68 research outputs found
The R\'enyi entropy of L\'evy distribution
The equivalence between non-extensive C. Tsallis entropy and the extensive
entropy introduced by Alfr\'ed R\'enyi is discussed. The R\'enyi entropy is
studied from the perspective of the geometry of the Lebesgue and generalised,
exotic Lebesgue spaces. A duality principle is established. The R\'enyi entropy
for the L\'evy distribution, in the domain when the nunerical methods fails, is
approximated by asymptotic expansion for the large values of the R\'enyi
parameter.Comment: 10 pages, 0 figure
Minimum Dissipation Principle in Nonlinear Transport
We extend Onsager's minimum dissipation principle to stationary states that
are only subject to local equilibrium constraints, even when the transport
coefficients depend on the thermodynamic forces. Crucial to this generalization
is a decomposition of the thermodynamic forces into those that are held fixed
by the boundary conditions, and the subspace which is orthogonal with respect
to the metric defined by the transport coefficients. We are then able to apply
Onsager and Machlup's proof to the second set of forces. As an example we
consider two-dimensional nonlinear diffusion coupled to two reservoirs at
different temperatures. Our extension differs from that of Bertini et al. in
that we assume microscopic irreversibility and we allow a nonlinear dependence
of the fluxes on the forces.Comment: 20 pages, 1 figur
New Class of Generalized Extensive Entropies for Studying Dynamical Systems in Highly Anisotropic Phase Space
Starting from the geometrical interpretation of the R\'enyi entropy, we
introduce further extensive generalizations and study their properties. In
particular, we found the probability distribution function obtained by the
MaxEnt principle with generalized entropies. We prove that for a large class of
dynamical systems subject to random perturbations, including particle transport
in random media, these entropies play the role of Liapunov functionals. Some
physical examples, which can be treated by the generalized R\'enyi entropies
are also illustrated.Comment: 13 pages, 0 figure
Symmetry Group and Group Representations Associated to the Thermodynamic Covariance Principle (TCP)
We describe the Lie group and the group representations associated to the
nonlinear Thermodynamic Coordinate Transformations (TCT). The TCT guarantee the
validity of the Thermodynamic Covariance Principle (TCP) : {\it The nonlinear
closure equations, i.e. the flux-force relations, everywhere and in particular
outside the Onsager region, must be covariant under TCT}. In other terms, the
fundamental laws of thermodynamics should be manifestly covariant under
transformations between the admissible thermodynamic forces, i.e. under TCT.
The TCP ensures the validity of the fundamental theorems for systems far from
equilibrium. The symmetry properties of a physical system are intimately
related to the conservation laws characterizing that system. Noether's theorem
gives a precise description of this relation. We derive the conserved
(thermodynamic) currents and, as an example of calculation, a simple system out
of equilibrium where the validity of TCP is imposed at the level of the kinetic
equations is also analyzed.Comment: 35 pages, 6 figure
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