Coarsening dynamical systems: dynamic scaling, universality and mean-field theories
Chapman, Craig K.
Coarsening dynamical systems: dynamic scaling, universality and mean-field theories.
PhD thesis, University of Glasgow.
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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 mean-field 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 power-law motion rules. We show that in the limit as the power-law exponent approaches infinity, this family of models approaches a probabilistic min-driven model. This min-driven model is then analysed through a mean-field type model, which yields a prediction of the universal scaling function.
We also consider the conserved Kuramoto-Sivashinsky (CKS) equation and provide, in particular, a critique of the effective dynamics derived by Politi and ben-Avraham. We consider several different numerical methods for solving the CKS equation, both on fixed and adaptive grids, before settling on an implicit-explicit 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 length-based CDS does not capture the effective dynamics of the CKS equation.
Finally, we consider a faceted CDS derived from a one-dimensional geometric partial differential equation. Unusually, an obvious one-point mean-field theory for this CDS is not present. As a result, we consider the two-point distribution of facet lengths. We derive a mean-field evolution equation governing the two-point distribution, which serves as a two-dimensional generalisation of the LSW theory. Through consideration of the two-point theory, we subsequently derive a non-trivial one-point sub-model which we analytically solve. Our predicted one-point distribution bears a significant resemblance to the LSW distribution and stands in reasonable agreement with the underlying faceted CDS.
||CDS, coarsening, coarsening dynamical system, LSW, LSW theory, CKS, conserved Kuramoto Sivashinsky, mean, field, mean field, mean-field, universality, dynamic, scaling, dynamic scaling, mean field theory, mean-field theory, mean field theories, mean-field theories, ballistic, ballistics, ballistic particle, ballistic particle model, paste, paste all, paste-all, paste all model, paste-all model, effective dynamics, one point, one-point, two point, two-point, aggregating, aggregation, evolution equation, hill, valley, hill-valley, fokker, planck, fokker-planck, fokker-planck equation, fokker planck equation
||Q Science > QA Mathematics
||College of Science and Engineering > School of Mathematics and Statistics > Mathematics
||Watson, Dr. Stephen J.
|Date of Award:
Mr Craig Chapman
||Copyright of this thesis is held by the author.
||08 Mar 2012
||10 Dec 2012 14:05
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