Equilibrium and Stability via Maximum Entropy

The first row of boxes shown in Fig. 3.1 depicts a number of identical systems differing only in their internal energies, Ey, volumes, Vy, and mass contents, n . The boundaries of the systems allow the exchange of these quantities between the systems upon contact. The second row of boxes in Fig. 3.1 illustrates this situation. All (sub-)systems combined form an isolated system. We ask the following question What can be said about the quantities Xy, where x represents E, V or n, after we bring the boxes into contact and allow the exchanges to occur  

According to our experience the exchange is an irreversible spontaneous process and therefore relation Eq. (1.50) applies to the entropy of the overall system. We can expand the entropy of the combined systems, S, in a Taylor series in the variables Ey, Vy, and Hy with respect to its maximum, i.e. 

Hentschke, Thermodynamics. Undergraduate Lecture Notes in Physics, 

In this sense we shall use the expression of equilibrium. For all practical purposes equilibrium is understood in a local sense, i.e. the time scale underlying the process of interest is much shorter than the time scale underlying other processes influencing the former. In the case at hand equilibrium means that all variables, Ey, Vv, and Hv, have assumed the values V°, and n% corresponding to maximum entropy. However, we may impose deviations from these values in each subsystem, AXv, as illustrated in the bottom part of Fig. 3.1. The long dashed line indicates the equilibrium value(s), which is the same in all (identical) systems. The short dashed lines indicate the imposed deviations from equilibrium in each system, Ax . Because the whole system is isolated, we have the condition(s) 

Equation (3.1) is nothing but a Taylor expansion of S to second order in the Ax , which we can freely and independently adjust except for the condition(s) Eq.(3.2). 

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