Mapping the master equation of 3-state systems on the 2-sphere

Abstract

Motivated by information geometry, we map 3-state stochastic systems to the 2-sphere in order to write the master equations as a dynamical system on a 2-sphere in terms of angular coordinates. We provide a framework for finding the angular coordinates for the equilibrium points of the dynamical system, which involves obtaining the physical solutions of a cubic equation. When performing linearization on the dynamical system, the eigenvalues of the dynamical system are shown to always have a negative real-part. Hence, mapping the master equation to the 2-sphere can provide a way to interpret spiral trajectories on a sphere in terms of stochastic thermodynamics.