General Equation For The Nonequilibrium Reversible-irreversible Coupling Research Articles

Thermodynamically admissible 13-moment equations

Grad's 13-moment equations describe transport in mildly rarefied gases well, but are not properly embedded into nonequilibrium thermodynamics since they are not accompanied by a formulation of the second law. In this work, the Grad-13 equations are embedded into the framework of GENERIC (general equation for the nonequilibrium reversible–irreversible coupling), which demands additional contributions in the equations to guarantee thermodynamic structure. As GENERIC building blocks, we use a Poisson matrix for the basic convection behavior and antisymmetric friction matrices to correct for additional convective transport terms. The ensuing GENERIC-13 equations completely match the Grad-13 equations up to second-order terms in the Knudsen number and fulfill all thermodynamic requirements.

A Case Study of Non-Fourier Heat Conduction Using Internal Variables and GENERIC

Abstract Applying simultaneously the methodology of non-equilibrium thermodynamics with internal variables (NET-IV) and the framework of General Equation for the Non-Equilibrium Reversible–Irreversible Coupling (GENERIC), we demonstrate that, in heat conduction theories, entropy current multipliers can be interpreted as relaxed state variables. Fourier’s law and its various extensions—the Maxwell–Cattaneo–Vernotte, Guyer–Krumhansl, Jeffreys type, Ginzburg–Landau (Allen–Cahn) type and ballistic–diffusive heat conduction equations—are derived in both formulations. Along these lines, a comparison of NET-IV and GENERIC is also performed. Our results may pave the way for microscopic/multiscale understanding of beyond-Fourier heat conduction and open new ways for numerical simulations of heat conduction problems.

A simple microswimmer model inspired by the general equation for nonequilibrium reversible-irreversible coupling.

A simple mean-field microswimmer model is presented. The model is inspired by the nonequilibrium thermodynamics of multi-component fluids that undergo chemical reactions. These thermodynamics can be rigorously described in the context of the GENERIC (general equation for the nonequilibrium reversible-irreversible coupling) framework. More specifically, this approach was recently applied to non-ideal polymer solutions [T. Indei and J. D. Schieber, J. Chem. Phys. 146, 184902 (2017)]. One of the species of the solution is an unreactive polymer chain represented by the bead-spring model. Using this detailed description as inspiration, we then make several simplifying assumptions to obtain a mean-field model for a Janus microswimmer. The swimmer model considered here consists of a polymer dumbbell in a sea of reactants. One of the beads of the dumbbell is allowed to act as a catalyst for a chemical reaction between the reactants. We show that the mean-squared displacement (MSD) of the center of mass of this Janus dumbbell exhibits ballistic behavior at time scales at which the concentration of the reactant is large. The time scales at which the ballistic behavior is observed in the MSD coincide with the time scales at which the cross-correlation between the swimmer's orientation and the direction of its displacement exhibits a maximum. Since the swimmer model was inspired by the GENERIC framework, it is possible to ensure that the entropy generation is always positive, and therefore, the second law of thermodynamics is obeyed.

A Data-Driven Learning Method for Constitutive Modeling: Application to Vascular Hyperelastic Soft Tissues.

We address the problem of machine learning of constitutive laws when large experimental deviations are present. This is particularly important in soft living tissue modeling, for instance, where large patient-dependent data is found. We focus on two aspects that complicate the problem, namely, the presence of an important dispersion in the experimental results and the need for a rigorous compliance to thermodynamic settings. To address these difficulties, we propose to use, respectively, Topological Data Analysis techniques and a regression over the so-called General Equation for the Nonequilibrium Reversible-Irreversible Coupling (GENERIC) formalism (M. Grmela and H. Ch. Oettinger, Dynamics and thermodynamics of complex fluids. I. Development of a general formalism. Phys. Rev. E 56, 6620, 1997). This allows us, on one hand, to unveil the true “shape” of the data and, on the other, to guarantee the fulfillment of basic principles such as the conservation of energy and the production of entropy as a consequence of viscous dissipation. Examples are provided over pseudo-experimental and experimental data that demonstrate the feasibility of the proposed approach.

