A generalized action with dissipative potentials

Abstract

In this chapter a lagrangian with dissipative (e.g., Onsager’s) potentials is constructed for the field description of irreversible heat conducting fluids, off local equilibrium. The effect of thermal inertia is also taken into account following the approach developed in Chaps. 4 – 5 (resting on the coefficient g of statistical mechanics rather than the coefficient θ of Chap. 6). Extremum conditions of action yield Clebsch representations of temperature, chemical potential, velocities and generalized momenta, including thermal momenta, extensions of those introduced in Chaps. 4 – 6. The basic question asked is: To what extent may irreversibility, represented by a given form of the entropy source, influence the analytical form of the conservation laws for the energy and momentum? By incorporating the entropy source into the action functional we obtain Nöther’s energy for a fluid with irreversible heat flow, which leads to a fundamental equation and extended Hamiltonian dynamics obeying the second law. We show that while in the case of the equal Onsager’s potentials this energy coincides numerically with the classical energy E, it contains an extra term (vanishing along the path) still contributing to an irreversible evolution. Components of the energy-momentum tensor preserve all terms standardly regarded as “irreversible” (heat flux, tangential stresses, etc.) generalized to the case when thermodynamics includes the state gradients and the thermal phase, η, as a new thermodynamic variable. This variable, has already been discussed in Chaps. 4 – 6 in the context of reversible processes, hence it is known to us; it is the Lagrange multiplier of the entropy balance. Here it is shown to be crucial for consistent treatment of irreversible processes via an action formalism. Observing that the results strongly depend on the functional form of the entropy source we ultimately end up with a hypothesis postulating that embedding the first and second laws in the context of the extremal behavior of action under irreversible conditions may imply accretion of a new term to the classical energy.