Kinetic derivation of a quasi-hydrodynamic system of equations on the base of nonextensive statistics

摘要

Applying the formalism of the Tsallis nonadditive statistical mechanics, a derivation of hydrodynamic and quasi-hydrodynamic equations is considered based on the generalized BGK kinetic equation. In particular, those equations are intended for construction of more appropriate macroscopic and mesoscopic models of transport systems related to the so-called anomalous systems where the corresponding phase space possesses a complicated (fractal) structure. The classic Boltzmann-Gibbs statistics (or Maxwell's distribution of velocities in the case of kinetic theory) is violated in such systems and hence the additivity of extensive variables related to it does not hold either. The application of the approach developed in this paper does not change the structure of hydrodynamic and quasi-hydrodynamic equations, but the modified thermal and calorific state equations and also the transfer coefficients contain two additional free parameters, which are the parameter q of the non-additivity of the system and the fractional dimension D of the phase space. Along with the smoothing parameter τ, these parameters can be determined empirically in each particular case from statistical or experimental data, which allows us to simulate the actual variable traffic situation within a continual approach both in regular and crisis cases.