Space partitioning basics

The morphology of a region evolves spato-temporally if its shape and/or size, and/or position changes in real space.

Above figure shows morphology change indicators. Sub-space ω partitions space Ω in the volume ratio VOL(ω)/VOL(Ω). Each ω is characterized by a set of boundaries having elements ξn. Thick arrow denotes sub-space mobility and thin dotted arrow denotes sub-space boundary mobility. For reasons of simplicity, we will address sub-space mobility as the centroid of the sub-space having a definite motion in real space. This definiteness shall be associated with a non-random motion. This mobility will therefore be simply addressed as "centroidal mobility". This could be either translational or rotational. This could be associated to the Global coordinate system. Along with centroidal mobility, the boundaries may have a mobility of their own, defined by boundary migration velocity (V.ξn). This could be defined in the global coordinate system and could be referred to as local coordinate system attached to each ξn. In space partitions evolving temporally (along ti) based on minimization principles involving random processes/samplings, spatio-temporal evolution may be bound to happen. For stable ensembles, the spatio-temporal evolutions will only be characterized by fluctuations. For evolving systems, both a definite mobility and fluctuation will be present. Following these discussions, three types of spatio-temporal evolution of space paritioning that could be achieved using PXO are described below:

1. Sub-space centroid and sub-space boundary mobility (ωcb mobility) t_i denotes temporal slice, x-y is the real space coordinate system. Here, boundary mobility is contributing to an increase in VOL(ω)/VOL(Ω) and the sub-space is growing temporally. Alternatively, it could also shrink if all V.ξn are otherwise. If all V.ξn senses are not outward from the sub-space centroid, the sub-space suffers anisotropic changes in morphology.

2. Sub-space boundary isotropic mobility (ωbi mobility) Here, there is no net sub-space centroid mobility. Only grain boundary mobility, characterized by definite motion and fluctuations are present. The characteristic here is the monotonosity in VOL(ω)/VOL(Ω) over the time interval considered.

3. Sub-space boundary anisotropic mobility (ωba mobility) Here, there is no net sub-space centroid mobility. Only grain boundary mobility, characterized by definite motion and fluctuations are present. VOL(ω)/VOL(Ω) may increase or decrease depending on senses of V.ξn elements, over the time interval considered.