Nonequilibrium thermodynamics local equilibrium assumption

The theory treating near-equilibrium phenomena is called the linear nonequilibrium thermodynamics. It is based on the local equilibrium assumption in the system and phenomenological equations that linearly relate forces and flows of the processes of interest. Application of classical thermodynamics to nonequilibrium systems is valid for systems not too far from equilibrium. This condition does not prove excessively restrictive as many systems and phenomena can be found within the vicinity of equilibrium. Therefore equations for property changes between equilibrium states, such as the Gibbs relationship, can be utilized to express the entropy generation in nonequilibrium systems in terms of variables that are used in the transport and rate processes. The second law analysis determines the thermodynamic optimality of a physical process by determining the rate of entropy generation due to the irreversible process in the system for a required task. 

Sect. 2.10.5 for further details). For Kn < 0.01 the medium can be considered as a continuum and the fluid dynamic transport equations are valid. In the case of liquids, the break-down of the continuum hypothesis manifests in anomalous diffusion mechanisms [64]. Moreover, the local equilibrium assumption implies that the thermodynamic formulas derived from classical equilibrium thermodynamics may be applied locally for nonequilibrium systems [13, 88, 89]. When this is done, we have established a fluid dynamic continuum theory in which all intensive thermodynamic variables like T, p become functions of position r and time t thus T(t, r), pit, r) [88, 89]. The extensive variables like S, H are replaced by mass specific variables that also become functions of position r and time t thus s(t, r), h(t, r). 

At equilibrium, or in the linear nonequilibrium domain, equation (9) is a direct result of the second law of thermodynamics, and inequality (8) is derived as a sufficient criterion for stability. In the nonlinear range of nonequilibrium thermodynamics, as long as the local equilibrium assumption remains valid, inequality (8) is assumed and becomes the starting point from which equation (9) is derived as a sufficient stability criterion. Far from equilibrium, if a(82s)... 

The kinetic theory leads to the definitions of the temperature, pressure, internal energy, heat flow density, diffusion flows, entropy flow, and entropy source in terms of definite integrals of the distribution function with respect to the molecular velocities. The classical phenomenological expressions for the entropy flow and entropy source (the product of flows and forces) follow from the approximate solution of the Boltzmann kinetic equation. This corresponds to the linear nonequilibrium thermodynamics approach of irreversible processes, and to Onsager s symmetry relations with the assumption of local equilibrium. 

The second fact that is implicit in macroscopic or continuum laws is the idea of local thermodynamic equilibrium. For example, when we write the Fourier law of heat conduction, it is inherently assumed that one can define a temperature at any point in space. This is a rather severe assumption since temperature can be defined only under thermodynamic equilibrium. The question that we might ask is the following. If there is thermodynamic equilibrium in a system, then why should there be any net transport of energy Thus, we implicitly resort to the concept of local thermodynamic equilibrium, where we assume that thermodynamic equilibrium can be defined over a volume which is much smaller than the overall size of the system. What happens when the size of the object becomes on the order of this volume Obviously, the macroscopic or continuum theories break down and new laws based on nonequilibrium thermodynamics need to be formulated. This chapter focuses on developing more generalized theories of transport which can be used for nonequilibrium conditions. This involves going to the root of the macroscopic or continuum theories. 

These observations pose an interesting question because the results reported in Refs. (Santamaria Holek, 2005 2009 2001) were obtained under the assumption of local equilibrium in phase space, that is, at the mesoscale. It seems that for systems far from equilibrium, as those reported in (Sarman, 1992), the validity of the fundamental h5q)othesis of linear nonequilibrium thermodynamics can be assumed at the mesoscale. After a reduction of the description to the physical space, this non-Newtonian dependedence of the transport coefficients on the shear rate appears. This point will be discussed more thoroughly in the following sections when analyzing the formulation of non-Newtonian constitutive equations. 

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