Langevin equation stochastic dynamics

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

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. 

The friction coefficient determines the strength of the viscous drag felt by atoms as they move through the medium its magnitude is related to the diffusion coefficient, D, through the relation Y= kgT/mD. Because the value of y is related to the rate of decay of velocity correlations in the medium, its numerical value determines the relative importance of the systematic dynamic and stochastic elements of the Langevin equation. At low values of the friction coefficient, the dynamical aspects dominate and Newtonian mechanics is recovered as y —> 0. At high values of y, the random collisions dominate and the motion is diffusion-like. 

The basic equation of motion for stochastic dynamics is the Langevin equation. 

The concept of the TS trajectory was first introduced [37] in the context of stochastically driven dynamics described by the Langevin equation of motion... 

Constraints may be introduced either into the classical mechanical equations of motion (i.e., Newton s or Hamilton s equations, or the corresponding inertial Langevin equations), which attempt to resolve the ballistic motion observed over short time scales, or into a theory of Brownian motion, which describes only the diffusive motion observed over longer time scales. We focus here on the latter case, in which constraints are introduced directly into the theory of Brownian motion, as described by either a diffusion equation or an inertialess stochastic differential equation. Although the analysis given here is phrased in quite general terms, it is motivated primarily by the use of constrained mechanical models to describe the dynamics of polymers in solution, for which the slowest internal motions are accurately described by a purely diffusive dynamical model. 

Keywords stochastic dynamics, generalized Langevin equation, nonstationary and colored friction... 

If Xe is somewhat larger, then there may arise an effective time scale Xr > Xe, with 5, < Xr sueh that the environment has some memory of the particle s previous history and therefore responds accordingly. This is the regime of the generalized Langevin equation (GLE) with colored friction. - In all these cases, the environment is sufficiently large that the particle is unable to affect the environment s equilibrium properties. Likewise, the environment is noninteracting with the rest of the universe such that its properties are independent of the absolute time. All of these systems, therefore, describe the dynamics of a stochastic particle in a stationary —albeit possibly colored— environment. 

Use of this v (t) as the friction (t) in the generalized Langevin equation provides a complete specification of a nonlocal stationary stochastic dynamics with the exponential friction jo. 

In this section, we discuss briefly how the Langevin equation, which is a stochastic equation, can be derived from the molecular equations of motion. The stochastic model described by the Langevin equation has been of great use in interpreting a large number of experiments and physical systems. The stochastic model is extremely simple but, as always, its ultimate justification rests on the molecular dynamical laws. 

In order to compute Eq. (158), write the Langevin equations governing the dynamics of the Brownian oscillator. In the present situation that leads us to consider three time-dependent stochastic variables S(t), Q(f), and v(t), described by the following three equations ... 

In all these experimental studies, the particle was in a viscous fluid and therefore the equations of motion of the particle were well approximated by a stochastic Langevin equation. In 2007, a capture experiment was carried out in a viscoelastic solvent where this approximation no longer applies. It was shown that despite this, the experiments validated the ES FR, and therefore could not be consider just a special property of Brownian dynamics. Blickle et a/. verified the fluctuation relation for the work (or dissipation function) for a system where the trap potential was not harmonic. 

We are interested in the case s -C 1 because if s is large, then the dynamics tends to become more and more irregular and eventually well-approximated by stochastic dynamics (such as the Langevin equation) and a statistical description will be valid. 

In order to understand how the algorithm actually works and to construct an explicit expression for the error it is not convenient to work with the metadynamics equations (12) in their full generality. Instead, we notice that the finite temperature dynamics of the collective variables satisfies, under rather general conditions, a stochastic differential equation [54,55]. Furthermore, in real systems the quantitative behavior of metadynamics is perfectly reproduced by the Langevin equation in its strong friction limit [56]. This is due to the fact that all the relaxation times are usually much smaller than the typical diffusion time in the CV space. Hence, we model the CVs evolution with a Langevin t3rpe dynamics ... 

The continuous limit of a simple random walk model leads to a stochastic dynamic equation, first discussed in physics in the context of diffusion by Paul Langevin. The random force in the Langevin equation [44], for a simple dichotomous process with memory, leads to a diffusion variable that scales in time and has a Gaussian probability density. A long-time memory in such a random force is shown to produce a non-Gaussian probability density for the system response, but one that still scales. 

---