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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Notes
- 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 \(\beta^{-1} A : D_{z}\).
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Pavliotis, G.A. (2014). Derivation of the Langevin Equation. In: Stochastic Processes and Applications. Texts in Applied Mathematics, vol 60. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1323-7_8
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