  In the case of fully chaotic systems the distribution of the
Poincare'recurrence times is an exponential whose decay rate is the
Kolmogorov-Sinai(KS) entropy.We address the discussion of the same problem, the
connection between dynamics and thermodynamics,in the case of sporadic
randomness,using the Manneville map as a prototype of this class of processes.
We explore the possibility of relating the distribution of Poincare' recurrence
times to `thermodynamics',in the sense of the KS entropy,also in the case of an
inverse power law. This is the dynamic property that Zaslavsly [Phys.Today(8),
39(1999)] finds to be responsible for a striking deviation from ordinary
statistical mechanics under the form of Maxwell's Demon effect. We show that
this way of estabi- lishing a connection between thermodynamics and dynamics is
valid only in the case of strong chaos. In the case of sporadic randomness,
resulting at long times in the Levy diffusion processes,the sensitivity to
initial conditions is initially an inverse pow erlaw,but it becomes exponential
in the long-time scale, whereas the distribution of Poincare times keeps its
inverse power law forever. We show that a nonextensive thermodynamics would
imply the Maxwell's Demon effect to be determined by memory and thus to be
temporary,in conflict with the dynamic approach to Levy statistics. The
adoption of heuristic arguments indicates that this effect,is possible, as a
form of genuine equilibrium,after completion of the process of memory erasure.
