# AMaSiS 2021 - Abstract

**Kangabire, Alain**

*Stochastic simulation of continuous time random walks: asymptotic rate coefficients in diffusion-limited relaxations*

Coauthor: Matthew A. Grayson

Northwestern University, UK

In amorphous semiconductors, polymers, and composites, relaxation rates are known to relax according to a slower-than-exponential decay rate which can be explained through the microscopic theory of continuous time random walks (CTRW) [1]. However, simulations of such systems are currently restricted to analytical wait-time distribution functions, and it is necessary to develop a computational CTRW formalism that can handle more generic wait time distributions that represent real physical systems. Here, a reaction limited by standard diffusion is simulated stochastically following the wait-time distribution formalism from CTRW theory. A step-by-step simulation of the diffusive random walk reveals the fraction of surviving reactants P(t) as a function of time, and the time-dependent unimolecular reaction rate coefficient K(t). The accuracy of the simulation is confirmed by comparing to analytical expressions from the continuum limit and to the asymptotic solution from Fickian diffusion. A transient feature observed at the start of the reaction is shown to be related to the initial separation of the walkers from the reaction sites and can be used to calibrate this separation distance. Shot noise is shown to be the dominant noise source in the simulation, and its amplitude is calibrated. Within the uncertainty of the noise, the simulated reaction rate coefficient is within 1% of the known analytical value using 10^{7} walkers. The stochastic simulations presented here can be generalized to model anomalous diffusion-limited reactions in regimes where the governing wait-time distributions yield no analytical solution.

**Acknowledgments**: This work was supported by NSF DMR-1729016.

**References**

[1] E. W. Montroll and G. H. Weiss, Random Walks on Lattices II, *Journal of Mathematical Physics*, **6** (1965), 167-181.