Coarse -graining dynamics of interacting particle systems
This thesis is concerned with coarse-graining dynamics of interacting particle systems. We study two different coarse-graining approximations of the microscopic process. In the first part of the thesis we study coarse-graining schemes for stochastic many body microscopic models focusing on the dynamics for the spin adsorption/desorption mechanism. We show that these coarse-graining schemes which are derived using cluster expansion are able to describe complex phenomena. We also study the role of multi-body interactions in coarse-graining schemes for achieving higher order accuracy. The multi-body interactions are often not included in coarse-graining schemes as they can be computational expensive. On the other hand the numerical experiments show that the inclusion of multi-body interactions is critical in accurately reproducing dynamical properties such as rare events and switching times. Here we propose strategies to compress multi-body interactions within a specified error tolerance, making such corrections computationally feasible.
In the second part of the thesis we present an alternate way of modeling the noise in the CGMC/MC simulations for the diffusion mechanisms. Computational efficiency has been a motivation for Langevin approximations of the microscopic process in many fields. The Langevin approximation we derive is based upon the coarse-grained Markov process, so it automatically inherits the coarse-graining. We show that the long time behavior of the Langevin approximation is asymptotically equivalent to the long time behavior of the microscopic process. We establish this connection using a calculus of variations perspective and large deviations principles. We derive a time dependent action functional for the Langevin approximation using a Taylor series expansion of the drift and diffusion coefficients. We show that, using Γ-convergence arguments, asymptotically this action functional is equivalent to the action functional of the mesoscopic limit of the microscopic diffusion process. In the end, we show that the CGSDE model is capable of further enhancing the CPU savings achieved by CGMC and also that it is independent of the temperature and the interaction potential radius and other parameters of the problem.