<Langevin dynamics> describes a dynamical variable driven by deterministic forces and random noise. <Underdamped Langevin dynamics> retains inertia, whereas <Overdamped Langevin dynamics> neglects it. At <thermal equilibrium>, the <fluctuation-dissipation relation for a Langevin particle> relates <Gaussian white noise> strength to drag and <temperature>.

Langevin dynamics

Write $E(t)=\delta A/\delta x(t)$ for the <functional derivative> of the <action>. Under <time reversal in classical mechanics>, $\dot x$ changes sign whereas $E$ is even by assumption. Thus the forward and reversed <Gaussian white noise> histories associated with the same geometric path are
$$
f_F=\zeta\dot x-E,\qquad f_B=-\zeta\dot x-E,
\qquad f_F^2-f_B^2=-4\zeta\dot xE.
$$
The <Onsager--Machlup path probability> therefore gives
$$
\log\frac{\mathbb P_F}{\mathbb P_B}
=-\frac1{2\sigma^2}\int_{t_1}^{t_2}(f_F^2-f_B^2)\,dt,
\qquad
\boxed{\frac{\mathbb P_F}{\mathbb P_B}
=\exp\!\left[\frac{2\zeta}{\sigma^2}\int_{t_1}^{t_2}\dot x\frac{\delta A}{\delta x(t)}\,dt\right]}.
$$
Here the <path probabilities> are densities conditional on their respective initial states; they are not separately normalized bridges conditioned on both endpoints. The noise-to-path <Jacobian determinant> must be treated with a consistent discretization: it cancels when it is invariant under <time reversal in classical mechanics>. This is the usual additive-noise <Langevin dynamics> convention, including a constant-mass <Underdamped Langevin dynamics> or the <time-reversal invariance of a path Jacobian> in midpoint <Overdamped Langevin dynamics>. A completely arbitrary velocity-dependent <Lagrangian> would require specifying that measure rather than deducing its cancellation solely from the parity of $E$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 344 / 1 / a / Solution

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 Incoming links: Langevin dynamics