The macroscopic study of a system evolving close to equilibrium states can be based upon the systematic approach of irreversible thermodynamics, the principles of which are summarized in § 14.2.6: identifying the processes involved — relaxation towards equilibrium, transport, forced regime, chemical kind of system, and so on; analyzing the system in terms of weakly coupled sub-systems, each of which is practically at equilibrium; finding the macroscopic state variables and the conserved quantities; writing down the conservation equations, the local equations of state, and the equations describing the linear response of the fluxes to the affinities; using symmetry and invariance laws and Onsager relations and checking that the dissipation is positive; finally, solving the coupled evolution equations which we have obtained.
These elements, postulated by non-equilibrium thermodynamics, can be deduced from the microscopic equations of motion and the methods of statistical mechanics, provided the evolution is sufficiently slow. The proof of the conservation laws gives us the microscopic interpretation of the fluxes. The study of the dynamics involves, together with the microscopic description through a density operator \( \widehat D \) which evolves according to the Liouville-von Neumann equation, a mesoscopic description through a simplified density operator \( \widehat D_0 \) which follows \( \widehat D \) in its motion and which contains only information about the macroscopic, relevant variables. The relevant statistical entropy \( S(\widehat D_0 ) \), which is the missing information associated with these variables, can be identified with the entropy of thermodynamics; its increase reflects a leak of information towards the other, irrelevant variables. The reduction, or contraction, of the description from \( \widehat D \) to \( \widehat D_0 \) is the subject of the projection method; it associates the macroscopic quasi-equilibrium regime with a short-memory approximation for the dynamics of the irrelevant microscopic variables. The linear regime corresponds, moreover, to a weak coupling between subsystems.
The use of symmetries and invariances enables us to analyze the structure of the macroscopic dynamic equations, to reduce the number of independent response coefficients, and to connect various effects with one another. We illustrated the general method by studying in that spirit diffusion, heat or electric conduction, thermoelectric effects, and hydrodynamics. We proved, especially, the macroscopic laws governing the thermal and mechanical behaviour of Newtonian fluids, starting solely from the general principles of non-equilibrium thermodynamics.
KeywordsDensity Operator Local Equilibrium Response Coefficient Global Equilibrium Intensive Variable
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