functional integration and its applications (Proc. Internat. Conf., London, 1974), pages 97–123. Clarendon Press, Oxford, 1975.
[141]
K. Lindenberg and B. J. West. The nonequilibrium statistical mechanics of open and closed systems. VCH Publishers Inc., New York, 1990.MATH
[161]
R.M. Mazo. Brownian motion, volume 112 of International Series of Monographs on Physics. Oxford University Press, New York, 2002.
[169]
H. Mori. A continued-fraction representation of the time-correlation functions. Progress of Theoretical Physics, 34(3):399–416, 1965.CrossRefMathSciNet
[179]
[187]
E Pollak. Theory of Activated Rate-processes—a new derivation of Kramers expression. Journal of Chemical Physics, 85(2):865–867, 1986.CrossRefMathSciNet
[188]
E. Pollak, H. Grabert, and P. Hanggi. Theory of activated rate processes for arbitrary frequency dependent friction: Solution of the turnover problem. The Journal of Chemical Physics, 91(7):4073–4087, 1989.CrossRef
[193]
M. Reed and B. Simon. Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. Academic Press, New York, 1975.
[201]
L. Rey-Bellet. Ergodic properties of markov processes. In S. Attal, A. Joye, and C.-A. Pillet, editors, Open Quantum Systems II, volume 1881 of Lecture Notes in Mathematics, pages 1–39. Springer, 2006.
[202]
L. Rey-Bellet. Open classical systems. In Open quantum systems. II, volume 1881 of Lecture Notes in Math., pages 41–78. Springer, Berlin, 2006.
[203]
L. Rey-Bellet and L. E. Thomas. Exponential convergence to non-equilibrium stationary states in classical statistical mechanics. Comm. Math. Phys., 225(2):305–329, 2002.CrossRefMATHMathSciNet
[220]
C. Soize. The Fokker-Planck equation for stochastic dynamical systems and its explicit steady state solutions, volume 17 of Series on Advances in Mathematics for Applied Sciences. World Scientific Publishing Co. Inc., River Edge, NJ, 1994.
[223]
H. Spohn. Large Scale Dynamics of Interacting Particle Systems. Springer, Heidelberg, 1991.CrossRef
[245]
R. Zwanzig. Nonlinear generalized Langevin equations. J. Stat. Phys., 9(3):215–220, 1973.CrossRef
[246]
R. Zwanzig. Nonequilibrium statistical mechanics. Oxford University Press, New York, 2001.MATH
Metadata
Title
Derivation of the Langevin Equation
Author
Grigorios A. Pavliotis
Copyright Year
2014
Publisher
Springer New York
Book
Stochastic Processes and Applications

Print ISBN: 978-1-4939-1322-0

Electronic ISBN: 978-1-4939-1323-7

Copyright Year: 2014

DOI
https://doi.org/10.1007/978-1-4939-1323-7_8