Structure-preserving integrators for dissipative systems based on reversible- irreversible splitting.

We study the optimal design of numerical integrators for dissipative systems, for which there exists an underlying thermodynamic structure known as GENERIC (general equation for the nonequilibrium reversible–irreversible coupling). We present a frame-work to construct structure-preserving integrators by splitting the system into reversible and irreversible dynamics. The reversible part, which is often degenerate and reduces to a Hamiltonian form on its symplectic leaves, is solved by using a symplectic method (e.g. Verlet) with degenerate variables being left unchanged, for which an associated modified Hamiltonian (and subsequently a modified energy) in the form of a series expansion can be obtained by using backward error analysis. The modified energy is then used to construct a modified friction matrix associated with the irreversible part in such a way that a modified degeneracy condition is satisfied. The modified irreversible dynamics can be further solved by an explicit midpoint method if not exactly solvable. Our findings are verified by various numerical experiments, demonstrating the superiority of structure-preserving integrators over alternative schemes in terms of not only the accuracy control of both energy conservation and entropy production but also the preservation of the conformal symplectic structure in the case of linearly damped systems.

A GENERIC formalism for Korteweg-type fluids: I. A comparison with classical theory

We investigated a mathematical model for Korteweg-type fluids, formulated in the form of the General equation for the non-equilibrium reversible–irreversible coupling (GENERIC) as proposed by Grmela and Öttinger, in comparison with the classical work of Dunn and Serrin. Interstitial working introduced by Dunn and Serrin reappeared in the formulation and proved to be necessary if we claim that the work done by the Korteweg stress is ‘isentropic’. Indeed, the GENERIC formalism provides a fairly simple derivation of the governing equations with the interstitial working for Korteweg-type fluids. In the accompanying paper, and within this formalism, we derived a higher-order model including the second-order gradient of the density. We also discussed the relation of the interstitial working to the microforces introduced by Gurtin.

Scientific Machine Learning for Coarse-Grained Constitutive Models

We present here a review on some of our latest works concerning the development of thermodynamics-aware machine learning strategies for the data-driven construction of constitutive models. We suggest a methodology constructed upon three main ingredients. (i) the employ of manifold learning strategies to unveil the true dimensionality of data, thus pointing out the need for the definition of “internal” variables, different of the experimental ones. (ii) the process will be described by the so-called General Equation for the Non-Equilibrium Reversible-Irreversible Coupling (GENERIC). (iii) the precise form of the GENERIC terms will be unveiled by regression of data.

Kluitenberg–Verhás Rheology of Solids in the GENERIC Framework

Abstract The internal variable methodology of non-equilibrium thermodynamics, with a symmetric tensorial internal variable, provides an important rheological model family for solids, the so-called Kluitenberg–Verhás model family [Cs. Asszonyi et al., Contin. Mech. Thermodyn. 27, 2015]. This model family is distinguished not only by theoretical aspects but also on experimental grounds (see [Cs. Asszonyi et al., Period. Polytech., Civ. Eng. 60, 2016] for plastics and [W. Lin et al., Rock Engineering in Difficult Ground Conditions (Soft Rock and Karst), Proceedings of Eurock’09, 2009; K. Matsuki, K. Takeuchi, Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 30, 1993; K. Matsuki, Int. J. Rock Mech. Min. Sci. 45, 2008] for rocks). In this article, we present and discuss how the internal variable formulation of the Kluitenberg–Verhás model family can be presented in the non-equilibrium thermodynamical framework GENERIC (General Equation for the Non-Equilibrium Reversible–Irreversible Coupling) [H. C. Öttinger, Beyond Equilibrium Thermodynamics, 2005; M. Grmela, J. Non-Newton. Fluid Mech. 165, 2010; M. Grmela, H. C. Öttinger, Phys. Rev. E 56, 1997; H. C. Öttinger, M. Grmela, Phys. Rev. E 56, 1997], for the benefit of both thermodynamical methodologies and promising practical applications.

