  When relaxation towards an equilibrium or steady state is exponential at
large times, one usually considers that the associated relaxation time $\tau$,
i.e., the inverse of that decay rate, is the longest characteristic time in the
system. However that need not be true, and in particular other times such as
the lifetime of an infinitesimal perturbation can be much longer. In the
present work we demonstrate that this paradoxical property can arise even in
quite simple systems such as a chain of reactions obeying mass action kinetics.
By mathematical analysis of simple reaction networks, we pin-point the reason
why the standard relaxation time does not provide relevant information on the
potentially long transient times of typical infinitesimal perturbations.
Overall, we consider four characteristic times and study their behavior in both
simple chains and in more complex reaction networks taken from the publicly
available database "Biomodels." In all these systems involving mass action
rates, Michaelis-Menten reversible kinetics, or phenomenological laws for
reaction rates, we find that the characteristic times corresponding to
lifetimes of tracers and of concentration perturbations can be much longer than
$\tau$.
