182 research outputs found
Generalized molecular chaos hypothesis and H-theorem: Problem of constraints and amendment of nonextensive statistical mechanics
Quite unexpectedly, kinetic theory is found to specify the correct definition
of average value to be employed in nonextensive statistical mechanics. It is
shown that the normal average is consistent with the generalized
Stosszahlansatz (i.e., molecular chaos hypothesis) and the associated
H-theorem, whereas the q-average widely used in the relevant literature is not.
In the course of the analysis, the distributions with finite cut-off factors
are rigorously treated. Accordingly, the formulation of nonextensive
statistical mechanics is amended based on the normal average. In addition, the
Shore-Johnson theorem, which supports the use of the q-average, is carefully
reexamined, and it is found that one of the axioms may not be appropriate for
systems to be treated within the framework of nonextensive statistical
mechanics.Comment: 22 pages, no figures. Accepted for publication in Phys. Rev.
Nonextensive Thermostatistics and the H-Theorem
The kinetic foundations of Tsallis' nonextensive thermostatistics are
investigated through Boltzmann's transport equation approach. Our analysis
follows from a nonextensive generalization of the ``molecular chaos
hypothesis". For , the -transport equation satisfies an -theorem
based on Tsallis entropy. It is also proved that the collisional equilibrium is
given by Tsallis' -nonextensive velocity distribution.Comment: 4 pages, no figures, corrected some typo
Verschraenkung versus Stosszahlansatz: Disappearance of the Thermodynamic Arrow in a High-Correlation Environment
The crucial role of ambient correlations in determining thermodynamic
behavior is established. A class of entangled states of two macroscopic systems
is constructed such that each component is in a state of thermal equilibrium at
a given temperature, and when the two are allowed to interact heat can flow
from the colder to the hotter system. A dilute gas model exhibiting this
behavior is presented. This reversal of the thermodynamic arrow is a
consequence of the entanglement between the two systems, a condition that is
opposite to molecular chaos and shown to be unlikely in a low-entropy
environment. By contrast, the second law is established by proving Clausius'
inequality in a low-entropy environment. These general results strongly support
the expectation, first expressed by Boltzmann and subsequently elaborated by
others, that the second law is an emergent phenomenon that requires a
low-entropy cosmological environment, one that can effectively function as an
ideal information sink.Comment: 4 pages, REVTeX
Boltzmann's H-theorem, its limitations, and the birth of (fully) statistical mechanics
A comparison is made of the traditional Loschmidt (reversibility) and Zermelo
(recurrence) objections to Boltzmann's H-theorem, and its simplified variant in
the Ehrenfests' 1912 wind-tree model. The little-cited 1896 (pre-recurrence)
objection of Zermelo (similar to an 1889 argument due to Poincare) is also
analysed. Significant differences between the objections are highlighted, and
several old and modern misconceptions concerning both them and the H-theorem
are clarified. We give particular emphasis to the radical nature of Poincare's
and Zermelo's attack, and the importance of the shift in Boltzmann's thinking
in response to the objections as a whole.Comment: 40 page
Generalized Statistics Variational Perturbation Approximation using q-Deformed Calculus
A principled framework to generalize variational perturbation approximations
(VPA's) formulated within the ambit of the nonadditive statistics of Tsallis
statistics, is introduced. This is accomplished by operating on the terms
constituting the perturbation expansion of the generalized free energy (GFE)
with a variational procedure formulated using \emph{q-deformed calculus}. A
candidate \textit{q-deformed} generalized VPA (GVPA) is derived with the aid of
the Hellmann-Feynman theorem. The generalized Bogoliubov inequality for the
approximate GFE are derived for the case of canonical probability densities
that maximize the Tsallis entropy. Numerical examples demonstrating the
application of the \textit{q-deformed} GVPA are presented. The qualitative
distinctions between the \textit{q-deformed} GVPA model \textit{vis-\'{a}-vis}
prior GVPA models are highlighted.Comment: 26 pages, 4 figure
The relativistic statistical theory and Kaniadakis entropy: an approach through a molecular chaos hypothesis
We have investigated the proof of the theorem within a manifestly
covariant approach by considering the relativistic statistical theory developed
in [G. Kaniadakis, Phy. Rev. E {\bf 66}, 056125, 2002; {\it ibid.} {\bf 72},
036108, 2005]. As it happens in the nonrelativistic limit, the molecular chaos
hypothesis is slightly extended within the Kaniadakis formalism. It is shown
that the collisional equilibrium states (null entropy source term) are
described by a power law generalization of the exponential Juttner
distribution, e.g., , with
, where is a scalar,
is a four-vector, and is the four-momentum. As a simple example, we
calculate the relativistic power law for a dilute charged gas under
the action of an electromagnetic field . All standard results are
readly recovered in the particular limit .Comment: 7 pages; to be published in EPJ
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