Stochastic continuum mechanics - A thermodynamic-limit-free alternative to statistical mechanics: Equilibrium of isothermal ideal isotropic uniform fluid
"We do not know how to study finite systems in any clean way; that is, the thermodynamic limit is inevitable." The lack of the clean way stressed in this sentence of Resibois and de Leener means the following. To study finite systems, statistical mechanics and kinetic theory offer nothing but the formalism based on the thermodynamic limit (TDL) which in essence disagrees with notion of finite system. The present work proposes the model (almost-equilibrium in a certain sense) enabling one to construct continuous equilibrium descriptions of fluids, discrete multiparticle systems, with no application of TDL. For simplicity, the fluids are assumed to be isothermal, (rheologically) ideal, isotropic and uniform. The core of the model is nonlinear It (o) over tilde 's stochastic differential equation (ISDE) for the fluid-particle velocity. The continuous equilibrium description is based on the stationary probability density corresponding to this equation. The construction is described as a simple analytical recipe formulated in terms of quadratures and includes the velocity (or momentum) relaxation times which can be determined theoretically, experimentally, or as the results of numerical simulations and depending on the specific nature of the fluid. The recipe can be applied to an extremely wide range of fluids of the above class. It is illustrated by the derivations of the Maxwell-Boltzmann and Fermi-Dirac descriptions for the classical and fermion fluids in arbitrary space domain, bounded or unbounded. In the particular, TDL case, the derived results are in a complete agreement with those of statistical mechanics. (C) 2002 Elsevier Science Ltd. All rights reserved.
isothermal ideal isotropic uniform fluid descriptions relaxation time the maxwell-boltzmann and fermi-dirac thermodynamic-limit-free modelling nonlinear ito's stochastic differential equation