Non-negative Entropy Production Research Articles

A stabilised Total Lagrangian Element-Free Galerkin method for transient nonlinear solid dynamics

AbstractThis paper presents a new stabilised Element-Free Galerkin (EFG) method tailored for large strain transient solid dynamics. The method employs a mixed formulation that combines the Total Lagrangian conservation laws for linear momentum with an additional set of geometric strain measures. The main aim of this paper is to adapt the well-established Streamline Upwind Petrov–Galerkin (SUPG) stabilisation methodology to the context of EFG, presenting three key contributions. Firstly, a variational consistent EFG computational framework is introduced, emphasising behaviours associated with nearly incompressible materials. Secondly, the suppression of non-physical numerical artefacts, such as zero-energy modes and locking, through a well-established stabilisation procedure. Thirdly, the stability of the SUPG formulation is demonstrated using the time rate of Hamiltonian of the system, ensuring non-negative entropy production throughout the entire simulation. To assess the stability, robustness and performance of the proposed algorithm, several benchmark examples in the context of isothermal hyperelasticity and large strain plasticity are examined. Results show that the proposed algorithm effectively addresses spurious modes, including hour-glassing and spurious pressure fluctuations commonly observed in classical displacement-based EFG frameworks.

Port-metriplectic neural networks: thermodynamics-informed machine learning of complex physical systems

We develop inductive biases for the machine learning of complex physical systems based on the port-Hamiltonian formalism. To satisfy by construction the principles of thermodynamics in the learned physics (conservation of energy, non-negative entropy production), we modify accordingly the port-Hamiltonian formalism so as to achieve a port-metriplectic one. We show that the constructed networks are able to learn the physics of complex systems by parts, thus alleviating the burden associated to the experimental characterization and posterior learning process of this kind of systems. Predictions can be done, however, at the scale of the complete system. Examples are shown on the performance of the proposed technique.

On Regularized Systems of Equations for Gas Mixture Dynamics with New Regularizing Velocities and Diffusion Fluxes

We deal with multidimensional regularized systems of equations for the one-velocity and one-temperature inert gas mixture dynamics consisting of the balance equations for the mass of components and the momentum and total energy of the mixture, with diffusion fluxes between the components as well as the viscosity and heat conductivity terms. The regularizations are kinetically motivated and aimed at constructing conditionally stable symmetric in space discretizations without limiters. We consider a new combined form of regularizing velocities containing the total pressure of the mixture. To confirm the physical correctness of the regularized systems, we derive the balance equation for the mixture entropy with the non-negative entropy production, under generalized assumptions on the diffusion fluxes. To confirm nice regularizing properties, we derive the systems of equations linearized at constant solutions and provide the existence, uniqueness and L2-dissipativity of weak solutions to an initial-boundary problem for them. For the original systems, we also discuss the related Petrovskii parabolicity property and its important corollaries. In addition, in the one-dimensional case, we also present the special three-point and symmetric finite-difference discretization in space of the regularized systems and prove that it inherits the entropy correctness property. We also give results of numerical experiments confirming that the discretization is able to simulate well various dynamic problems of contact between two different gases.

Entropy Principle and Shock-Wave Propagation in Continuum Physics

According to second law of thermodynamics, the local entropy production must be nonnegative for arbitrary thermodynamic processes. In 1996, Muschik and Ehrentraut observed that such a constraint can be fulfilled in two different ways: either by postulating a suitable form of the constitutive equations, or by selecting among the solutions of the systems of balance laws those which represent physically realizable thermodynamic processes. Hence, they proposed an amendment to the second law which assumes that reversible process directions in state space exist only in correspondence with equilibrium states. Such an amendment allowed them to prove that the restriction of the constitutive equations is the sole possible consequence of non-negative entropy production. Recently, Cimmelli and Rogolino revisited the classical result by Muschik and Ehrentraut from a geometric perspective and included the amendment in a more general formulation of the second law. Herein, we extend this result to nonregular processes, i.e., to solutions of balance laws which admit jump discontinuities across a given surface. Two applications of these results are presented: the thermodynamics of an interface separating two different phases of a Korteweg fluid, and the derivation of the thermodynamic conditions necessary for shockwave formation. Commonly, for shockwaves the second law is regarded as a restriction on the thermodynamic processes rather than on the constitutive equations, as only perturbations for which the entropy continues to grow across the shock can propagate. We prove that this is indeed a consequence of the general property of the second law of thermodynamics that restricts the constitutive equations for nonregular processes. An analysis of shockwave propagation in different thermodynamic theories is developped as well.

