Evolution of pattern complexity in the Cahn–Hilliard theory of phase separation

Probabilistic and numerical validation of homology computations for nodal domains

“…These results have established rigorous lower bounds on the duration of the linear regime. In contrast, our results provide an upper bound on the onset of nonlinear behavior in the Cahn-Hilliard equation, and complement our findings in [14].…”

“…This stochastic partial differential equation has been proposed as a model for phase separation in metallic alloys and produces complicated patterns; see for example [3,4,5,7,26] and the references therein. As we mentioned in the Introduction, computational homology can be used to quantify these complicated structures [14], and the question of choosing the correct discretization size M for the homology computations is of utmost importance. Notice that if we are interested in the evolution of (4.3) originating at a random field, then for any time t > 0 the solution u(t, ·) is a random field over G. In general, however, the coefficients in the Fourier expansion of this random field will be neither Gaussian nor independent.…”

“…Again, these are typically high-dimensional systems for which there is considerable interest in understanding the temporal changes of the low-dimensional nodal domains, i.e., patterns, and homological methods appear to provide new techniques through which one can explore these problems [13,14,18]. Notice, however, that in this setting the topology of the level sets which define the nodal domains changes as a function of time, and thus, there is no fixed manifold X for which one is trying to compute the homology.…”

Microstructural Characterization of Hard Ceramics

“…While there is also a long history of applying the mathematical tools of topology to microstructures, there are still relatively few examples. (Gameiro et al, 2005, Steele, 1972, Mecke, 1996, Mecke and Sofonea, 1997, Mendoza et al, 2004, Wanner et al, 2010 In two dimensions, there are two topological metrics that measure the number of independent pieces of the network (referred to as B0) and the number of closed loops (referred to as B1). (Wanner et al, 2010) In the context of plane sections of grain boundary networks, B0 measures groups of grain boundaries not connected to the rest of the network and B1 measures continuous, closed paths of grain boundaries or the number of grains.…”

An Impact of Stochastic Dynamic Boundary Conditions on the Evolution of the Cahn-Hilliard System

“…The function f is the derivative of a logarithmic potential which is usually approximated by a polynomial with strictly positive dominant coefficient. Theoretical results on asymptotic dynamical behavior of the system (1.1), under either Neumann or periodic boundary conditions, can be founded, for example, in the survey [24] or the book [30], and [9,2,18].…”