When aiming at the formulation of a complete system of balance and constitutive equations one is immediately faced with fundamental issues of nonequilibrium thermodynamics. If a derivation of the constitutive equations of continuum mechanics from kinetic theory or molecular dynamics is desired then one needs also the background of nonequilibrium statistical mechanics. In this context one has to deal with fundamental problems concerning the general formulation of time-evolution equations on different levels and the passage from a given to a more coarse-grained level. We have recently developed a general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) by empirical arguments and by identifying the common structure of successful models in nonequilibrium thermodynamics. A derivation of GENERIC from Hamilton's equations of motion by means of projection operators leads to microscopic expressions for the four building blocks occurring in GENERIC.
Many important examples of nonequilibrium systems have been expressed in the GENERIC form: hydrodynamics, polymer kinetic theory (including hydrodynamic interaction, rigid constraints, reptation models, and polymer heat conductivity), polymer blend rheology (Doi-Ohta model), nonisothermal rheology, chemical reactions, and Boltzmann’s equation (highly nonlinear, no local equilibrium). New results include: generalized reptation models, modifications of the pom-pom model, a consistency check for moment closure approximations for liquid crystal polymers, equations for relativistic hydrodynamics and cosmology, and improved versions of Dissipative Particle Dynamics (DPD) and Smoothed Particle Hydrodynamics (SPH). Moreover, the bracket formalism of Beris and Edwards, which includes the linear thermodynamics of irreversible processes, and the matrix model of Jongschaap have been reproduced. The relationships to extended irreversible thermodynamics, modern classical irreversible thermodynamics, the invariant manifold method and rational thermodynamics have been elaborated.
We use fortran, c++ and c-compilers and ksh and perl-scripts to run our jobs. Results are animated using SciAn, inventor, AVS and showcase (Silicon Graphics). Sometimes we make use of the NAG-libraries, postprocessing is achieved with the help of the symbolic language such as Mathematica. Our parameter space for the problems at hand is large enough to allow for non-parallel computations with maximum efficiency. For the asgard architecture, we reformulate our problems to rely on low communication speed between processors.
During the last 15 years a bracket formalism of dissipative continuum physics has been developed which resulted in a formulation which is shortly denoted as GENERIC. GENERIC has been applied to different problems of continuum thermodynamics, often in this way, that a well-known problem was reformulated in GENERIC formalism. To learn some more about the GENERIC procedure we consider a gas which is contained in a cylinder closed by a piston moving with friction. We treat this simple discrete system with rational non-equilibrium thermodynamics by using Liu's procedure and, for comparison, also with the GENERIC formalism. Both different procedures yield in the same results, especially in the same entropy production. Differences, similarities and fundamental presuppositions of both formalisms are discussed and compared.
Physica A 285 (2000) 448-466.
We recognize some universal features of macroscopic dynamics describing the approach of a well-established level of description (that is, successfully tested by experimental observations) to equilibrium. The universal features are collected in a general equation for the nonequilibrium reversible-irreversible coupling (abbreviated as GENERIC). In this paper we formulate a GENERIC, derive properties of its solutions, and discuss their physical interpretation. The relation of the GENERIC with thermodynamics is most clearly displayed in a formulation that uses contact structures. The GENERIC is also discussed in the presence of noise. In applications we either search for new governing equations expressing our insight into a particular complex fluid or take well-established governing equations and cast them into the form of a GENERIC. In the former case we obtain the governing equations as particular realizations of the GENERIC structure; in the latter case we justify the universality of the GENERIC and derive some properties of solutions. Both types of applications are discussed mainly in the following paper [Phys. Rev. E 56, 6633 (1997)].
Phys. Rev. E 56 (1997) 6620-6632.
For a number of well-known time-evolution equations for nonequilibrium systems we extract a common structure from these equations, referred to as a general equation for the nonequilibrium reversible-irreversible coupling (GENERIC). This fundamental structure is determined by four building blocks, two "potentials" (total energy and entropy) and two "matrices." We illustrate for various examples how three of the four building blocks can be determined in a rather straightforward manner so that, within our GENERIC approach to nonequilibrium dynamics, understanding of a given nonequilibrium system is reduced to determining a single "metric matrix," or friction matrix, either empirically or by more microscopic considerations. In formulating nonisothermal polymer kinetic theories, we show how the general structure provides a clear distinction between spring potentials of energetic and entropic origins in the various time-evolution equations.