Non-equilibrium thermodynamics of diffusion in fluctuating potentials

A positive rate of entropy production at steady-state is a distinctive feature of truly non-equilibrium processes. Exact results, while being often limited to simple models, offer a unique opportunity to explore the thermodynamic features of these processes in full detail. Here we derive analytical results for the steady-state rate of entropy production in single particle systems driven away from equilibrium by the fluctuations of an external potential of arbitrary shapes. Subsequently, we provide exact results for a diffusive particle in a harmonic trap whose potential stiffness varies in time according to both discrete and continuous Markov processes. In particular, studying the case of a fully intermittent potential allows us to introduce an effective model of stochastic resetting for which it is possible to obtain finite non-negative entropy production. Altogether, this work lays the foundation for a non-equilibrium thermodynamic theory of fluctuating potentials, with immediate applications to stochastic resetting processes, fluctuations in optical traps and fluctuating interactions in living systems.

Mixture model for thermo-chemo-mechanical processes in fluid-infused solids

This work presents a new thermodynamically consistent model for thermo-chemo-mechanical processes in open systems comprised of nonlinear elastic solids that are infused with reactive fluids and are undergoing large local strains. The interactions between different constituents of the material system are captured through locally homogenized mixture theory. The development of the formulation starts from constituent-wise balance equations for mass, momentum, and energy. The constitutive relations are derived to enforce non-negative entropy production and satisfy the second law of thermodynamics for the open system. The deformation due to thermal and chemical effects is carried out through a multiplicative split of the deformation gradient. Further assumptions on the fluid and solid material responses are introduced to specialize the model to a class of thermo-chemo-mechanical processes that arise in thermal oxidation of metallic and ceramic materials, lithiation of battery anodes, and other related problems. The model is implemented in the standard Galerkin finite element method using eight node hexahedral elements. A set of test cases involving fully coupled thermo-mechanical, chemo-mechanical, and thermo-chemical effects are analyzed. Numerical results obtained are in reasonable agreement with the experimental data for thermo-chemo-mechanical problems concerning the thermal oxidation of silicon carbide and FeCrAlY bond coat material in thermal barrier coatings.

Nonequilibrium thermodynamics for soft matter made easy(er)

We use straightforward energy and entropy balances to test the thermodynamic consistency of microstructural rheological models. The method utilizes the same mathematical methods as classical transport phenomena, so it is much simpler to use than the much more rigorous GENERIC formalism. The cost of this simplicity is that fewer restrictions are actually tested than those in either the single-generator or the two-generator formalisms. The proposed test does provide, however, a separation of energy and entropy, leading to an interesting internal energy balance. More importantly, it leads to two requirements for non-negative entropy production: one closely related to a virtual work argument, important during flow, and a second that guarantees adherence to the second law of thermodynamics during microstructural relaxation. These criteria do not appear to be in conflict with the requirements of the more rigorous formulations and are much simpler to implement. Several illustrative examples are given with models using the conformation tensor level of description. As expected, the models that use a relaxation function that is proportional to the free energy gradient are straightforward to check. These include the Hookean dumbbell, the FENE-P, and the Giesekus models, which are shown to satisfy the first and second laws. With a little more work, models with relaxation functions not driven by the free energy gradient only can also be checked for thermodynamic compliance with the proposed formalism. Examples of these are the GLaMM and Rolie–Poly models; they violate the first and second laws of thermodynamics, respectively. A recently proposed FENE-mode model is also checked, which superficially satisfies both laws, but fails to have an analytic free energy. Finally, the application of the formalism to a nonlinear dumbbell model that uses the probability density for chain conformations is also illustrated. In that case, satisfaction of the fluctuation-dissipation theorem and a positive mobility guarantee compliance with the first and second laws.

Свойства агрегированной квазигидродинамической системы уравнений гомогенной газовой смеси с общей регуляризующей скоростью

We study a quasi-hydrodynamic system of equations for a homogeneous (with common velocity and temperature) multicomponent gas mixture in the absence of chemical reactions, with a regularizing velocity common for the components. We derive the entropy balance equation with a non-negative entropy production taking into account the diffusion fluxes of the mixture components. In the absence of diffusion fluxes, a system of equations linearized at a constant solution is constructed by a new technique, In the absence of diffusion fluxes, a system of equations linearized on a constant solution is constructed by a new technique. It is reduced to a symmetric form, the L^2-dissipativity of its solutions is proved, and a degeneration (with respect to the densities of the mixture components) of the parabolicity property for the original system is established. Actually, the system has the composite type. The obtained properties strictly reflect its physical correctness and dissipative nature of the quasi-hydrodynamic regularization.

