  We compare the relation between dispersion and dissipation for two random
variables that can be used to characterize the precision of a Brownian clock.
The first random variable is the current between states. In this case, a
certain precision requires a minimal energetic cost determined by a known
thermodynamic uncertainty relation. We introduce a second random variable that
is a certain linear combination of two random variables, each of which is the
time a stochastic trajectory spends in a state. Whereas the first moment of
this random variable is equal to the average probability current, its
dispersion is generally different from the dispersion associated with the
current. Remarkably, for this second random variable a certain precision can be
obtained with an arbitrarily low energy dissipation, in contrast to the
thermodynamic uncertainty relation for the current. As a main technical
achievement, we provide an exact expression for the dispersion related to the
time that a stochastic trajectory spends in a cluster of states for a general
unicyclic network.