Phys. Rev. E 56 (1997) 6633-6655.
By using projection operator methods, we derive the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) that was recently obtained by empirical arguments. We find microscopic expressions for the building blocks of GENERIC, and we generally derive the rules for passing from any given level of description to a more macroscopic one.
Phys. Rev. E 57 (1998) 1416-1420.
In recent years, several seemingly distinct approaches have been developed which aim to describe the mechanics and thermodynamics of complex fluids under dynamical conditions. Two of these approaches of decidedly different flavor are the Matrix Model of thermodynamically driven systems developed by Jongschaap and the GENERIC formalism of Grmela and Öttinger. Herein, we examine the interrelationships between these two alternate approaches on an abstract level and work out direct connections for two specific examples of thermodynamically driven systems: a gas housed in a cylinder closed by a piston subject to external force and potential fields, and a dilute solution of Hookean dumbbells in a flow field. It is demonstrated that the Matrix Model derives from the GENERIC formalism when the global thermodynamic system is split into two parts: a smaller open thermodynamic system with fewer internal variables, and its environment composed of external driving forces which are determined from the variables which are neglected in the smaller system. Thus the Matrix Model provides an explicit answer to the question of how to handle driven systems within the more global GENERIC formalism. It is also shown that the Matrix Model requires a prerequisite knowledge of the laws of motion for a given system, whereas these are explicitly imbedded in the GENERIC structure.
J. Non-Equilib. Thermodyn. 22 (1997) 356-373.
We express Boltzmann's kinetic equation in the form of the recently proposed general equation for the nonequilibrium reversible-irreversible coupling (GENERIC). This GENERIC formulation demonstrates that no dissipative potential is required for representing Boltzmann's kinetic equation in a general framework for nonequilibrium systems.
J. Non-Equilib. Thermodyn. 22 (1997) 386-391.
The model of Doi and Ohta for multiphase flow is developed and analyzed from a thermodynamic perspective using the GENERIC formalism. A procedure for obtaining a set of "thermodynamically consistent" transport equations is illustrated. The results demonstrate that the Doi-Ohta model is thermodynamically consistent, and is thus a valid set of transport equations for multiphase flow. Steady state results are compared against "exact" simulation results to assess the model's accuracy. The derivation demonstrates the ability of the GENERIC formalism to improve the theoretical basis of models for transport in complex fluids.
AIChE Journal 45 (1999) 1169-1181.
In the context of an imperfect fluid with heat flow we show that a recently developed formalism for nonequilibrium systems (GENERIC: general equation for the nonequilibrium reversible-irreversible coupling) is compatible with the structure of special relativity. An additional generalized thermal force variable related to the heat flux needs to be introduced. By doing so, common features of GENERIC and extended irreversible thermodynamics are revealed, and a new set of equations for relativistic imperfect fluids is formulated. These equations are compared to the classical first-order theory of Eckart and the second-order theory of Israel. The GENERIC generators of reversible and irreversible dynamics, energy and entropy, occur as the time components of the energy-momentum tensor and the entropy current four-vector, respectively.
Physica A 259 (1998) 24-42.
A detailed comparison is made between single and double generator formalisms for the thermodynamics and mechanics of complex fluids, expressed in either bracket or operator form. In the first part of this paper (Part I), we found a complete equivalence between single and double generator formalisms applied to macroscopic systems with respect to the physics described and the practical utility of the formalisms. For microscopic systems the conclusion is different: the double generator formalism is more natural to implement, and for some physical systems may be absolutely essential to the system description. As in Part I, the interrelationships between these two alternate approaches are examined and direct connections are determined for some specific examples of fluid systems: a nonisothermal dilute polymer solution, with and without a polymer contribution to the heat transport, and Boltzmann's kinetic equation. The splitting of the GENERIC dissipation matrix into mechanical and thermodynamic submatrices in the linear regime, as described in Part I, remains valid at the microscopic level. Furthermore, this splitting allows for the mathematical statement of a double generator dissipation bracket, which may subsequently be transformed to one involving only a single generator for many systems of interest. The Boltzmann equation appears to require the use of two generators since no thermodynamic potential can be defined for the system.
