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A lower bound for topological entropy of generic non Anosov symplectic diffeomorphisms
We prove that a generic symplectic diffeomorphism is either Anosov or
the topological entropy is bounded from below by the supremum over the smallest
positive Lyapunov exponent of the periodic points. We also prove that
generic symplectic diffeomorphisms outside the Anosov ones do not admit
symbolic extension and finally we give examples of volume preserving
diffeomorphisms which are not point of upper semicontinuity of entropy function
in topology
Mechanical Proof of the Second Law of Thermodynamics Based on Volume Entropy
In a previous work (M. Campisi. Stud. Hist. Phil. M. P. 36 (2005) 275-290) we
have addressed the mechanical foundations of equilibrium thermodynamics on the
basis of the Generalized Helmholtz Theorem. It was found that the volume
entropy provides a good mechanical analogue of thermodynamic entropy because it
satisfies the heat theorem and it is an adiabatic invariant. This property
explains the ``equal'' sign in Clausius principle () in a purely
mechanical way and suggests that the volume entropy might explain the ``larger
than'' sign (i.e. the Law of Entropy Increase) if non adiabatic transformations
were considered. Based on the principles of microscopic (quantum or classical)
mechanics here we prove that, provided the initial equilibrium satisfy the
natural condition of decreasing ordering of probabilities, the expectation
value of the volume entropy cannot decrease for arbitrary transformations
performed by some external sources of work on a insulated system. This can be
regarded as a rigorous quantum mechanical proof of the Second Law. We discuss
how this result relates to the Minimal Work Principle and improves over
previous attempts. The natural evolution of entropy is towards larger values
because the natural state of matter is at positive temperature. Actually the
Law of Entropy Decrease holds in artificially prepared negative temperature
systems.Comment: 17 pages, 1 figur
Entropy of gravitating systems: scaling laws versus radial profiles
Through the consideration of spherically symmetric gravitating systems
consisting of perfect fluids with linear equation of state constrained to be in
a finite volume, an account is given of the properties of entropy at conditions
in which it is no longer an extensive quantity (it does not scale with system's
size). To accomplish this, the methods introduced by Oppenheim [1] to
characterize non-extensivity are used, suitably generalized to the case of
gravitating systems subject to an external pressure. In particular when, far
from the system's Schwarzschild limit, both area scaling for conventional
entropy and inverse radius law for the temperature set in (i.e. the same
properties of the corresponding black hole thermodynamical quantities), the
entropy profile is found to behave like 1/r, being r the area radius inside the
system. In such circumstances thus entropy heavily resides in internal layers,
in opposition to what happens when area scaling is gained while approaching the
Schwarzschild mass, in which case conventional entropy lies at the surface of
the system. The information content of these systems, even if it globally
scales like the area, is then stored in the whole volume, instead of packed on
the boundary.Comment: 16 pages, 11 figures. v2: addition of some references; the stability
of equilibrium configurations is readdresse
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