Noncanonical equilibrium, dynamic

VIII. Dynamic Versus Thermodynamic Approach to Noncanonical Equilibrium... 

VIII. DYNAMIC VERSUS THERMODYNAMIC APPROACH TO NONCANONICAL EQUILIBRIUM... 

In this section we plan to discuss the dynamic versus the thermodynamic approach to noncanonical equilibrium. We plan to show that the thermodynamic method, in the form of an information approach, does not yield satisfactory result. The dynamic approach is more promising than the information approach, even if it involves difficult conceptual problems that have not yet been solved. We argue, however, that the strange response of fluctuations with inverse power law correlation to perturbation, illustrated in Section VII, might turn out to be useful in settling these difficult conceptual problems. 

However, this derivation of Levy statistics does not seem to reproduce satisfactorily all the properties that emerged from the illustration of superdiffusion processes of Section VI. We think that this formal way of proceeding does not address directly what seems to us a still unsolved issue of modem physics, this being the connection between dynamics and thermodynamics. As pointed out by the authors of Ref. 80, a satisfactory connection between a noncanonical form of equilibrium and thermodynamics should satisfactorily address the problem of thermal contact between a system at equilibrium in a canonical state and a system at equilibrium in a noncanonical condition. In other words, we have to establish which is the temperature of Levy system, if we believe that this form of noncanonical equilibrium is compatible with thermodynamic equilibrium. 

Before ending this section, we want to notice that a completely dynamical approach to the noncanonical equilibrium, of the WS form, is still missing. The adoption of the same dynamical models as those used in Ref. 49 to establish the existence of a form of linear response violating the Green-Kubo theory seems to be impractical for numerical reasons. A more accessible dynamic model was studied in Ref. 89, resulting, however, in an unstable form of noncanonical equilibrium, although characterized by an extremely extended lifetime. With all these warnings in mind, we reach the conclusion that a noncanonical form of equilibrium of the WS type cannot be ruled out. 

---