Continuous-time random walk for open systems: Fluctuation theorems and counting statistics

We consider continuous-time random walks (CTRW) for open systems that exchange energy and matter with multiple reservoirs. Each waiting time distribution (WTD) for times between steps is characterized by a positive parameter alpha, which is set to alpha=1 if it decays at least as fast as t-2 at long times and therefore has a finite first moment. A WTD with alpha<1 decays as t-alpha-1. A fluctuation theorem for the trajectory quantity R, defined as the logarithm of the ratio of the probability of a trajectory and the probability of the time reversed trajectory, holds for any CTRW. However, R can be identified as a trajectory entropy change only if the WTDs have alpha=1 and satisfy separability (also called "direction time independence"). For nonseparable WTDs with alpha=1, R can only be identified as a trajectory entropy change at long times, and a fluctuation theorem for the entropy change then only holds at long times. For WTDs with 0<alpha<1 no meaningful fluctuation theorem can be derived. We also show that the (experimentally accessible) nth moments of the energy and matter transfers between the system and a given reservoir grow as tnalpha at long times.