Thermodynamic noncanonical equilibrium

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. 

At least in the traditional domains of chemical engineering and in the traditional core of instructions that chemical engineers receive during their education, fluid mechanics (transport phenomena) has played a key role. Also one of the principal motivations for creating nonequilibrium thermodynamics was an attempt to make fluid mechanics manifestly compatible with equilibrium thermodynamics. Even the noncanonical Hamiltonian structures that play such an important role in the multiscale nonequilibrium thermodyna mics presented in Section 3 have been first discovered... 

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