Higher-order generalized hydrodynamics: Foundations within a nonequilibrium statistical ensemble formalism

Abstract

Construction, in the framework of a nonequilibrium statistical ensemble formalism, of a higher-order generalized hydrodynamics, also referred to as mesoscopic hydrothermodynamics, that is, covering phenomena involving motion of fluids displaying variations short in space and fast in time-unrestricted values of Knudsen numbers, is presented. In that way, an approach is provided enabling the coupling and simultaneous treatment of the kinetics and hydrodynamic levels of descriptions. It is based on a complete thermostatistical approach in terms of the densities of matter and energy and their fluxes of all orders covering systems arbitrarily driven away from equilibrium. The set of coupled nonlinear integrodifferential hydrodynamic equations is derived. They are the evolution equations of the Gradlike moments of all orders, derived from a generalized kinetic equation built in the framework of the nonequilibrium statistical ensemble formalism. For illustration, the case of a system of particles embedded in a fluid acting as a thermal bath is fully described. The resulting enormous set of coupled evolution equations is of unmanageable proportions, thus requiring in practice to introduce an appropriate description using the smallest possible number of variables. We have obtained a hierarchy of Maxwell times, associated to the set of all the higher-order fluxes, which have a particular relevance in the process of providing criteria for establishing the contraction of description.

Nonlinear higher-order hydrodynamics: Fluids under driven flow and shear pressure

“…(B3), (B15), and (49), the barycentric velocity vðr; tÞ and the drift velocity v n ðr; tÞ are related by the expression x ¼ m À1 þ M À1 : Hence, in Eq. (C1) we do have the contribution to the scattering operator of the form of Maxwell's one, but two other contributions are present: we do no enter the cumbersome expressions form then, suffice to say that J ð2Þ n contains the divergence of the pressure tensor (the divergence of the second order flux) and the gradient of the concentration, as it should according to mesoscopic irreversible thermodynamics: 54 These terms, as shown in Ref. 33, lead to contributions of Burnett and super-Burnett type.…”

“…The construction of the nonlinear higher-order thermohydrodynamics here presented is based, as noticed, in a nonequilibrium statistical ensemble formalism (NESEF). 41,54,55 and for the sake of completeness, we here very briefly review its foundations. For such purpose, first it needs be noticed that for systems away from equilibrium, several important points need be carefully taken into account in each case under consideration:…”

Sobre modelagem matemática e formalismos estatísticos de sistemas complexos

“…O "NESEF" tem sido aplicado com sucesso ao estudo de várias situações de não-equilíbrio na física de semicondutores [7] e polímeros [83], bem como em estudos de comportamento complexo de sistemas bosônicos em biopolímeros [8] e sistemas de fônons [84,85]. Também pode ser notado que a teoria cinética não-linear baseada no "NESEF" fornece, em casos limites particulares, generalizações de longo alcance das equações de L. Boltzmann [86] e H. Mori [87], juntamente com fundamentos estatísticos para a Termodinâmica Irreversível Mesoscópica [88], e uma hidrodinâmica de ordem superior [89][90][91][92][93].…”

Higher‐order generalized hydrodynamics of carriers and phonons in semiconductors in the presence of electric fields: Macro to nano

“…The construction of a Mesoscopic Hydro-Thermodynamics for the description of the movement of matter and energy in fluids under nonequilibrium thermodynamic conditions and at the classical mechanical level based on a generalized moments approach method to the solution of a NESEF-based generalized Boltzmann equation [29], is described elsewhere [30,31].…”