  We introduce a family of Maxwellian Demons for which correlations among
information bearing degrees of freedom can be calculated exactly and in compact
analytical form. This allows one to precisely determine Demon functional
thermodynamic operating regimes, when previous methods either misclassify or
simply fail due to approximations they invoke. This reveals that these Demons
are more functional than previous candidates. They too behave either as
engines, lifting a mass against gravity by extracting energy from a single heat
reservoir, or as Landauer erasers, consuming external work to remove
information from a sequence of binary symbols by decreasing their individual
uncertainty. Going beyond these, our Demon exhibits a new functionality that
erases bits not by simply decreasing individual-symbol uncertainty, but by
increasing inter-bit correlations (that is, by adding temporal order) while
increasing single-symbol uncertainty. In all cases, but especially in the new
erasure regime, exactly accounting for informational correlations leads to
tight bounds on Demon performance, expressed as a refined Second Law of
Thermodynamics that relies on the Kolmogorov-Sinai entropy for dynamical
processes and not on changes purely in system configurational entropy, as
previously employed. We rigorously derive the refined Second Law under minimal
assumptions and so it applies quite broadly---for Demons with and without
memory and input sequences that are correlated or not. We note that general
Maxwellian Demons readily violate previously proposed, alternative such bounds,
while the current bound still holds.

Statistical Mechanics;Dynamical Systems;Chaotic Dynamics;Biological Physics;Chemical Physics