The Extended Thermodynamics for the fractional statistics
M. Trovato
Dipartimento di Matematica e Informatica, Università di Catania, Italy
In classical mechanics, the introduction of a maximum entropy principle (MEP) has
proven to be very fruitful in solving the closure problem to any degree of approximation
in Extended Thermodynamics (ET) [1, 2]. The aim of this talk is to present a rigorous
formulation of the ET in the framework of fractional exclusion statistics [3, 4, 5] for different
dimensions of the phase space.
In particular, in the framework of a local theory: (i) by using the H-theorem we construct a correct formulation of the collisional terms for systems of identical particles, by
generalizing the Uehling-Uhlenbeck approach; (ii) by introducing the MEP, we calculate
explicitly the entropy, the flux entropy, and the closure relations for a set of hydrodynamic
balance equations (HD) obtained for an arbitrary number of moments of the distribution
function; (iii) we obtain a virial expansion and a Sommerfeld expansion, evaluated to all
orders, both for the chemical potential and for the equation of state.
Analogously in the framework of a nonlocal theory we present a rigorous formulation of
the Quantum Extended Thermodynamics (QET) by using an arbitrary number of moments.
Here, the main difficulties concern with: (i) the definition of a proper quantum entropy that
includes the fractional exclusion statistics; (ii) the formulation of a global quantum MEP
(QMEP) [5, 6, 7] that allows one to obtain the corresponding Wigner distribution function;
(iii) the generalization of the Lagrange multipliers and the corresponding closure relations
for a quantum gas.
In particular, all the results available in the literature [4, 8, 9] are generalized in terms
of both the fractional statistics and a nonlocal description for a quantum gas. Finally,
some examples of closed quantum systems are reported, gradient quantum corrections are
explicitly given and, classical results are recovered when h̄ tends to zero.
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