Langevin equation time evolution

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... 

The specific form of the short-time transition probability depends on the type of dynamics one uses to describe the time evolution of the system. For instance, consider a single, one-dimensional particle with mass m evolving in an external potential energy V(q) according to a Langevin equation in the high-friction limit... 

Third, Eq. (31) shows that A is nondistributive, and determines fluctuations. Since there is a flucmation, we can expect that the time evolution in Eq. (34) may be related to a stochastic process. Indeed, one can show that the time evolution (34) is identical to the time evolution generated by the set of Langevin equations for the stochastic operators aj(r), a (r) (see Ref. 14) ... 

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 ... 

As discussed in Section 8.2.1, the Langevin equation (8.13) describes a Markovian stochastic process The evolution of the stochastic system variable x(Z) is determined by the state of the system and the bath at the same time t. The instantaneous response of the bath is expressed by the appearance of a constant damping coefficient y and by the white-noise character of the random force 7 (Z). 

We have already noted the difference between the Langevin description of stochastic processes in terms of the stochastic variables, and the master or Fokker-Planck equations that focus on their probabilities. Still, these descriptions are equivalent to each other when applied to the same process and variables. It should be possible to extract information on the dynamics of stochastic variables from the time evolution of their probabihty distribution, for example, the Fokker-Planck equation. Here we show that this is indeed so by addressing the passage time distribution associated with a given stochastic process. In particular we will see (problem 14.3) that the first moment of this distribution, the mean first passage time, is very useful for calculating rates. 

Let x t) be the temporal variable of the excitable system which is also subject to external noise. A corresponding differential equation for the time evolution of x t) is called a Langevin-equation and includes random parts. We specify to the situation where randonmess is added linearly to modify the time derivative of x t), i.e. 

The procedure for constructing kinetic equations using the generalized Langevin equation is well known - one uses as variables in this description the p/ioje-space density fields. We could of course simply use the solute phase-space fields, (7.1), and follow the methods of Section V to obtain a formal kinetic equation for their time evolution. This procedure... 

No calculations yet have tackled this problem in all its complexity. Typically, it is assumed that the photodissociation produces some initial distribution of pairs, and then the subsequent time evolution of the unreacted pair probability is calculated. Even this more modest program has been carried out only at the diffusion and Langevin equation levels. We briefly comment on these results, since they indicate the magnitude of the solvent and velocity relaxation effects. 

The time evolution of beads at position is calculated by the Langevin equation as follows ... 

Approximate time evolution equation for the system of interest that are used in molecular dynamics simulations can now be obtained. If the total Liouvillian is replaced by in Eq. (36) and use is made of Eqs. (29) and (30), a generalized Langevin equation follows ... 

If the evolution of the system over times At j>l/ is of interest (thermalization of velocities in each time step— high friction limit ), the friction term is dominant and the inertia term on the left hand side of the Langevin equation can be neglected. Then the equation becomes first order ... 

In this method, one considers that the interactions of the proton transfer chain with the rest of the protein and the solvent generate friction and random forces. These processes are characterized by phenomenological parameters evaluated by generating molecular dynamics trajectories, which are also used to build the PMF. The PMF and the phenomenological parameters are then introduced into the Langevin equation to simulate the time evolution of the protonation state of the chain." A transit time can be defined and compared with the experimental data. ... 

Needless to say, an FP equation can incorporate variables other than a Langevin velocity for example, Kramers equation is a bivariate FP which involves both spatial coordinates and velocity. In a slightly more abstract sense, one might consider the time evolution of a probability density P(y, t) where, for simplicity, y is taken to be a continuous scalar on the real line. It turns out that a "legitimate FP equation can be written as (Fox, 1978 Van Kampen, 1981)... 

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