Discrete variable representation of the Smoluchowski equation using a sinc basis set

Abstract

We present a new general framework for solving the monodimensional Smoluchowski equation using a discrete variable representation (DVR) based on the so called sinc basis set. The reliability of our implementation is assessed by comparing the convergence of diffusive operator eigenvalues calculated using our method and using a simple finite difference scheme for some model diffusive problems. The results here presented open encouraging possibilities for dealing with more complicated systems, where additional coordinate dependent terms in the equation or multidimensional treatments are needed and traditional methods often become unfeasible.

General Approach to Coupled Reactive Smoluchowski Equations: Integration and Application of Discrete Variable Representation and Generalized Coordinate Methods to Diffusive Problems

“…Here we have shown the limit in which both diffusion and reaction take place, that is one of the most problematic case where space-dependent sink term is present. The reliability of the method in the case of free diffusion and/or no reactivity has been shown in our previous work, ref.,16 guaranteeing the performance also in other simpler limits.…”

“…We have merged in a modular code all our previous developments on the solution of the one-dimensional Smoluchowski equation and molecular diffusion field. These are respectively, the robust numerical approach rooted in the discrete variable representation (DVR) in order to solve the equation16 and the variable diffusion tensor along a one-dimensional generalized coordinate considered as a diffusive process 17…”

Toward a General Yet Effective Computational Approach for Diffusive Problems: Variable Diffusion Tensor and DVR Solution of the Smoluchowski Equation along a General One-Dimensional Coordinate

“…Our goal is to determine D GG ( s ) from the integrated calculation of the potential scan along the generalized coordinate with the calculation of the diffusion tensor along the same path. In some cases we shall use these two pieces of information to set up and solve a one-dimensional Smoluchowski equation for the generalized coordinate treated as a diffusive one and making use of the novel DVR solving approach 15. The interested reader can find all the details about the DVR implementation in that reference; here we just briefly recall the one-dimensional Smoluchowski equation that reads33–35 ∂∂tp(x,t)=−Γp(x,t) where p ( x, t ) is the probability density of finding a value of x at time t , with stationary limit lim t→+∞ p(x,t) = p eq (x) ; in our case x could be the generalized coordinate s or the internal coordinate scanned with DiTe.…”

“…As stated before, they are essential ingredients to set up and solve the Smoluchowski equation for the generalized coordinate treated as a stationary diffusive process. Similarly to what we have done in ref 15. and in the past works of Moro and Nordio, 16,42 we use the two first lowest and non zero eigenvalues and the equilibrium populations in order to calculate the kinetic constants for the transitions between the three stable conformations (eqn (27) in Appendix B).…”

“…Although the specific applications presented in the following employ rigid or relaxed scans, the implementation permits also the use of IRC or PCA5 paths. The final goal is to build a reliable framework for setting up and solving the one-dimensional Smoluchowski equation along any generalized coordinate, making use of a quite powerful solution method based on the so called Discrete Variable Representation (DVR) 15. Once both the main pieces of information (dissipation and energetics) are given and after the diffusive problem solved, some specific quantities ( e.g.…”