Michal Pavelka

Charles University in Prague | CUNI · Faculty of Mathematics and Physics

The general equation of nonequilibrium reversible-irreversible coupling (GENERIC) is studied in light of time-reversal transformation. It is shown that Onsager-Casimir reciprocal relations are implied by GENERIC in the near-equilibrium regime. A general structure which gives the reciprocal relations but which is valid also far from equilibrium is i...

Exergy analysis, which provides means of calculating efficiency losses in industrial devices, is reviewed, and the area of its validity is carefully discussed. Consequently, a generalization is proposed, which holds also beyond the area of applicability of exergy analysis. The generalization is formulated within the framework of classical irreversi...

We provide an extensively discussed formulation of theory of mixtures of inter-acting fluids. This careful discussion leads to clarification of questions whether kinetic energy of diffusion should be defined to be part of internal energy or not, whether potential energy should be included into internal energy or not, how studying time-reversal pari...

In this paper, we present neural networks learning mechanical systems that are both symplectic (for instance particle mechanics) and non-symplectic (for instance rotating rigid body). Mechanical systems have Hamiltonian evolution, which consists of two building blocks: a Poisson bracket and an energy functional. We feed a set of snapshots of a Hami...

In this paper, we present a Hamiltonian and thermodynamic theory of heat transport on various levels of description. Transport of heat is formulated within kinetic theory of polarized phonons, kinetic theory of unpolarized phonons, hydrodynamics of polarized phonons, and hydrodynamics of unpolarized phonons. These various levels of description are...

We propose a conformal generalization of the reversible Vlasov equation of kinetic plasma dynamics, called conformal kinetic theory. To arrive at this formalism, we start with the conformal Hamiltonian dynamics of particles and lift this to the dynamical formulation of the associated kinetic theory. This theory represents a simple example of a geom...

In this paper, we present neural networks learning mechanical systems that are both symplectic (for instance particle mechanics) and non-symplectic (for instance rotating rigid body). Mechanical systems have Hamiltonian evolution, which consists of two building blocks: a Poisson bracket and an energy functional. We feed a set of snapshots of a Hami...

This paper investigates the applicability of smoothed particle hydrodynamics in simulations of superfluid helium-4. We devise a new approach based on Hamiltonian mechanics suitable for simulating thermally driven and weakly compressible flows with free surfaces. The method is then tested in three cases, including a simulation of the fountain effect...

The Eulerian distortion field is an essential ingredient for the continuum modeling of finite elastic and inelastic deformations of materials; however, its relation to finer levels of description has not yet been established. This paper provides a definition of the Eulerian distortion field in terms of the arrangement of the constituent microscopic...

Smoothed Particle Hydrodynamics (SPH) methods are advantageous in simulations of fluids in domains with free boundary. Special SPH methods have also been developed to simulate solids. However, there are situations where the matter behaves partly as a fluid and partly as a solid, for instance, the solidification front in 3D printing, or any system i...

Despite the fact that the theory of mixtures has been part of non-equilibrium thermodynamics and engineering for a long time, it is far from complete. While it is well formulated and tested in the case of mechanical equilibrium (where only diffusion-like processes take place), the question how to properly describe homogeneous mixtures that flow wit...

The General Equation for Non-Equilibrium Reversible–Irreversible Coupling (GENERIC) provides the structure of mesoscopic multiscale dynamics that guarantees the emergence of equilibrium states. Similarly, a lift of the GENERIC structure to iterated cotangent bundles, called a rate GENERIC, guarantees the emergence of the vector fields that generate...

This paper contains a fully geometric formulation of the General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC). Although GENERIC, which is the sum of Hamiltonian mechanics and gradient dynamics, is a framework unifying a vast range of models in non-equilibrium thermodynamics, it has unclear geometric structure due to the d...

This paper describes an energy-preserving and globally time-reversible code for weakly compressible smoothed particle hydrodynamics (SPH). We do not add any additional dynamics to the Monaghan's original SPH scheme at the level of ordinary differential equation, but we show how to discretize the equations by using a corrected expression for density...

In this paper, we present a Hamiltonian and thermodynamic theory of heat transport on various levels of description. Transport of heat is formulated within kinetic theory of polarized phonons, kinetic theory of unpolarized phonons, hydrodynamics of polarized phonons, and hydrodynamics of unpolarized phonons. These various levels of description are...

Smoothed Particle Hydrodynamics (SPH) methods are advantageous in simulations of fluids in domains with free boundary. Special SPH methods have also been developed to simulate solids. However, there are situations where the matter behaves partly as a fluid and partly as a solid, for instance, the solidification front in 3D printing, or any system i...

