Langevin stochastic equation

Wolf and Villain (WV) [8], Villain, Lai, and Das Sanna (VLD) [9] and others [10]. The rate equations for the film growth resulting from these models are modified versions of the Langevin stochastic equation [10] for the variation with time of the interface height. 

Depending on the desired level of accuracy, the equation of motion to be numerically solved may be the classical equation of motion (Newton s), a stochastic equation of motion (Langevin s), a Brownian equation of motion, or even a combination of quantum and classical mechanics (QM/MM, see Chapter 11). 

The set of equations (50) can be formally considered as generalized Langevin equations if the operator Fj(t) can be interpreted as a stochastic quantity in the statistical mechanical sense. If the memory function does not correlate different solute modes, namely, if Kjj =Sjj Kj, then a Langevin-type equation follows for each mode ... 

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 a stochastic approach, one replaces the difficult mechanical equations by stochastic equations, such as a diffusion equation, Langevin equation, master equation, or Fokker-Planck equations.5 These stochastic equations have fewer variables and are generally much easier to solve than the mechanical equations, One then hopes that the stochastic equations include the significant aspects of the physical equations of motion, so that their solutions will display the relevant features of the physical motion. 

The autocorrelation function k is not a delta function, and (t) is some well-defined, non-singular stochastic function, so that (7.1) is also well-defined. Such Langevin-like equations have been amply discussed in the literature. The fact... 

The theory of Brownian motion is a particular example of an application of the general theory of random or stochastic processes [2]. Since Kramers approach is based on a more general stochastic equation than the Langevin equation, we have reviewed some of the fundamental ideas and methods of the theory of stochastic processes in Appendix H. 

Here F°(f) is the Langevin stochastic force, the average and the autocorrelation of which are given by the equations ... 

Next, Fig. 5 shows a typical center of mass trajectory for the case of a full chaotic internal phase space. The eyecatching new feature is that the motion is no more restricted to some bounded volume of phase space. The trajectory of the CM motion of the hydrogen atom in the plane perpendicular to the magnetic field now closely resembles the random motion of a Brownian particle. In fact, the underlying equation of motion at Eq. (35) for the CM motion is a Langevin-type equation without friction. The corresponding stochastic Lan-gevin force is replaced by our intrinsic chaotic force — e B x r). A main characteristic of random Brownian motion is the diffusion law, i.e. the linear dependence of the travelled mean-square distance on time. We have plotted in Fig. 6 for our case of a chaotic force for 500 CM trajectories the mean-square distance as a function of time. Within statistical accuracy the plot shows a linear dependence. The mean square distance

of the CM after time t, therefore, obeys the diffusion equation... 

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. 

The problem can also be approached by considering the Langevin equation rather than the Fokker-Planck equation.In the Langevin description, the motion of the particle is given by the stochastic equations... 

Thus far, we have described the time-dependent nature of polymerizing environments both through stochastic [49-51] and lattice [52,53] models capable of addressing this kind of dynamics in a complex environment. The current article focuses on the former approach, but now rephrases the earlier justification of the use of the irreversible Langevin equation, iGLE, to the polymerization problem in the context of kinetic models, and specifically the chemical stochastic equation. The nonstationarity in the solvent response due to the collective polymerization of the dense solvent now appears naturally. This leads to a clear recipe for the construction of the requisite terms in the iGLE. Namely the potential of mean force and the friction kernel as described in Section 3. With these tools in hand, the iGLE is used... 

For stochastic pathways, such as those generated according to Langevin s equation of motion, the path probability functional can be... 

The remainder of this section is devoted to the derivation of Eq.[54]. Besides the mathematics we also define the range of applicability of simulations based on the Nernst-Planck equation. The starting point for deriving the Nernst-Planck equation is Langevin s equation (Eq. [45]). A solution of this stochastic differential equation can be obtained by finding the probability that the solution in phase space is r, v at time t, starting from an initial condition ro, Vo at time = 0. This probability is described by the probability density function p r, v, t). The basic idea is to find the phase-space probability density function that is a solution to the appropriate partial differential equation, rather than to track the individual Brownian trajectories in phase space. This last point is important, because it defines the difference between particle-based and flux-based simulation strategies. 

In Eq. [57], At is a time interval chosen to satisfy two criteria (1) During At the position and velocity do not change appreciably, and (2) the stochastic term in Langevin s equation must undergo many fluctuations. Equation [57] states the Markovian nature of p r,v,t). 

The LB method and its improved versions are widely used for the effident treatment of polymer solution dynamics. In the application to polymer solution dynamics, the polymer itself is still treated on a partide-based CG level using, for example, a bead-spring model, while the solvent is treated on the level of a discretized Boltzmann equation. The two parts are coupled by a simple dissipative point-partide force, and the system is driven by Langevin stochastic forces added to both the fluid and the polymers. In this approach, the hydrodynamics of the low-molecular-weight solvent is correctly captured, and the HI between polymer segments, which is mediated by the hydrodynamic flow generated within the solvent through the motion of the polymer, is present in the simulation without explidt... 

Abstract We numerically study the pair-annihilation process of magnetic vortices in two-dimensional type-II superconductors. The dynamics for interacting vortices is described in terms of the Langevin-type stochastic equation. Carrying out the Langevin dynamics simulation, we find that the power-law behavior of the mean distance among vortices exhibits a crossover phenomenon as... 

One is the time-dependent Ginzburg-Landau equation which is described by a complex order parameter and vector potential. The other is the Langevin-type stochastic equation of motion for magnetic vortices in two and three dimensions, which is described in terms of vortex position variables. 

The relation between the Langevin equations (3.49) [assuming -correlated gaussian random forces obeying (3.59, 61)] and the stochastic equation (3.29) has now to be established. The constitutive quantities of the latter equation are the moments, see (3.23, 28),... 

At finite temperature, stochastic fluctuations of the membrane due to thermal motion affect the dynamics of vesicles. Since the calculation of thermal fluctuations under flow conditions requires long times and large membrane sizes (in order to have a sufficient range of undulation wave vectors), simulations have been performed for a two-dimensional system in the stationary tank-treading state [213]. For comparison, in the limit of small deviations from a circle, Langevin-type equations of motion have been derived, which are highly nonlinear due to the constraint of constant perimeter length [213]. 

Let us first explain what we mean here for generalized Langevin equation (GLE). For this, we summarize rather well-known concepts of the theory of Brownian motion and thermal fluctuations cast as stochastic processes, generated by linear stochastic equations with additive noise. 

Hamiltonian, but in practice one often begins with a phenomenological set of equations. The set of macrovariables are chosen to include the order parameter and all otlier slow variables to which it couples. Such slow variables are typically obtained from the consideration of the conservation laws and broken synnnetries of the system. The remaining degrees of freedom are assumed to vary on a much faster timescale and enter the phenomenological description as random themial noise. The resulting coupled nonlinear stochastic differential equations for such a chosen relevant set of macrovariables are collectively referred to as the Langevin field theory description. 

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 two sources of stochasticity are conceptually and computationally quite distinct. In (A) we do not know the exact equations of motion and we solve instead phenomenological equations. There is no systematic way in which we can approach the exact equations of motion. For example, rarely in the Langevin approach the friction and the random force are extracted from a microscopic model. This makes it necessary to use a rather arbitrary selection of parameters, such as the amplitude of the random force or the friction coefficient. On the other hand, the equations in (B) are based on atomic information and it is the solution that is approximate. For ejcample, to compute a trajectory we make the ad-hoc assumption of a Gaussian distribution of numerical errors. In the present article we also argue that because of practical reasons it is not possible to ignore the numerical errors, even in approach (A). 

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