Langevin equation stochastic difference

Stochastic dynamics The stochastic dynamics (SD) method is a further extension of the original molecular dynamics method. A space-time trajectory of a molecular system is generated by integration of the stochastic Langevin equation which differs from the simple molecular dynamics equation by the addition of a stochastic force R and a frictional force proportional to a friction coefficient g. The SD approach is useful for the description of slow processes such as diffusion, the simulation of electrolyte solutions, and various solvent effects. 

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 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 ... 

It is important to emphasize, however, that our model is different from the Langevin equation, which is a stochastic differential equation. Our model has no noise in the limit of small time steps in which the numerical errors approach zero. The noise we introduce is numerical. Once we filter the rapid oscillations, it is impossible for us to recover the tme trajectory using only the low-frequency modes. The noise in the SDE approach is introduced when we approximate a differential equation by a finite difference formula and filter out high-frequency motions. 

There are two forms of phenomenological equations for describing Brownian motion the Smoluchowski equation and the Langevin equation. These two equations, essentially the same, look very different in form. The Smoluchowski equation is derived from the generalization of the diffusion equation and has a clear relation to the thermodynamics of irreversible processes. In Chapters 6 and 7, its application to the elastic dumbbell model and the Rouse model to obtain the rheological constitutive equations will be discussed. In contrast, the Langevin equation, while having no direct relation to thermodynamics, can be applied to wider classes of stochastic processes. In this chapter, it will be used to obtain the time-correlation function of the end-to-end vector of a Rouse chain. 

There are many problems that would require so much computer time that their study by the previous method would not be possible. For example polyelectrolyte solutions, or motions of particles in membranes would not be susceptible to study because of wide separations in the time scales for different dynamic processes characterizing solute and solvent or because the property of interest evolves so slowly that an excessively long trajectory would be required. The study of these systems requires a different approach. A beginning was made many years ago by Simon,who studied the melting of DNA by solving the coupled set of stochastic Langevin equations on a computer. This required an assumption about the statistical distribution of random forces. The precise values of the forces were then sampled from this distribution. 

The relaxational part of the scattering law in a symmetric double well potentials is calculated assuming that the anharmonic interaction between different modes acts like stochastic forces on a given mode, giving rise to a Langevin equation with a friction term The scattering law Sfeiax(Q>w) due to the relaxa-... 

Different approaches have been proposed to justify a Butler-Volmer behavior. In 1986, Gurevich and Kharkats proposed a stochastic approach to ion-transfer reactions [116-118], Here, the main equation is the Langevin equation ... 

Figure 18.1 Regimes of the problem space for multiscale stochastic simulations of chemical reaction kinetics. The r-axis represents the number of molecules of reacting species, x, and the ) -axis measures the frequency of reaction events, A. The threshold variables demarcate the partitions of modeling formalisms. In area I, the number of molecules is so small and the reaction events are so infrequent that a discrete-stochastic simulation algorithm, like the SSA, is needed. In contrast, in area V, which extends to infinity, the thermodynamic limit assumption becomes vahd and a continuous-deterministic modehng formalism becomes valid. Other areas admit different modehng formalisms, such as ones based on chemical Langevin equations, or probabilistic steady-state assumptions.

The stochastic model of ion transport in liquids emphasizes the role of fast-fluctuating forces arising from short (compared to the ion transition time), random interactions with many neighboring particles. Langevin s analysis of this model was reviewed by Buck [126] with a focus on aspects important for macroscopic transport theories, namely those based on the Nernst-Planck equation. However, from a microscopic point of view, application of the Fokker-Planck equation is more fruitful [127]. In particular, only the latter equation can account for local friction anisotropy in the interfacial region, and thereby provide a better understanding of the difference between the solution and interfacial ion transport. 

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. 

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. 

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