Despite the fact that the theory of mixtures has been part of non-equilibrium thermodynamics and engineering for a long time, it is far from complete. While it is well formulated and tested in the case of mechanical equilibrium (where only diffusion-like processes take place), the question how to properly describe homogeneous mixtures that flow wit...

This paper contains a fully geometric formulation of the General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC). Although GENERIC, which is the sum of Hamiltonian mechanics and gradient dynamics, is a framework unifying a vast range of models in non-equilibrium thermodynamics, it has unclear geometric structure, due to the...

The General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) provides structure of mesoscopic multiscale dynamics that guarantees emergence of equilibrium states. Similarly, a lift of the GENERIC structure to iterated cotangent bundles, called a rate GENERIC, guarantees emergence of the vector fields that generate the approac...

This paper describes an energy-preserving and globally time-reversible code for weakly compressible smoothed particle hydrodynamics (SPH). We do not add any additional dynamics to the Monaghan's original SPH scheme at the level of ordinary differential equation, but we show how to discretize the equations by using a corrected expression for density...

Within the framework of natural configurations developed by Rajagopal and Srinivasa, evolution within continuum thermodynamics is formulated as evolution of a natural configuration linked with the current configuration. On the other hand, withing the General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) framework, the evol...

Density gradient theory describes the evolution of diffuse interfaces in both mixtures and pure substances by minimization of the total free energy, which consists of a non-convex bulk part and an interfacial part. Minimization of the bulk free energy causes phase separation while building up the interfacial free energy (proportional to the square...

How to properly describe continuum thermodynamics of binary mixtures where each constituent has its own momentum? The Symmetric Hyperbolic Thermodynamically Consistent (SHTC) framework and Hamiltonian mechanics in the form of the General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) provide two answers, which are similar b...

Superfluid helium-4 is characterized by extremely small values of kinematic viscosity, and its thermal conductivity can be huge, orders of magnitude larger than that of water or air. Additionally, quantum vortices may exist within the fluid. Therefore, its behavior cannot be explained by using the classical tools of Newtonian fluid mechanics, and,...

We propose a novel approach for learning the evolution that employs differentiable neural networks to approximate the full GENERIC structure. Instead of manually choosing the fitted parameters, we learn the whole model together with the evolution equations. We can reconstruct the energy and entropy functions for the system under various assumptions...

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 la...

Particles are a widespread tool for obtaining information from fluid flows. When Eulerian data are unavailable, they may be employed to estimate flow fields or to identify coherent flow structures. Here we numerically examine the possibility of using particles to capture the dynamics of isolated vortex rings propagating in a quiescent fluid. The an...

Experimental data bases are typically very large and high dimensional. To learn from them requires to recognize important features (a pattern), often present at scales different to that of the recorded data. Following the experience collected in statistical mechanics and thermodynamics, the process of recognizing the pattern (the learning process)...

The classical mass action law is put into the context of multiscale thermodynamics. The emergence of time irreversibility in mechanics becomes in chemical kinetics the emergence of time reversibility in the completely irreversible gradient time evolution of chemically reacting systems. As in mechanics, this type of symmetry breaking is a tool for e...

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 la...

A multiscale theory of interacting continuum mechanics and thermodynamics of mixtures of fluids, electrodynamics, polarization, and magnetization is proposed. The mechanical (reversible) part of the theory is constructed in a purely geometric way by means of semidirect products. This leads to a complex Hamiltonian system with a new Poisson bracket,...

Originally derived by Walther Nernst more than a century ago, the Nernst equation for the open-circuit voltage is a cornerstone in the analysis of electrochemical systems. Unfortunately, the assumptions behind its derivation are often overlooked in the literature, leading to incorrect forms of the equation when applied to complex systems (for examp...

We find a relation between two approaches to non-equilibrium continuum mechanics and thermodynamics. The framework of natural configuration developed by Rajagopal and Srinivasa, where the reference, natural, and current configurations of the continuum are linked together, is compared with the General Equation for Non-Equilibrium Reversible-Irrevers...

The lack-of-fit statistical reduction, developed and formulated first by Bruce Turkington, is a general method taking Liouville equation for probability density (detailed level) and transforming it to reduced dynamics of projected quantities (less detailed level). In this paper the method is generalized. The Hamiltonian Liouville equation is replac...

We place the Landau theory of critical phenomena into the larger context of multiscale thermodynamics. The thermodynamic potentials, with which the Landau theory begins, arise as Lyapunov like functions in the investigation of the relations among different levels of description. By seeing the renormalization-group approach to critical phenomena as...

An experimental study of turbulent vortex rings in superfluid 4He - Volume 889 - P. Švančara, M. Pavelka, M. La Mantia

The standard way towards continuum thermodynamics based on conservation laws is discussed. Modelling of complex materials, where detailed state variables are necessary, requires insight beyond the conservation laws to determine the evolution equations for the detailed variables in a unique way. Several such approaches are discussed, including the n...

