Two-Member Markov Processes toward an Equilibrium from a Continuum of Initial States

Dynamics of two-member Markov processes is formulated based on the binomial probability. Sets of initial states are then sought such that the final state reaches an equilibrium. On the two-parameter phase plane, such initial states are found to exhibit diverse geometric configurations depending on the source probability. Those initial-state boundaries undergo phase transitions ranging over pills, stripes, circles, ellipses, lemons, and even fuzzy shapes. These results are quite helpful in understanding several physical phenomena involving photons, electrons, and atoms. For convenience of discussion, deformations of vortices are taken as an example.