Hyperbolicity of a model for polyatomic gases in relativistic extended thermodynamics

The hyperbolicity of a model for relativistic extended thermodynamics of polyatomic gases is here proved for every time direction $$\xi _\alpha $$ ; moreover, an expression for the production terms of the field equation is proposed which encloses as particular cases the variant of the Anderson–Witting model already known in the literature, but also a new variant of the Marle model. With these expressions, it is shown that the field equations satisfy automatically the requirement of a non-negative entropy production.

Closure conditions for a one temperature non-equilibrium multi-component model of baer-nunziato type

A class of non-equilibrium models for compressible multi-component uids is investigated. These models are subject to the choice of interfacial pressures and interfacial velocity as well as relaxation terms for velocity, pressure and chemical potentials. Sufficient conditions are derived for these quantities that ensure meaningful physical properties such as a non-negative entropy production and thermodynamical stability as well as mathematical properties such as hyperbolicity. For the relaxation of chemical potentials a three-component model gas-water-vapor is considered.

Stochastic dynamics Ⅱ: Finite random dynamical systems, linear representation, and entropy production

<p style='text-indent:20px;'>We study finite state random dynamical systems (RDS) and their induced Markov chains (MC) as stochastic models for complex dynamics. The linear representation of deterministic maps in RDS is a matrix-valued random variable whose expectation corresponds to the transition matrix of the MC. The instantaneous Gibbs entropy, Shannon-Khinchin entropy of a step, and the entropy production rate of the MC are discussed. These three concepts, as key anchoring points in applications of stochastic dynamics, characterize respectively the uncertainties of a system at instant time <inline-formula><tex-math id="M1">\begin{document}$ t $\end{document}</tex-math></inline-formula>, the randomness generated in a step in the dynamics, and the dynamical asymmetry with respect to time reversal. The stationary entropy production rate, expressed in terms of the cycle distributions, has found an expression in terms of the probability of the deterministic maps with single attractor in the maximum entropy RDS. For finite RDS with invertible transformations, the non-negative entropy production rate of its MC is bounded above by the Kullback-Leibler divergence of the probability of the deterministic maps with respect to its time-reversal dual probability.

Beyond Onsager-Casimir Relations: Shared Dependence of Phenomenological Coefficients on State Variables.

Phenomenological coefficients in linear nonequilibrium thermodynamics have been considered to be independent, apart from restrictions due to the Onsager-Casimir reciprocal relations and the requirement to have non-negative entropy production. Recently, it has been shown that functional constraints between these coefficients may hold, restricting their dependence on state variables, especially in the case of coupled phenomena. Here we demonstrate that such restrictions require only mild assumptions on the system of interest and are, in fact, much more constraining than previously reported. Such constraints vastly reduce the set of plausible models for constitutive relations and allow for simpler experimental determinations of dependencies in coupled systems. These results may also clarify inconsistencies in the literature regarding constitutive models used that do not obey these thermodynamic constraints.

An Onsager Singularity Theorem for Turbulent Solutions of Compressible Euler Equations

We prove that bounded weak solutions of the compressible Euler equations will conserve thermodynamic entropy unless the solution fields have sufficiently low space-time Besov regularity. A quantity measuring kinetic energy cascade will also vanish for such Euler solutions, unless the same singularity conditions are satisfied. It is shown furthermore that strong limits of solutions of compressible Navier-Stokes equations that are bounded and exhibit anomalous dissipation are weak Euler solutions. These inviscid limit solutions have non-negative anomalous entropy production and kinetic energy dissipation, with both vanishing when solutions are above the critical degree of Besov regularity. Stationary, planar shocks in Euclidean space with an ideal-gas equation of state provide simple examples that satisfy the conditions of our theorems and which demonstrate sharpness of our $L^3$-based conditions. These conditions involve space-time Besov regularity, but we show that they are satisfied by Euler solutions that possess similar space regularity uniformly in time.

