Oscillatory and dissipative dynamics of complex probability in non-equilibrium stochastic processes

TL;DR

This paper explores how introducing complex transition rates in non-equilibrium stochastic processes leads to oscillatory and dissipative behaviors in complex probabilities, revealing new dynamical phenomena depending on system dimensionality and perturbations.

Contribution

It introduces complex transition rates into classical master equations, uncovering oscillatory and dissipative dynamics in complex probabilities, a novel approach in non-equilibrium stochastic analysis.

Findings

Persistent oscillations occur with purely imaginary transition rates.

Oscillatory behavior depends on the dimensionality of the state space.

Perturbations can switch between dissipation and persistence in complex probabilities.

Abstract

For a Markov and stationary stochastic process described by the well-known classical master equation, we introduce complex transition rates instead of real transition rates to study the pre-thermal oscillatory behaviour in complex probabilities. Further, for purely imaginary transition rates we obtain persistent infinitely long lived oscillations in complex probability whose nature depends on the dimensionality of the state space. We also take a peek into cases where we perturb the relaxation matrix for a dichotomous process with an oscillatory drive where the relative sign of the angular frequency of the drive decides whether there will be dissipation in the complex probability or not.