Extended irreversible thermodynamic

D. Jou, Extended Irreversible Thermodynamics, Springer-Verlag, New York (1996). 

D. Jou, J. Casas-Vazquez, G. Lebon, Extended Irreversible Thermodynamics, third enlarged ed. (2001), Springer, Berlin, 1993. 

After 1960 two new approaches to the thermodynamics of irreversible processes emerged, rational thermodynamics and extended irreversible thermodynamics [12], The latter formulation was based on similar assumptions to that of Natanson. 

Jou, D., Casas-Vazques, J., Lebon, G. Extended Irreversible Thermodynamics, 2nd edn. Springer, BerUn (1993). 2001 (3rd edition)... 

Other types of such principles were also proposed phenomenological relations between flows and forces in classical irreversible thermodynamics [10-19], the use of pressure (flow of momentum) as independent variables in equihbrium thermodynamics and more generally the use of flows as independent variables in extended irreversible thermodynamics [20-23]. Cf. also internal variables in Sect. 2.3. 

We call the hyperbolic system (2.17) and (2.18) a reaction-Cattaneo system. Eu and Al-Ghoul have derived such systems from generalized hydrodynamic theory [9, 7, 8, 6]. Reaction-Cattaneo systems can also be obtained from extended irreversible thermodynamics [223], see for example [282]. If we differentiate (2.17) with respect to t and (2.18) with respect to x and eliminate mixed second derivatives, we obtain the so-called reaction-telegraph equation. 

It should be emphasized that the above development is a lowest-order development in that only the first corrections to the local equilibrium values of the internal energy and heat capacity have been used, This is consistent with current practice in fluid dynamics and nonequilibiium thermodynamics. Further refinements can be developed if it is desired to make contact with some of the recent work in extended irreversible thermodynamics [36, 37]. 

Subsequent efforts have been made to extend the domain of validity of nonequilibrium thermodynamics by incorporating newer concepts of extended irreversible thermodynamics (BIT) and non-equilibrium molecular dynamics (NEMD). However, these are limited applicability as discussed in Appendices I-III of Part One. 

It may be noted that entropy is not a function of gradients in this formalism [1]. Gradients are taken into consideration in extended irreversible thermodynamics as discussed in Appendix II. 

Complex systems such as solutions of macromolecules, magnetic hysteresis bodies, visco-elastic fluids, polarizable media require some extra variables in the fundamental equation of Gibbs. Dissipative fluxes (heat, diffusion, viscous pressure tensor and viscous pressure) are included in the Gibbs function in new formalism. In the formalism of extended irreversible thermodynamics (EIT), the dissipative fluxes are the independent variables in addition to classical variables of thermostatics [1]. 

Thus, in the formalism of extended irreversible thermodynamics by inclusion of gradients in the basic formalism of linear thermodynamics of irreversible processes, a small correction in the local entropy due to flow appears. With this modification, the new formalism can be applied to the systems such as shock waves, where there are large gradients. 

Here T is the local-equilibrium temperature. In extended irreversible thermodynamics fluxes are independent variables. The kinetic temperature associated to the three spatial directions of along the flow, along the velocity gradient, and perpendicular to the previous to directions may be different from each other. To define temperature from the entropy is the most fundamental definition, and the nonequilibrium temperature may come from the derivative of a nonequilibrium entropy du/dS) -p. Effective nonequilibrium temperature may be defined from the fluctuation-dissipation theorem relating response function and correlation function. 

The different temperatures may be related amongst themselves and the knowledge of one of the different temperatures allows us to obtain the values of the other. They do not coincide because the different temperatures are related to different aspects of the system. Extended irreversible thermodynamics temperature depends not only on the energy but also on the energy flux. Therefore, the concept of temperature has open... 

Based on the extended irreversible thermodynamics, the Gibbs equation for a simple single-component fluid in the presence of a viscous pressure tensor P (up to the second order in P ) is... 

Jou, D., Extended Irreversible Thermodynamics, 1996, New York Springer-Verlago. 

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