Langevin equation numerical solutions

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. 

Hynes et al. [298] and later Schell et al. [272] have developed a numerical simulation method for the recombination of iodine atoms in solution. The motions of iodine atoms was governed by a Langevin equation, though spatially dependent friction coefficients could be introduced to increase solvent structure. The force acting on iodine atoms was obtained from the mutual potential energy of interaction, represented by a Morse potential and the solvent static potential of mean force. The solvent and iodine atoms were regarded as hard spheres. The probability of reaction was calculated by following many trajectories until reaction had occurred or was most improbable. The importance of the potential of... 

In order to complete the above analysis, one needs to solve the full non-Markovian Langevin equation (NMLE) with the frequency-dependent friction for highly viscous liquids to obtain the rate. This requires extensive numerical solution because now the barrier crossing dynamics and the diffusion cannot be treated separately. However, one may still write phenomenologically the rate as [172],... 

The numerical solution of both the fractional Fokker-Planck equation in terms of the Griinwald-Letnikov scheme used to find a discretized approximation to the fractional Riesz operator exhibits reliable convergence, as corroborated by direct solution of the corresponding Langevin equation. 

The space- and time-dependent generalized Langevin equation (4) is a phenomenological equation. The exact numerical dynamics for any given simulation is not identical to the numerical solution of an STGLE. However, the numerical solution of the equations of motion of a system of hundreds or thousands of particles is in many senses a black box. One may get some numbers, but the dynamics is so complicated that there is very little useful additional information. On the other hand, because so much more is known about the solution of STGLEs it is very useful to try and map the complex dynamics onto an... 

Baneijee, D., B. C. Bag, S. K. Banik, and D. S. Ray. 2004. Solution of quantum Langevin equation Approximations, theoretical and numerical aspects. Journal of Chemical Physics 120(19) 8960-8972. 

Fig. 8. Numerical solution of the Langevin equation (eq. (18)) in the region of thermal explosion. Left part Individual realizations of the process of explosion featuring the considerable dispersion of ignition times. Right part Probability distribution of ignition times, illustrating further the random character of explosion...

This model has proved to be successful in the study of chemical reactions. As an example, it mimics properly the photolysis of iodine in xenon. Some of the properties of interfacial reactions, such as the recombination of H-H can also be reproduced. Even if the set of generalized Langevin equations used here is in limited number, it requires a numerical solution. 

Brownian dynamics is nothing but the numerical solution of the Smoluchowski equation. The method exploits the mathematical equivalence between a Fokker-Planck type of equation and the corresponding Langevin... 

With the proper choice of the operator F in (3), motional constraints in the form of an ordering potential can also he introduced into these discrete models. In a different model of a single particle undergoing rotational diffusion in the mean field of an ordering potential, the BD trajectories are obtained hy direct numerical solution of the Langevin equation ... 

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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