Stability analysis for a thermodynamically consistent model of relativistic fluid dynamics

In relativistic fluid mechanics, positive entropy production is known to be insufficient for guaranteeing stability. Much stronger criteria for thermodynamic admissibility have become available in nonequilibrium thermodynamics. We here perform a linear stability analysis for a model of relativistic hydrodynamics that is based on the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) framework of nonequilibrium thermodynamics. Assuming a quadratic entropy function near equilibrium, we find stability for the entire range of physically meaningful model parameters for relativistic fluid dynamics based on GENERIC. The search for thermodynamic admissibility moreover reveals a fundamental difference between liquids and gases in relativistic fluid dynamics.

GENERIC Integrators: Structure Preserving Time Integration for Thermodynamic Systems

Abstract Thermodynamically admissible evolution equations for non-equilibrium systems are known to possess a distinct mathematical structure. Within the GENERIC (general equation for the non-equilibrium reversible–irreversible coupling) framework of non-equilibrium thermodynamics, which is based on continuous time evolution, we investigate the possibility of preserving all the structural elements in time-discretized equations. Our approach, which follows Moser’s [1] construction of symplectic integrators for Hamiltonian systems, is illustrated for the damped harmonic oscillator. Alternative approaches are sketched.

Hydrodynamically Coupled Brownian Dynamics: A coarse-grain particle-based Brownian dynamics technique with hydrodynamic interactions for modeling self-developing flow of polymer solutions.

We present a novel coarse-grain particle-based simulation technique for modeling self-developing flow of dilute and semi-dilute polymer solutions. The central idea in this paper is the two-way coupling between a mesoscopic polymer model and a phenomenological fluid model. As our polymer model, we choose Responsive Particle Dynamics (RaPiD), a Brownian dynamics method, which formulates the so-called "conservative" and "transient" pair-potentials through which the polymers interact besides experiencing random forces in accordance with the fluctuation dissipation theorem. In addition to these interactions, our polymer blobs are also influenced by the background solvent velocity field, which we calculate by solving the Navier-Stokes equation discretized on a moving grid of fluid blobs using the Smoothed Particle Hydrodynamics (SPH) technique. While the polymers experience this frictional force opposing their motion relative to the background flow field, our fluid blobs also in turn are influenced by the motion of the polymers through an interaction term. This makes our technique a two-way coupling algorithm. We have constructed this interaction term in such a way that momentum is conserved locally, thereby preserving long range hydrodynamics. Furthermore, we have derived pairwise fluctuation terms for the velocities of the fluid blobs using the Fokker-Planck equation, which have been alternatively derived using the General Equation for the Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) approach in Smoothed Dissipative Particle Dynamics (SDPD) literature. These velocity fluctuations for the fluid may be incorporated into the velocity updates for our fluid blobs to obtain a thermodynamically consistent distribution of velocities. In cases where these fluctuations are insignificant, however, these additional terms may well be dropped out as they are in a standard SPH simulation. We have applied our technique to study the rheology of two different concentrations of our model linear polymer solutions. The results show that the polymers and the fluid are coupled very well with each other, showing no lag between their velocities. Furthermore, our results show non-Newtonian shear thinning and the characteristic flattening of the Poiseuille flow profile typically observed for polymer solutions.

GENERIC and Speed-Gradient Principle

Abstract In this paper we investigate the relationship between GENERIC (general equation for the nonequilibrium reversible-irreversible coupling) framework originated in nonequilibrium thermodynamics and the Speed-Gradient (SG) Principle originated in control theory. GENERIC is known as a general structure for the various time-evolution equations for nonequilibrium systems. The SG-principle is successfully used to determine dynamics of entropy-driven systems from the perspective of MaxEnt principle. Based on the SG-principle a time-evolution equations can be derived in a general way. We consider several examples of SG-principle application and show its correspondence to GENERIC framework. The relation between GENERIC and Fokker-Planck equations has already been established by H.C. Ottinger et al. The result of this paper can also be treated to demonstrate the way how Fokker-Planck equations and the SG-principle relate.