Continuum mechanics can be formulated in the Lagrangian frame (addressing motion of individual continuum particles) or in the Eulerian frame (addressing evolution of fields in an inertial frame). There is a canonical Hamiltonian structure in the Lagrangian frame. By transformation to the Eulerian frame we find the Poisson bracket for Eulerian conti...

Reduction of a mesoscopic dynamical theory to equilibrium thermodynamics brings to the latter theory the fundamental thermodynamic relation (i.e. entropy as a function of the thermodynamic state variables). The reduction is made by following the mesoscopic time evolution to its conclusion, i.e. to fixed points at which the time evolution ceases to...

Let (M,J) be a dynamical model of macroscopic systems and (N,K) a less microscopic model (i.e. a model involving less details) of the same macroscopic systems; M and N are manifolds, J are vector fields on M, and K are vector fields on N. Let P be the phase portrait corresponding to (M,J) (i.e. P is the set of all trajectories in M generated by a f...

What is the maximum voltage of a cell with a given electrochemical reaction? The answer to this question has been given more than a century ago by Walther Nernst and bears his name. Unfortunately, the assumptions behind the answer have been forgotten by many authors, which leads to wrong forms of the Nernst relation. Such mistakes can be overcome b...

The lack-of-fit statistical reduction, developed and formulated first by Bruce Turkington, is a general method for taking Liouville equation for probability density (detailed level) and transforming it to reduced dynamics of projected quantities. In this paper the method is generalized. The Hamiltonian Liouville equation is replaced by an arbitrary...

Geometrical and algebraic aspects of the Hamiltonian realizations of the Euler’s fluid and the Vlasov’s plasma are investigated. A purely geometric pathway (involving complete lifts and vertical representatives) is proposed, which establishes a link from particle motion to evolution of the field variables. This pathway is free from Poisson brackets...

Experimental data bases are typically very large and high dimensional. To learn from them requires to recognize important features (a pattern), often present at scales different to that of the recorded data. Following the experience collected in statistical mechanics and thermodynamics, the process of recognizing the pattern (the learning process)...

Any physical system can be regarded on different levels of description varying by how detailed the description is. We propose a method called Dynamic MaxEnt (DynMaxEnt) that provides a passage from the more detailed evolution equations to equations for the less detailed state variables. The method is based on explicit recognition of the state and c...

Continuum mechanics can be formulated in the Lagrangian frame (where properties of continuum particles are addressed) or in the Eulerian frame (where fields livein an inertial frame). There is a canonical Hamiltonian structure in the Lagrangian frame. By transformation to the Eulerian frame we find the Poisson bracket for Eulerian continuum mechani...

Imagine a freely rotating rigid body. The body has three principal axes of rotation. It follows from mathematical analysis of the evolution equations that pure rotations around the major and minor axes are stable while rotation around the middle axis is unstable. However, only rotation around the major axis (with highest moment of inertia) is stabl...

Heat conduction is investigated on three levels: equilibrium, Fourier, and Cattaneo. The Fourier level is either the point of departure for investigating the approach to equilibrium or the final stage in the investigation of the approach from the Cattaneo level. Both investigations bring to the Fourier level an entropy and a thermodynamics. In the...

Fluid mechanics and electrodynamics are two theories of Hamiltonian nature, which are coupled through the Lorentz force. Besides the fields of electric displacement and magnetic field, there are also the fields of polarization and magnetization, which are interacting with both matter and electromagnetic field. We propose a geometrical construction...

Continuum mechanics with dislocations, with the Cattaneo type heat conduction, with mass transfer, and with electromagnetic fields is put into the Hamiltonian form and into the form of the Godunov type system of the first order, symmetric hyperbolic partial differential equations (SHTC equations). The compatibility with thermodynamics of the time r...

Imagine a freely rotating rigid body. The body has three principal axes of rotation. It follows from mathematical analysis of the evolution equations that pure rotations around the major and minor axes are stable while rotation around the middle axis is unstable. However, only rotation around the major axis (with highest moment of inertia) is stabl...

Heat conduction is investigated on three levels: equilibrium, Fourier, and Cattaneo. The Fourier level is either the point of departure for investigating the approach to equilibrium or the final stage in the investigation of the approach from the Cattaneo level. Both investigations bring to the Fourier level an entropy and a thermodynamics. In the...

Reformulating constitutive relation in terms of gradient dynamics (being derivative of a dissipation potential) brings additional information on stability, metastability and instability of the dynamics with respect to perturbations of the constitutive relation, called CR-stability. CR-instability is connected to the loss of convexity of the dissipa...

