Langevin formulation

To make quantitative progress, we use a Langevin formulation. This amounts to an overdamped harmonic oscillator driven by a noise term (Kardar, 1994). The generic form is... 

We underline that the usage of stochastic methods in many particle physics was initiated by Albert Einstein in 1905 working on heavy particles immersed in liquids and which are thus permanently agitated by the molecules of the surrounding liquid. Whereas Einstein formulated an evolution law for the probability P(r, t) to And the particle in a certain position r at time t Paul Langevin formulated a stochastic equation of motion, i.e. a stochastic differential equation for the time dependent position r t) itself. 

Linearized Reference System We first formulate the following linearized Langevin system at some reference point (e.g., X", AT" + V") ... 

We further discuss how quantities typically measured in the experiment (such as a rate constant) can be computed with the new formalism. The computations are based on stochastic path integral formulation [6]. Two different sources for stochasticity are considered. The first (A) is randomness that is part of the mathematical modeling and is built into the differential equations of motion (e.g. the Langevin equation, or Brownian dynamics). The second (B) is the uncertainty in the approximate numerical solution of the exact equations of motion. 

Relations (2.46) and (2.47) are equivalent formulations of the fact that, in a dense medium, increase in frequency of collisions retards molecular reorientation. As this fact was established by Hubbard within Langevin phenomenology [30] it is compatible with any sort of molecule-neighbourhood interaction (binary or collective) that results in diffusion of angular momentum. In the gas phase it is related to weak collisions only. On the other hand, the perturbation theory derivation of the Hubbard relation shows that it is valid for dense media but only for collisions of arbitrary strength. Hence the Hubbard relation has a more general and universal character than that originally accredited to it. 

A non-linear, multi-variate Langevin model wherein the diffusion coefficients depend on 4 can be formulated for scalar mixing (Fox 1999). 

A completely unambiguous formulation of the Langevin equation may, however, be constructed by relating the random Langevin forces to random functions /m(f) of the type used in our discussion of the Stratonovich SDE, by taking... 

We now instead calculate the drift velocity and diffusivity by directly integrating the traditional formulation of the Langevin equation in terms of random forces, and compare the results to those obtained above by rewriting the Langevin equation as a standard Stratonovich SDE. As in the analysis of the Stratonovich SDE, we calculate the first and second moments of an increment AX (f) = Z (f) — X (0) by integrating Eq. (2.262) from a known initial condition at f = 0. 

The inertialess Langevin equation for the Cartesian bead positions of a constrained system may be formulated as a limit of a set of ODEs containing rapidly varying constraint forces, which must be chosen so as keep the instantaneous bead velocities always tangent to the constraint surface. We consider a Langevin equation... 

S.A. Adelman, J.D. Doll Generalized Langevin equation approach for atom/solid-surface scattering General formulation for classical scattering off harmonic solids. J. Chem. Phys. 64, 2375-2388 (1976)... 

Schlick and Olson recently developed such an algorithm that permits larger time steps. The implicit Euler integration scheme is combined with the Langevin dynamics formulation, which contains frictional and random... 

We begin with an abstract of the physics that underlies the kinetics of bond dissociation and structural transitions in a liquid environment. Developed from Einstein s theory of Brownian motion, these well-known concepts take advantage of the huge gap in time scale that separates rapid thermal impulses in liquids (< 10 s) from slow processes in laboratory measurements (e.g. from 10 s to min in the case of force probe tests). Three equivalent formulations describe molecular kinetics in an overdamped liquid environment. The first is a microscopic perspective where molecules behave as particles with instantaneous positions or states x(t) governed by an overdamped Langevin equation of motion,... 

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