Reexamination of multi-component non-ideal polymer solution based on the general equation for nonequilibrium reversible-irreversible coupling

We reexamine nonequilibrium thermodynamics of multi-component fluids that undergo chemical reactions and reveal how to describe it in the context of GENERIC (general equation for the nonequilibrium reversible-irreversible coupling) framework. In the former part of this paper, we study monomer solutions. The independent variables are chosen so that the symmetry in the thermodynamic description of the mixed solution is kept, while avoiding the redundancy in the independent variables that describe the system. In the latter part of this paper, we apply this approach to polymer solutions. One of the species of the solution is the unreactive polymer chain represented by the bead-spring model. The polymer solution is neither dilute nor ideal. We construct the solution entropy so that the contributions from mixing and chain conformation are fully separated. The pressure tensor derived from such entropy with the help of the degeneracy requirement is slightly different from that derived previously for a dilute polymer solution. We discuss how the difference occurs.

Essential equivalence of the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) and steepest-entropy-ascent models of dissipation for nonequilibrium thermodynamics.

By reformulating the steepest-entropy-ascent (SEA) dynamical model for nonequilibrium thermodynamics in the mathematical language of differential geometry, we compare it with the primitive formulation of the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) model and discuss the main technical differences of the two approaches. In both dynamical models the description of dissipation is of the "entropy-gradient" type. SEA focuses only on the dissipative, i.e., entropy generating, component of the time evolution, chooses a sub-Riemannian metric tensor as dissipative structure, and uses the local entropy density field as potential. GENERIC emphasizes the coupling between the dissipative and nondissipative components of the time evolution, chooses two compatible degenerate structures (Poisson and degenerate co-Riemannian), and uses the global energy and entropy functionals as potentials. As an illustration, we rewrite the known GENERIC formulation of the Boltzmann equation in terms of the square root of the distribution function adopted by the SEA formulation. We then provide a formal proof that in more general frameworks, whenever all degeneracies in the GENERIC framework are related to conservation laws, the SEA and GENERIC models of the dissipative component of the dynamics are essentially interchangeable, provided of course they assume the same kinematics. As part of the discussion, we note that equipping the dissipative structure of GENERIC with the Leibniz identity makes it automatically SEA on metric leaves.

Dynamics of complex fluid-fluid interfaces

EXTENDED ABSTRACT Surface rheological properties like surface shear viscosities, or surface dilatational moduli often play an important role in the stability and dynamic behavior of emulsions, foam, biological fluids, liquid jets, coatings flows, or immiscible polymer blends [1]. This is particularly true when the interfaces in these systems have a complex microstructure, for example, when the surface active components stabilizing the interface form a 2d gel phase, a 2d glass phase, or 2d (liquid) crystalline phase. Such 2d mesophases are typically formed when interfaces are stabilized by colloidal particles, proteins, protein aggregates, proteinpolymer complexes, or amphiphilic polymers [1]. Applied deformations induce changes in the microstructure of the interface, and the resulting changes in surface rheological properties (such as surface shear thinning, shear thickening, or thixotropic behavior) affect the behavior of the multiphase system on a macroscopic scale. Most currently available constitutive models for the surface extra stress tensor either do not account for the strain (rate) dependence of surface rheological properties, or are appropriate only for infinitesimally small rates, where departures from linear behavior are very small [1]. In this paper we will discuss recent advances in the development of nonlinear constitutive equations for the stress-deformation behavior of fluidfluid interfaces in the framework of nonequilibrium thermodynamics. We will focus on two frameworks: the classical irreversible thermodynamics (CIT) framework, and the general-equation-for-the-nonequilibrium-reversible-irreversiblecoupling (GENERIC) framework. We will illustrate the construction of surface rheological constitutive equations within these two frameworks for a specific example: interfaces stabilized by anisotropic colloidal particles, in the dilute particle concentration regime. In both frameworks we construct models describing the effect of microstructural changes on the nonlinear response of an interface to a deformation through a dependence of the surface stress tensor on a set of scalar and a tensorial structural variables. We present the time evolution equations for these structural variables, and evaluate the ability of these types of models to describe the shear thinning behavior typically observed experimentally for such interfaces. We compare the models in both simple and oscillatory shear. We find that both frameworks allow us to construct nonlinear expressions for the surface extra stress tensor capable of describing shear thinning behavior, but the CIT model gives realistic predictions only for small departures from equilibrium, whereas the GENERIC framework allows us to create models valid also far from equilibrium. Besides giving more accurate predictions for the shear thinning behavior the GENERIC model also predicts the existence of in-plane normal stresses (normal to the direction of flow), and effect of which its existence has been hypothesized, but that has so far not been observed experimentally. These results show that microstructural models developed using nonequilibrium thermodynamic frameworks provide a valuable tool for the analysis of the highly nonlinear dynamics of multiphase systems with complex liquid-liquid interfaces.