J. Non-Equilib. Thermodyn. 23 (1998) 334-350.
We formulate a set of Lorentz-covariant equations for an imperfect fluid with viscosity, dilatational viscosity, and anisotropic thermal conductivity that possess the full GENERIC structure of nonequilibrium thermodynamics. The GENERIC structure, which includes and goes beyond prior nonequilibrium generalizations of the second law of thermodynamics, is shown to provide a guideline for modifying previous phenomenological or kinetic-theory based equations of extended relativistic hydrodynamics. In the nonrelativistic limit, we discuss the form of the equations for viscous and viscoelastic fluids with anisotropic heat conduction.
Physica A 254 (1998) 433-450.
We formulate a set of equations for a self-gravitating imperfect fluid that satisfies all the principles of both general relativity and nonequilibrium thermodynamics, where the latter are condensed in the covariant version of the recently proposed general equation for the nonequilibrium reversible-irreversible coupling (GENERIC). In doing so, Einstein’s field equation is supplemented by fundamental and clearly structured transport equations for the sources of gravitational fields. The GENERIC framework determines the selection of the appropriate variables and the structure of the field equations compatible with the fundamental laws of thermodynamics. A nonzero cosmological constant cannot be ruled out by thermodynamic consistency criteria. In order to discuss the relationship to previous approaches, the simplified equations for bulk viscous cosmology are presented in some detail.
Phys. Rev. D 60 (1999) 103507.
Three different approaches to the relativistic thermodynamics of imperfect fluids are compared: the second-order causal theory of Israel and Stewart (IS), the phenomenological extension of the IS theory proposed by Maartens and Méndez (MM), and the recently developed GENERIC formalism of Grmela and Öttinger. All theories are applied to the case of dissipative cosmology, with bulk viscosity as the only dissipative phenomenon. The MM theory as well as GENERIC give an upper bound on the bulk viscous stress, whereas there is none in the IS theory. In a flat Robertson-Walker universe, the relationship between the different approaches is illustrated for the special case of a relativistic Boltzmann gas. Far away from equilibrium we find qualitatively different behavior, indicating that care should be taken when using the IS theory in this regime.
Phys. Rev. D 61 (2000) 023510.
The thermodynamics and mechanics of nonisothermal polymeric fluids are examined within the auspices of a new methodology wherein the laws of physics and principles of mechanics which are applicable to these thermodynamic systems are imbedded in a definite mathematical structure of a general, abstract equation. Such a concept allows new insight to be obtained concerning some aspects of nonisothermal flows of polymeric fluids, and permits a consistent expression and interpretation of other thermodynamic theories for these systems which have been developed over the past forty years. A major portion of this article is devoted to demonstrating the above statements, and in so doing some common misconceptions occurring in a significant fraction of the literature regarding this subject are exposed. The definite mathematical structure of the new methodology permits the thermodynamically consistent generalization of isothermal, incompressible models of polymeric fluids to nonisothermal, compressible conditions. Doing thus reproduces, corrects, and extends nonisothermal models which have been developed over the years, and also allows for simpler (but equivalent) representations of these models in terms of alternate variables with a clearer connection to the microstructure of the material than the stress tensor and heat flux vector fields. Furthermore, a generalization of the GENERIC structure is proposed that accomodates interactions between phenomena of differing parities, which impose antisymmetry upon the corresponding elements of the dissipative operator matrix.
Rheol. Acta 38 (1999) 117-136.
