Role of Topology in Relaxation of One-Dimensional Stochastic Processes

Stochastic processes describe dynamics of a wide variety of nonequilibrium phenomena, including chemical reaction, molecular motor and biological motion. In particular, master equations are often used to describe the dynamics of Markov processes and can be regarded as non-Hermitian Schrödinger equations.

In this talk, we will see the topological characterization of relaxation phenomena of one-dimensional stochastic processes with the aid of recent studies on the spectral theory of non-Hermitian topological phases accompanied by the non-Hermitian skin effect. To this end, we define a winding number of a master equation under the periodic boundary condition and theoretically show that it corresponds to a nonzero spectral gap under the open boundary condition (OBC) in the thermodynamic limit. We numerically confirm that the winding number corresponds to the system-size dependence of the divergent OBC relaxation time and the unconventional transient behavior called a cutoff phenomenon, where the relaxation does not occur until a certain time and then rapidly proceeds. These unconventional relaxation phenomena can be understood in terms of the so-called gap-discrepancy problems.