Near-equilibrium linear regime

We can now specify more precisely what is meant by the near-equilibrium linear regime. It means that Lqq, Lee, etc., may be treated as constants. Since T x) is a function of position, such an assumption is strictly not valid. It is valid only in the approximation that the change in T from one end of the system to another is small compared to the average T, i.e. if the average temperature is Tavg, then r(x) — Tavgl/ Tavg 1 for all x. Hence we may approximate and use Kr g in place of kT. ... 

The current frontiers for the subject of non-equilibrium thennodynamics are rich and active. Two areas dommate interest non-linear effects and molecular bioenergetics. The linearization step used in the near equilibrium regime is inappropriate far from equilibrium. Progress with a microscopic kinetic theory [38] for non-linear fluctuation phenomena has been made. Carefiil experiments [39] confinn this theory. Non-equilibrium long range correlations play an important role in some of the light scattering effects in fluids in far from equilibrium states [38, 39]. 

In the regime of linear kinetics near equilibrium, Eqn. 7-53 reduces to the approximate equation in Eqn. 7.54 ... 

Equations (12.27) and (12.64) show the stability of the nonequilibrium stationary states in light of the fluctuations Sev The linear regime requires P > 0 and dP/dt < 0, which are Lyapunov conditions, as the matrix (dAJdej) is negative definite at near equilibrium. 

The Navier-Stokes equations are valid when A is much smaller than the characteristic flow dimension L. When this condition is violated, the flow is no longer near equilibrium and the linear relations between stress and rate of strain and the no-slip velocity condition are no longer valid. Similarly, the linear relation between heat flux and temperature gradient and the no-jump temperature condition at a solid-fluid interface are no longer accurate when A is not much smaller than L. The different Knudsen number regimes are delineated in Fig. 2. 

In the previous section we have seen that the stationary states in the linear regime are also states that extremize the internal entropy production. We shall now consider the stability of these states, and also show that the entropy production is minimized. In Chapter 14 we saw that the fluctuations near the equilibrium state decrease the entropy and that the irreversible processes drive the system back to the equilibrium state of maximum entropy. As the system approaches the state of equilibrium, the entropy production approaches zero. The approach to equilibrium can be described not only as a steady increase in entropy to its maximum value but also as a steady decrease in entropy production to zero. It is this latter approach that naturally extends to the linear regime, close to equilibrium. 

Fig. 7-S. Reaction rate as a function of reaction affinity curve (a) = regime of linear kinetics near reaction equilibrium curve (b) = regime of nonlinear exponential kinetics away from reaction equilibrium v = reaction rate A= affinity.

The data reduction of vapor-pressure osmometry (VPO) follows to some extent the same relations as outlined above. However, from its basic principles, it is not an equilibrium method, since one measures the (very) small difference between the boiling point temperatures of the pure solvent drop and the polymer solution drop in a dynamic regime. This temperature difference is the starting point for determining solvent activities. There is an analogy to the boiling point elevation in thermodynamic equilibrium. Therefore, in the steady state period of the experiment, the following relation can be applied if one assumes that the steady state is sufficiently near the vapor-liquid equilibrium and linear non-equilibrium thermodynamics is valid ... 

Under these conditions, studying quantities such as Cp becomes problematic because their significance changes at an T interval near Tg, at which point the system falls out of equilibrium, and we are then faced with the problem of how to interpret a quantity such as Cp in a nonequilibrium state. How a measurement is performed affects the measured values. Thus, if we want to study well-defined equilibrium quantities in the liquid state, and still learn something about the GT, then we must study their dynamic behaviors. Note that in this situation these measurements are not in the linear-response regime (i.e., under nonequilibrium and nonlinear conditions). 

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