Hamiltonian Formulations

Abstract

Hamiltonian formulations have been identified at various levels of description: the microscopic one (classical mechanics of particles), the kinetic theory (based on a distribution function rather than on precise values for the mechanical variables of particles), and macroscopic approaches (as, for instance, hydrodynamics or equilibrium thermodynamics) [4.1 4.2 4.3 4.4 4.5]. Therefore, it is natural to ask whether such a structure is preserved in mesoscopic intermediate descriptions and more particularly in extended irreversible thermodynamics (EFT). This compatibility condition among different levels of description is especially useful in the non-linear domain. There are several reasons that militate in favour of a Hamiltonian description: it is attractive because of its conciseness and its physical content. Indeed, the whole set of balance equations is now expressed in terms of one or several generating functional which may be generally identified with a well-defined physical quantity such as the energy, the entropy, or the Gibbs free energy. Moreover, there exist many elegant results and powerful methods of solution typically developed for general Hamiltonian systems which can be of direct use in analysing the solutions of the basic equations of EIT. Finally, Hamiltonian techniques have a wide range of validity — they are not restricted to the linear regime — and provide supplementary restrictions which complement those of the second law as well as a link between thermodynamics and dynamics at several levels of description.