Continuous time random walks in closed and open single-molecule systems with microscopic reversibility

Continuous time random walk (CTRW) is studied with a new dynamic equation based on the age-structure of states. For a CTRW in a closed molecular system, two necessary conditions for microscopic reversibility are introduced: 1) independence of transition direction and waiting time for every state and 2) detailed balance among the transition probabilities. Together they are also sufficient condition. For a CTRW in an open system with explicit chemical energy input 1) still holds while 2) breaks down. Hence, CTRW models not satisfying 1) are either inconsistent with thermodynamics or cannot attain equilibrium due to hidden dissipation in non-Markovian states. Each CTRW defines a unique corresponding Markov process (cMP). The steady-state distribution of a CTRW equals that of the corresponding Markov process, and the two systems have the same steady-state flux, the same exit probabilities and the same mean trapping times. Mechanicity is discussed; a paradox observed by Kolomeisky and Fisher (J. Chem. Phys., 113 (2000) 10867) is resolved.