Stability Criteria, Instability, and Thermodynamic Theory

On the other hand, an open system possesses flow terms in the entropy variations (dgS) which relate to exchanges of energy and matter with the outside world. These terms do not have any definite sign—in 

The local equilibrium assumption was the basis on which the Brussels school developed a global thermodynamic theory. Use of this assumption makes possible the macroscopic evaluation of entropy production and entropy flow terms with macroscopic thermodynamic methods. The assumption states that there exists within each small mass element of the medium a state of local equilibrium for which the local entropy, s, is the same function of the local macroscopic variables as at equilibrium state (Glansdorff and Prigogine, 1971, p. 14). In other words, each small element of a system may be treated as a state near equilibrium but need not necessarily be at equilibrium. This does not mean that the system as a whole need be near equilibrium thus, neighboring local elements may differ in parameters (temperatures, chemical affinities, etc.) which are reflected in the function describing their local entropy. The additional assumption is made that the sum of the criteria of local stability for each element corresponds to the global stability criterion for the whole system. 

for each local equilibrium element, the entropy s is a function of local macroscopic variables, and the integration of the local entropy terms gives the entropy terms for the system. If we expand entropy about its reference value at steady state, Sq, taking the limit of small fluctuations (8) we obtain  

Higher-order terms of the expansion are not retained, since we are dealing only with small fluctuations. Since Sq is time independent, the variation of entropy with time is given by  

For systems both at equilibrium and removed from equilibrium, Einstein s theory of fluctuations indicates that the curvature of entropy (8 s) is a quantity which should be considered when seeking to describe the stability of a system. First-order terms, according to the stability theory of differential equations, are considered as general equilibrium conditions which reflect the irreversibility of the processes involved. Second-order terms, however, measure the ability to reestablish given boundary conditions when perturbations or fluctuations occur, and it is the sign of the curvature of entropy (8 s) which dictates stability for thermodynamic equilibrium. In the range for which the local equilibrium assumption remains valid. 

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