  We study non-normalizable quasi-equilibrium states (NNQE) arising from
anomalous diffusion. Initially, particles in contact with a thermal bath are
released from an asymptotically flat potential well, with dynamics that is
described by fractional calculus. For temperatures that are sufficiently low
compared to the potential depth, the properties of the system remain almost
constant in time. We use the fractional-time Fokker-Planck equation (FTFPE) and
continuous-time random walk approaches to calculate the ensemble averages of
observables. We obtain analytical estimates of the duration of NNQE, depending
on the fractional order, from approximate theoretical solutions of the FTFPE.
We study and compare two types of observables, the mean square displacement
typically used to characterize diffusion, and the thermodynamic energy. We show
that the typical time scales for stagnation depend exponentially on the
activation energy in units of temperature multiplied by a function of the
fractional exponent.