On Maxwell’s Relations of Thermodynamics for Polymeric Liquids away from Equilibrium

To describe complex systems deeply in the nonlinear regime, advanced formulations of nonequilibrium thermodynamics such as the extended irreversible thermodynamics (EIT), the matrix model, the generalized bracket formalism, and the GENERIC (=general equation for the nonequilibrium reversible−irreversible coupling) formalism consider generalized versions of thermodynamic potentials in terms of a few, well-defined position-dependent state variables (defining the system at a coarse-grained level). Straightforward statistical mechanics considerations then imply a set of equalities for its second derivatives with respect to the corresponding state variables, typically known as Maxwell’s relations. We provide here direct numerical estimates of these relations from detailed atomistic Monte Carlo (MC) simulations of an unentangled polymeric melt coarse-grained to the level of the chain conformation tensor, under both weak and strong flows. We also report results for the nonequilbrium (i.e., relative to the quiesc...

Constraints in nonequilibrium thermodynamics: General framework and application to multicomponent diffusion

We elaborate how holonomic constraints can be incorporated into the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) framework of nonequilibrium thermodynamics. Dirac's ideas for constructing constrained Poisson brackets are extended to dissipative brackets. The construction is presented such that it can be put into practice most readily. We illustrate the procedure by developing a symmetric thermodynamic description of diffusion in multicomponent systems and, as a further example, we impose an incompressibility constraint. As a consequence of its more elaborate and restrictive structure, GENERIC removes the ambiguities occurring in the classical thermodynamics of irreversible processes when one works with redundant variables.

A multi-scale model with GENERIC structure of polymeric melt with fiber suspensions

Based on elastic dumbbell model of polymer melt and tensor description of fiber orientation, a mathematic multi-scale model with a general equation for the non-equilibrium reversible-irreversible coupling (GENERIC) of polymeric melt with fiber suspensions is presented by the coupling of reversible and irreversible thermodynamics. And the model provides a general framework with homogeneous GENERIC structure for the multi-scale governing equations. Through analyzing the components of GENERIC structure, stress contributions from information on different scales can be achieved.

Dissipative electromagnetism from a nonequilibrium thermodynamics perspective

Dissipative effects in electromagnetism on macroscopic scales are examined by coarse-graining the microscopic Maxwell equations with respect to time. We illustrate a procedure to derive the dissipative effects on the macroscopic scale by using a Green-Kubo type expression in terms of the microscopic fluctuations and the correlations between them. The resulting macroscopic Maxwell equations are formulated within the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) framework, accounting also for inhomogeneous temperature.

Nonequilibrium thermodynamics for open systems

We develop the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) framework of nonequilibrium thermodynamics for open systems. A clear distinction between bulk and boundary contributions to the Poisson and dissipative brackets employed to generate reversible and irreversible contributions to time evolution from energy and entropy allows us to formulate the bulk equations as well as the exchange and interaction with the environment directly. The full brackets keep all the structure and hence the predictive power of the original GENERIC for isolated systems. The straightforward procedure is illustrated for hydrodynamics of open systems. Boltzmann's kinetic equation is discussed as a further example. In the Appendix, the thermodynamic treatment of surface excess variables at walls and their role in boundary conditions for the bulk variables is exemplified for a diffusion cell.