Navier-Stokes Equations for Generalized Thermostatistics

Tsallis has proposed a generalization of Boltzmann-Gibbs thermostatistics by introducing a family of generalized nonextensive entropy functionals with a single parameter $q$. These reduce to the extensive Boltzmann-Gibbs form for $q=1$, but a remarkable number of statistical and thermodynamic properties have been shown to be $q$-invariant -- that is, valid for any $q$. In this paper, we address the question of whether or not the value of $q$ for a given viscous, incompressible fluid can be ascertained solely by measurement of the fluid's hydrodynamic properties. We find that the hydrodynamic equations expressing conservation of mass and momentum are $q$-invariant, but that for conservation of energy is not. Moreover, we find that ratios of transport coefficients may also be $q$-dependent. These dependences may therefore be exploited to measure $q$ experimentally.