Mesoscopic nonequilibrium thermodynamics

Rubi and Perez-Madrid (2001) derived some kinetic equations of the Fokker-Planck type for polymer solutions. These equations are based on the fact that processes leading to variations in the conformation of the macromolecules can be described by nonequilibrium thermodynamics. The extension of this approach to the mesoscopic level is called the mesoscopic nonequilibrium thermodynamics, and applied to transport and relaxation phenomena and polymer solutions (Santamaria-Holek and Rubi, 2003). 

The concepts of meso-thermodynamics can be extended to some non-equilibrium phenomena. In particular, like the thermodynamic properties, transport coefficients, such as the diffusion coefficient, become spatially dependent at meso-scales. Moreover, away from equilibrium, generic long-range correlations emerge even in simple molecular fluids, making the famous concept of local equilibrium, at least, questionable. In this section we focus only on one application of mesoscopic nonequilibrium thermodynamics in fluids fluid phase separation. 

A different formahsm in which the diffusion of a Brownian gas in a fluid under stationary and non-stationary flow has been analyzed is mesoscopic nonequilibrium thermodynamics (MNET) (Perez-Madrid, 1994 Rubi Mazur, 1994 Rubi P rez-Madrid, 1999). This theory uses the nonequUibrium thermodynamics rules in the phase space of the system, and allows to derive Fokker-Planck equations that are coupled with the thermodynamic forces associated to the interaction between the system and the heat bath. The effects of this coupling on system s dynamics are not obvious. This is the case of Brownian motion in the presence of flow where, as we have discussed previously, both the diffusion coefficient and the chemical potential become modified by the presence of flow (Reguera Rubi, 2003a b Santamaria Holek, 2005 2009 2001). 

Using Eq. (47) and the rules of mesoscopic nonequilibrium thermodynamics, the resulting Fokker-Planck equation for the probability distribution is... 

Santamaria Holek, L, e. a. (2001). Diffusion in stationary flow from mesoscopic nonequilibrium thermodynamics, Phy. Rev. E 63 051106. 

The review is organized as follows In Section 2 we present the multiscale equilibrium thermodynamics in the setting of contact geometry. The time evolution (multiscale nonequilibrium thermodynamics) representing approach of a mesoscopic level LmeSoi to the level of equilibrium thermodynamics Leth is discussed in Section 3. A generalization in which the level Leth is replaced by another mesoscopic level LmesoZ is considered in Section 4. The notion of multiscale thermodynamics of systems arises in the analysis of this type of time evolution. 

In references (Santamaria Holek, 2005 2009 2001), the Smoluchowski equation was obtained by calculating the evolution equations for the first moments of the distribution function. These equations constitute the hydrodynamic level of description and can be obtained through the Fokker-Planck equation. The time evolution of the moments include relaxation equations for the diffusion current and the pressure tensor, whose form permits to elucidate the existence of inertial (short-time) and diffusion (long-time) regimes. As already mentioned, in the diffusion regime the mesoscopic description is carried out by means of a Smoluchowski equation and the equations for the moments coincide with the differential equations of nonequilibrium thermodynamics. 

In this chapter, we have discussed three important aspects of thermodynamics in the presence of flow. By considering different points of view ranging from kinetic and stochastic theories to thermod5mamic theories at mesoscopic and macroscopic levels, we addressed the effects of flow on transport coefficients like diffusion and viscosity, constitutive relations and equations of state. In particular, we focused on how some of these effects may be derived from the framework of mesoscopic nonequilibrium thermod5mamics. 

These theories are examples of mesoscopic or macroscopic models that lead to closed-form constitutive equations. Furthermore, they can all be described in the context of the single-generator bracket [175] or the GENERIC [167] formalisms of nonequilibrium thermodynamics. 

Another approach towards a thermodynamics of steady-state systems is presented by Santamaria-Holek et al.193 In this formulation a local thermodynamic equilibrium is assumed to exist. The probability density and associated conjugate chemical potential are interpreted as mesoscopic thermodynamic variables from which the Fokker-Planck equation is derived. Nonequilibrium equations of state are derived for a gas of shearing Brownian particles in both dilute and dense states. It is found that for low shear rates the first normal stress difference is quadratic in strain rate and the viscosity is given as a simple power law in the strain rate, in contrast to standard mode-coupling theory predictions (see Section 6.3). 

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