Symmetry in Nonlinear Mathematical Physics - 2009
Yuri Gordienko (G.V. Kurdyumov Institute of Metal Physics, Kyiv, Ukraine)
Kinetic model of defect substructure relaxation – non-Gaussian size distributions, scaling regimes and parameter-driven transition between them
Recently the relaxed defect substructures in solids with the self-affine geometry were experimentally observed on many scales and depicted by self-affine 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 scale-free distributions, those are different from the Gaussian distribution. The parameter-driven transition between different scaling regimes is demonstrated for these partial cases.