Title:

Coarsening dynamical systems : dynamic scaling, universality and meanfield theories

We study three distinct coarsening dynamical systems (CDS) and probe the underlying scaling laws and universal scaling functions. We employ a variety of computational methods to discover and analyse these intrinsic statistical objects. We consider meanfield type models, similar in nature to those used in the seminal work of Lifshitz, Slyozov and Wagner (LSW theory), and statistical information is then derived from these models. We first consider a simple particle model where each particle possesses a continuous positive parameter, called mass, which itself determines the particle’s velocity through a prescribed law of motion. The varying speeds of particles, caused by their differing masses, causes collisions to take place, in which the colliding particles then merge into a single particle while conserving mass. We computationally discover the presence of scaling laws of the characteristic scale (mean mass) and universal scaling functions for the distribution of particle mass for a family of powerlaw motion rules. We show that in the limit as the powerlaw exponent approaches infinity, this family of models approaches a probabilistic mindriven model. This mindriven model is then analysed through a meanfield type model, which yields a prediction of the universal scaling function. We also consider the conserved KuramotoSivashinsky (CKS) equation and provide, in particular, a critique of the effective dynamics derived by Politi and benAvraham. We consider several different numerical methods for solving the CKS equation, both on fixed and adaptive grids, before settling on an implicitexplicit hybrid scheme. We then show, through a series of detailed numerical simulations of both the CKS equation and the proposed dynamics, that their particular reduction to a lengthbased CDS does not capture the effective dynamics of the CKS equation. Finally, we consider a faceted CDS derived from a onedimensional geometric partial differential equation. Unusually, an obvious onepoint meanfield theory for this CDS is not present. As a result, we consider the twopoint distribution of facet lengths. We derive a meanfield evolution equation governing the twopoint distribution, which serves as a twodimensional generalisation of the LSW theory. Through consideration of the twopoint theory, we subsequently derive a nontrivial onepoint submodel which we analytically solve. Our predicted onepoint distribution bears a significant resemblance to the LSW distribution and stands in reasonable agreement with the underlying faceted CDS.