The Poissonian character of the reversible part of non-equilibrium dynamics is exploited here in order to determine a dynamically-consistent expression for a reptation model without the independent alignment assumption. It is shown that the previously proposed form by Doi and Edwards for such a model is compatible with the GENERIC formalism of nonequilibrium thermodynamics only after two changes are made: A) the production term in the evolution equation involves an average of the orientation dyadic uu over the entire internal phase space, and B) the extra stress tensor involves an additional term, qualitatively different from the one representing the original Doi-Edwards expression. The predictions of the new model for the stress after double shear-strain with flow reversal are shown to be realistic, demonstrating irreversibility effects, in contrast to the model with the independent alignment approximation.
J. Chem. Phys. 110 (1999) 6593-6596.
GENERIC is reviewed not only as a new general framework for modeling nonequilibrium systems, but also as a new way of thinking about nonequilibrium dynamics. This unified framework of nonequilibrium thermodynamics is shown to be deeply rooted in the ample accumulated experience with nonequilibrium systems and, provided that state variables with slow and fast time-evolution can be separated, the framework can actually be derived. In view of its natural capability of modeling systems on different levels of description, GENERIC is ideal for the highly topical attempts of "bridging scales" in science and engineering. The practical usefulness of GENERIC as a powerful tool in the phenomenological and structure-guided modeling of complex fluids is illustrated through two examples.
Applied Rheology 9 (1999) 17-26.
Starting from the quantum description of isolated systems on the microscopic level we derive the two-generator formulation of nonequilibrium thermodynamics (GENERIC) by means of the projection-operator technique. As a generalized canonical ensemble is employed, we obtain a convenient starting-point for practical calculations in nonequilibrium thermodynamics, in particular, in the classical limit. All dynamical material properties are contained in a canonical nonequilibrium correlation. However, the generalized canonical approach is inappropriate for systems with large fluctuations; possible steps towards a suitable generalization for quantum systems are discussed.
Phys. Rev. E62 (2000) 4720-4724.
Several seemingly distinct formalisms have been developed recently as an alternative to the modern conventional approach for describing the mechanics and thermodynamics of complex fluids under dynamical conditions. Two of these approaches for isolated thermodynamic systems are the single generator Bracket Formalism and the more recent GENERIC Formalism incorporating a second generating functional. The interrelationships between these two alternate approaches are examined on an abstract level and direct connections are determined for some specific examples of fluid systems: a compressible, nonisothermal, isotropic fluid, an arbitrary fluid with an unspecified internal variable, and a chemically reactive fluid system. In so doing, new physical insight is obtained regarding the thermodynamic origins of key elements of each formalism. For purely macroscopic systems, it is shown that the GENERIC degeneracy condition on the dissipative metric matrix ensures global energy conservation, produces a symmetric metric matrix, and allows this matrix to be split into purely thermodynamic and kinematical components for systems with linear dissipation. The last realization leads to the natural driving forces expressed in bracket form being constrained Volterra derivatives of the entropy functional with respect to the system variables. The use of this concept in the Bracket Formalism provides a symmetric, bilinear dissipation bracket without the need for an explicit energy correction term.
J. Non-Equilib. Thermodyn. 23 (1998) 301-333.
Focusing on the atomistic approach to the GENERIC framework of nonequilibrium thermodynamics, we first summarize the atomistic expressions for the inputs of this framework as previously derived by standard projection-operator methods. We then review to what extent GENERIC can be derived from atomistic principles. In the context of hydrodynamics, we illustrate how one can actually evaluate the atomistic expressions. This example shows what steps are involved in such calculations, and in which direction the formalism needs to be generalized in order to obtain a powerful tool for atomistic calculations for nonequilibrium systems of practical importance.
We discuss under which conditions the symmetries of atomistic systems are reflected on coarse-grained levels of description. A generalized canonical ensemble is constructed that possesses the proper symmetries even for representations of the symmetry group on the coarse-grained state space that are nonlinear or not induced by the atomistic symmetry transformations. Starting from such a relevant ensemble, the symmetries of the underlying systems are taken into account in deriving the two-generator formulation of nonequilibrium thermodynamics (GENERIC) by means of the standard projection-operator technique. The incorporation of Galilean invariance and its relation to the local-equilibrium assumption is illustrated for hydrodynamics and briefly discussed for complex fluids.