Sporadic randomness, Maxwell's demon and the poincare recurrence times

In the case of fully chaotic systems, the distribution of the Poincare recu rrence times is an exponential whose decay rate is the Kolmogorov-Sinai (KS ) entropy. We address the discussion of the same problem, the connection be tween dynamics and thermodynamics, in the case of sporadic randomness, usin g the Manneville map as a prototype of this class of processes. We explore the possibility of relating the distribution of Poincare recurrence times t o "thermodynamics", in the sense of the KS entropy, also in the case of an inverse power-law. This is the dynamic property that Zaslavsky [Physics Tod ay 52 (8) (1999) 39] 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 establishing a connection between thermodynamics and dynamics is valid only in the case of strong chaos, where both the sensiti vity to initial conditions and the distribution of the Poincare recurrence times are exponential. In the case of sporadic randomness, resulting at lon g times in the Levy diffusion processes, the sensitivity to initial conditi ons is initially a power-law, but it becomes exponential again in the long- time scale, whereas the distribution of Poincare recurrence times keeps, or gels, its inverse power-law nature forever, including the long-time scale where the sensitivity to initial condition becomes exponential. We show tha t a non-extensive version of thermodynamics would imply the Maxwell's Demon effect to be determined by memory, and thus to be temporary, in conflict w ith the dynamic approach to Levy statistics. The adoption of heuristic argu ments indicates that this effect is possible, as a form of genuine equilibr ium, after completion of the process of memory erasure. (C) 2001 Elsevier S cience Ltd. All rights reserved.