Convective variational approach to relativistic thermodynamics of dissipative fluids

Abstract

Using guidelines provided by Noether identities arising from a generalized variation procedure of convective type, a new (nonlinear and exactly self-consistent) category of relativistic thermodynamic models is developed for the systematic representation of viscous conducting fluid media (allowing for several independent charged or neutral chemical constituents). Apart from the provision of a set of dissipation coefficients of the usual (reactivity, resistivity, and viscosity) type, the specification of a particular model is determined just by giving the algebraic dependence a single ‘master function’, ͘ Λ say, on the relevant dynamical variables, which are supposed here to consist of an entropy current 4-vector and a set of particle current 4-vectors corresponding to the various chemical constituents, together with a set of symmetric (rank 3) viscosity tensors, which are considered as being dynamically independent of the corresponding current vectors except in the degenerate limit of linear viscosity. The master function is set up as a generalization of an ordinary lagrangian function, to which it reduces in the relevant non - dissipative limit, and, as in the conservative case, it is used for the construction of derived quantities in such a way that appropriate self-consistency conditions are satisfied as identities. In particular the relevant stress-momentum-energy tensor is obtained directly in terms of the independent variables and of their dynamical conjugates (whose role is hidden in the traditional approach as developed by Israel & Stewart), which are set of ordinary 4-momentum (not 4-momentum density) covectors associated with the independent currents, and a set of generalised Cauchy type strain (not strain - rate) tensors associated with the independent viscous stress contributions. The range of application of the category obtained in this way is intended to include that of the standard (Israel-Stewart) formalism to which it is expected to be effectively equivalent in the limit of sufficiently small deviations from thermodynamic equilibrium.