Consistent multiphase-field theory for interface driven multidomain dynamics

We present a new multiphase-field theory for describing pattern formation in multi-domain and/or multi-component systems. The construction of the free energy functional and the dynamic equations is based on criteria that ensure mathematical and physical consistency. We first analyze previous multiphase-field theories, and identify their advantageous and disadvantageous features. On the basis of this analysis, we introduce a new way of constructing the free energy surface, and derive a generalized multiphase description for arbitrary number of phases (or domains). The presented approach retains the variational formalism; reduces (or extends) naturally to lower (or higher) number of fields on the level of both the free energy functional and the dynamic equations; enables the use of arbitrary pairwise equilibrium interfacial properties; penalizes multiple junctions increasingly with the number of phases; ensures non-negative entropy production, and the convergence of the dynamic solutions to the equilibrium solutions; and avoids the appearance of spurious phases on binary interfaces. The new approach is tested for multi-component phase separation and grain coarsening.

Closure conditions for non-equilibrium multi-component models

A class of non-equilibrium models for compressible multi-component fluids in multi-dimensions is investigated taking into account viscosity and heat conduction. These models are subject to the choice of interfacial pressures and interfacial velocity as well as relaxation terms for velocity, pressure, temperature and chemical potentials. Sufficient conditions are derived for these quantities that ensure meaningful physical properties such as a non-negative entropy production, thermodynamical stability, Galilean invariance and mathematical properties such as hyperbolicity, subcharacteristic property and existence of an entropy–entropy flux pair. For the relaxation of chemical potentials, a two-component and a three-component models for vapor–water and gas–water–vapor, respectively, are considered.

Asymptotic behavior of BV solutions to hyperbolic systems of balance laws with relaxation

For hyperbolic systems of balance laws governing relaxation processes, in one space dimension, with source incurring nonnegative entropy production and satisfying a Kawashima-type condition, it is shown that when the initial data have small total variation on (-∞, ∞) and decay rapidly to zero, as |x| → ∞, then the Cauchy problem possesses a unique admissible BV solution, in the large, with total variation decaying to zero, as t → ∞.

A sharp interface model for void growth in irradiated materials

A thermodynamic formalism for the interaction of point defects with free surfaces in single-component solids has been developed and applied to the problem of void growth by absorption of point defects in irradiated metals. This formalism consists of two parts, a detailed description of the dynamics of defects within the non-equilibrium thermodynamic frame, and the application of the second law of thermodynamics to provide closure relations for all kinetic equations. Enforcing the principle of non-negative entropy production showed that the description of the problem of void evolution under irradiation must include a relationship between the normal fluxes of defects into the void surface and the driving thermodynamic forces for the void surface motion; these thermodynamic forces are identified for both vacancies and interstitials and the relationships between these forces and the normal point defect fluxes are established using the concepts of transition state theory. The latter theory implies that the defect accommodation into the surface is a thermally activated process. Numerical examples are given to illustrate void growth dynamics in this new formalism and to investigate the effect of the surface energy barriers on void growth. Consequences for phase field models of void growth are discussed.

Non-negative entropy production criteria for numerical simulations of flows in porous media

Non-negative entropy production criteria for numerical simulations of flows in porous media

Gyrokinetic Fokker-Planck Collision Operator

The gyrokinetic linearized exact Fokker-Planck collision operator is obtained in a form suitable for plasma gyrokinetic equations, for arbitrary mass ratio. The linearized Fokker-Planck operator includes both the test-particle and field-particle contributions, and automatically conserves particles, momentum, and energy, while ensuring non-negative entropy production. Finite gyroradius effects in both field-particle and test-particle terms are evaluated. When implemented in gyrokinetic simulations, these effects can be precomputed. The field-particle operator at each time step requires the evaluation of a single two-dimensional integral, and is not only more accurate, but appears to be less expensive to evaluate than conserving model operators.

Information entropy production for steady-state electrochemical systems

In this work, we develop non-equilibrium information theory formalism for the investigation of the steady-state heterogeneous electron transfer. In order to analyze the information transfer, we introduce the concept of information potential and information affinity. These new functions allow us to describe the transformation from the initial to final state by means of Fokker–Planck type equation. Accordingly, the transformation can be viewed as a diffusion process within the phase space. Also, we are able to determine the information entropy production. We show that the Shannon information entropy production is equal to zero for a reversible system. However, the change in the relative entropy or Kullback–Leibler measure of information distance is positive for an irreversible system. In all cases, we find that a non-negative information entropy production is always observed during the experiments. Here, we demonstrate the self-consistency of the non-equilibrium formalism in the information theory for studying steady-state electrochemical systems.