
Symmetry in Nonlinear Mathematical Physics  2009
Yuri Gordienko (G.V. Kurdyumov Institute of Metal Physics, Kyiv, Ukraine)
Kinetic model of defect substructure relaxation – nonGaussian size distributions, scaling regimes and parameterdriven transition between them
Abstract:
Recently the relaxed defect substructures in solids with the selfaffine geometry were experimentally observed on many scales and depicted by selfaffine measures. Here the general model for the aggregate growth by migration of components between aggregates is proposed to describe their formation. This idealized model of defect aggregate growth is based on minimum assumptions only to emphasize the main kinetic properties of relaxed defect aggregation by exchange of solitary defects. The assumptions allow us to reduce the master equation to Fokker–Planck equation with a diffusion and drift coefficients that are dependent on the rate of solitary defect exchange between defect aggregates, defect aggregate morphology, and kind of migration (ballistic, diffusive, etc.) of solitary defects. Two partial cases for typical defect configurations were considered to illustrate difference between their aggregation kinetics: model with minimum active surface ("pileup" of dislocat
ions) and maximum active surface ("wall" of dislocations). Their general group analysis is performed, symmetries of the governing equations are identified and two scaling regimes are determined. The exact solutions are found for these partial cases under typical initial and boundary conditions. It is shown that the different initial configurations of aggregates (uniform, Gaussian, etc.) finally evolve to scalefree distributions, those are different from the Gaussian distribution. The parameterdriven transition between different scaling regimes is demonstrated for these partial cases.

