Nonequilibrium Thermodynamics of Driven Systems

The time evolution that has been investigated in Section 3 is the time evolution seen experimentally in externally unforced macroscopic systems approaching, as t — oo, equilibrium states at which their behavior is seen experimentally to be well described by classical equilibrium thermodynamics. In other words, in Section 3, we have investigated the time evolution 

The Invariant manifold Meth cMi composed of equilibrium states (xi)eth(y ) e 

This manifold is invariant with respect to the time evolution taking place on Mi, and its elements are states on which the thermodynamic potential p(x, y) reaches its minimum if considered as a function of X eM.  

We recall that both the manifold Meth and the fundamental thermodynamic relations y are displayed on the manifold MN x,=0 (see the text following (6)). 

Experimental observations of the time evolution of externally unforced macroscopic systems on the level meSo l show that the level eth of classical equilibrium thermodynamics is not the only level offering a simplified description of appropriately prepared macroscopic systems. For example, if Cmeso is the level of kinetic theory (Sections 2.2.1, starting point. In order to see the approach 2.2.2, and 3.1.3) then, besides the level, also the level of fluid mechanics (we shall denote it here Ath) emerges in experimental observations as a possible simplified description of the experimentally observed time evolution. The preparation process is the same as the preparation process for Ath (i.e., the system is left sufficiently long time isolated) except that we do not have to wait till the approach to equilibrium is completed. If the level of fluid mechanics indeed emerges as a possible reduced description, we have then the following four types of the time evolution leading from a mesoscopic to a more macroscopic level of description (i) Mslow/ (ii) Aneso 2 - Ath, (ui) Aneso l - Aneso 2, and (iv) Aneso i — Aneso 2 — Ath- The first two are the same as (111). We now turn our attention to the third one, that is, 

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