Controlled-error Approximations for Surface Diffusion of Interacting Particles with Applications to Pattern Formation

Microscopic processes on surfaces such as adsorption, desorption, diffusion and reaction of interacting particles can be simulated using kinetic Monte Carlo (kMC) algorithms. Even though kMC methods are accurate, they are computationally expensive for large-scale systems. Hence approximation algorithms are necessary for simulating experimentally observed properties and morphologies. One such approximation method stems from the coarse graining of the lattice which leads to coarse-grained Monte Carlo (GCMC) methods while Langevin approximations can further accelerate the simulations. Moreover, sacrificing fine scale (i.e. microscopic) accuracy, mesoscopic deterministic or stochastic partial differential equations (SPDEs) are efficiently applied for simulating surface processes. In this paper, we are interested in simulating surface diffusion for pattern formation applications which is achieved by suitably discretizing the mesoscopic SPDE in space. The proposed discretization schemes which are actually Langevin-type approximation models are strongly connected with the properties of the underlying interacting particle system. In this direction, the key feature of our schemes is that controlled-error estimates are provided at three distinct time-scales. Indeed, (a) weak error analysis of mesoscopic observables, (b) asymptotic equivalence of action functionals and (c) satisfaction of detailed balance condition, control the error at finite times, long times and infinite times, respectively. In this sense, the proposed algorithms provide a “bridge” between continuum (S)PDE models and molecular simulations Numerical simulations, which also take advantage of acceleration ideas from (S)PDE numerical solutions, validate the theoretical findings and provide insights to the experimentally observed pattern formation through self-assembly. Such phenomena are characterized by a complex energy landscape where the role of noise is critical in the emergent behavior of the system. The stochastic fluctuations of the proposed algorithms are directly derived from the microscopic model allowing us to explore all experimentally observed pattern morphologies starting from a uniform initial state.