IWNET12
IWNET12
Quasi-linear versus potential-based formulations of force-ux relations
and the GENERIC for irreversible processes: comparisons and exam-
ples
Markus Hütter1, Bob Svendsen2
1 Materials Technology (MaTe), Department of Mechanical Engineering, Eindhoven University of Technology,
P. O. Box 513, NL5600 MB Eindhoven, The Netherlands
2 Material Mechanics, JARA, RWTH Aachen, Germany; Max-Planck Institute for Iron Research, Germany
Abstract
An essential part in modeling out-of-equilibrium dynamics is the formulation of irreversible dynamics. In the
latter, the major task consists in specifying the relations between thermodynamic forces and uxes. In the
literature, mainly two distinct approaches are used for the specication of force-ux relations. On the one hand,
quasi-linear relations are employed, which are based on the physics of transport processes and uctuation-
dissipation theorems [1,2]. On the other hand, force-ux relations are also often represented in potential form
with the help of a dissipation potential [3]. We address the question of how these two approaches are related.
The main result of this presentation states that the class of models formulated by quasi-linear relations is larger
than what can be described in a potential-based formulation. While the relation between the two methods is
shown in general terms, it is demonstrated also with the help of three examples: (i) heat conduction in rigid
bodies, (ii) homogeneous chemical reactions, and (iii) slippage in complex uids. In particular, whereas the
irreversible processes (i) and (ii) are dissipative, (iii) is not. Consequently, conditions for the existence of a
dissipation potential for the models of (i) and (ii) can be formulated. On the other hand, the potential for the
model of (iii) is identically zero. More generally, this is the case for the model of any irreversible process which
results in no dissipation. The nding that quasi-linear force-ux relations are more general than dissipation-
based ones also has ramications for the General Equation for Non-Equilibrium Reversible-Irreversible Coupling
(GENERIC: e.g., [4-6]). This framework has been formulated and used in two dierent forms, namely a quasi-
linear [5,6] and a dissipation-potential based [7-9] form, respectively, relating the irreversible evolution to the
entropy gradient. It is found that also in the case of GENERIC, the quasi-linear representation encompasses
a wider class of phenomena as compared to the dissipation-based formulation. Furthermore, it is found that a
potential exists for the irreversible part of the GENERIC if and only if one does for the underlying force-ux
relations.
References
[1] de Groot, S. R., Mazur, P., 1962. Non-equilibrium Thermodynamics. North-Holland.
[2] Lifshitz, E. M., Pitaevskii, L. P., 1981. Physical Kinetics. Vol. 10, Landau and Lifshitz Series on Theoretical
Physics. Pergamon Press.
[3] ’ilhavý, M., 1997. The Mechanics and Thermodynamics of Continuous Media. Springer.
[4] Grmela, M., Öttinger, H. C., 1997. Phys. Rev. E, 56, 66206632.
[5] Öttinger, H. C., Grmela, M., 1997. Phys. Rev. E, 56, 66336655.
[6] Öttinger, H. C., 2005. Beyond Equilibrium Thermodynamics. Wiley Interscience Publishers.
[7] Grmela, M., 2010. Adv. Chem. Engin., 39, 75129.
[8] Grmela, M., 2010. J. Non-Newtonian Fluid Mech., 165, 980986.
[9] Mielke, A., 2011. Cont. Mech. Thermodyn., 23, 233256.
E-mail: M.Huetter@tue.nl

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