  We know that maximal efficiency in physical systems is attained by reversible
processes. It is then interesting to see how irreversibility affects efficiency
in other systems, e.g., in a city. In this study, we focus on a cyclic process
of movements (home to workplace and back to home) in a city to investigate the
above question. To this end, we present a minimal model of the movements, along
with plausible definitions for the efficiency and irreversibility of the
process; more precisely, we take the inverse of the total travel time per
number of trips for efficiency and the relative entropy of the forward and
backward flow distributions for the process irreversibility. We perform
numerical simulations of the model for reasonable choices of the population
distribution, the mobility law, and the movement strategy. The results show
that the efficiency of movements is indeed negatively correlated with the above
measure of irreversibility. The structure of the network and the impact of the
flows on the travel times are the main factors here that affect the time
intervals of arriving to destinations and returning to origins, which are
usually larger than the time interval of the departures. This in turn gives
rise to diverging of the backward flows from the forward ones and results to
entropy (disorder or uncertainty) production in the system. The findings of
this study might be helpful in characterizing more accurately the city
efficiency and in better understanding of the main working principles of these
complex systems.