Landau damping is the tendency of solutions to the Vlasov equation towards spatially homogeneous distribution functions. The distribution functions, however, approach the spatially homogeneous manifold only weakly, and Boltzmann entropy is not changed by the Vlasov equation. On the other hand, density and kinetic energy density, which are integrals...

Despite the extensive attention that water transport in Nafion membranes has been subject to, several phenomenal still remain unexplained. It is for instance not clear why desorption rate is much faster than absorption. The purpose of this paper is to suggest an answer to that problem. A simple but thermodynamically consistent model is formulated w...

Vlasov kinetic theory is extended by adopting an extra one particle distribution function as an additional state variable characterizing the micro-turbulence internal structure. The extended Vlasov equation keeps the reversibility, the Hamiltonian structure, and the entropy conservation of the original Vlasov equation. In the setting of the extende...

The time evolution governed by the Boltzmann kinetic equation is compatible with mechanics and thermodynamics. The former compatibility is mathematically expressed in the Hamiltonian and Godunov structures, the latter in the structure of gradient dynamics guaranteeing the growth of entropy and consequently the approach to equilibrium. We carry all...

Reformulating a constitutive relation in terms of gradient dynamics (being derivative of a dis-sipation potential) brings additional information on stability, metastability and instability of the dynamics generated by the constitutive relation. Instability is connected to the loss of convexity of the dissipation potential, which makes the conjugate...

Thermodynamic fluxes (diffusion fluxes, heat flux, etc.) are often proportional to thermodynamic forces (gradients of chemical potentials, temperature, etc.) via the matrix of phenomenological coefficients. Onsager's relations imply that the matrix is symmetric, which reduces the number of unknown coefficients is reduced. In this article we demonst...

We compare two methods for modeling dissipative processes, namely gradient dynamics and entropy production maximization. Both methods require similar physical inputs–-how energy (or entropy) is stored and how it is dissipated. Gradient dynamics describes irreversible evolution by means of dissipation potential and entropy, it automatically satisfie...

The well-known Gouy-Stodola theorem states that a device produces maximum useful power when working reversibly, that is with no entropy production inside the device. This statement then leads to a method of thermodynamic optimization based on entropy production minimization. Exergy destruction (difference between exergy of fuel and exhausts) is als...

The conditions of existence of extra mass flux in single component dissipative non-relativistic fluids are clarified. By considering Galilean invariance we show that if total mass flux is equal to total momentum density, then mass, momentum, angular momentum and booster (center- of-mass) are conserved. However, these conservation laws may be fulfil...

Braun-Le Chatelier principle is a fundamental result of equilibrium thermodynamics, showing how stable equilibrium states shift when external conditions are varied. The principle follows from convexity of thermodynamic potential. Analogously, from convexity of dissipation potential it follows how steady non-equilibrium states shift when thermodynam...

Reversible part of evolution equations of physical systems is often generated by a Poisson bracket. We discuss geometric means of construction of Poisson brackets and their mutual coupling (direct, semidirect and matched-pair products) as well as projections of Poisson brackets to less detailed Poisson brackets. This way the Hamiltonian coupling be...

Reversible evolution of macroscopic and mesoscopic systems can be conve- niently constructed from two ingredients: an energy functional and a Poisson bracket. The goal of this paper is to elucidate how the Poisson brackets can be constructed and what additional features we also gain by the construction. In particular, the Poisson brackets governing...

Reduction of a mesoscopic level to a level with fewer details is made by the time evolution during which the entropy increases. An extension of a mesoscopic level is a construction of a level with more details. In particular, we discuss extensions in which extra state variables are found in the vector fields appearing on the level that we want to e...

Water and proton transport across a Nafion membrane are measured as functions of water activity and applied electric potential with a polymer electrolyte hydrogen pump. Water and proton transport across the membrane must match water and proton transport entering and leaving the electrode/membrane/vapor three phase interfaces at the anode and cathod...

Non-equilibrium thermodynamics, which serves as a framework for formulating evolution equations of macroscopic and mesoscopic systems, is briefly reviewed and further developed in this work. For example, the relation between the General Equation for the Nonequilibrium Reversible-Irreversible Coupling (GENERIC) and (ir)reversibility is elucidated, a...

Open circuit voltage of vanadium redox flow batteries is carefully calculated using equilibrium thermodynamics. This analysis reveals some terms in the Nernst relation which are usually omitted in literature. Due to the careful thermodynamic treatment, all uncertainties about the form of Nernst relation are removed except for uncertainties in activ...

Efficiency of hydrogen fuel cells is analyzed using a non-equilibrium theory of mixtures based on classical irreversible thermodynamics 1 . The efficiency depends on energy of fuel and on a function (map of efficiency losses) revealing how much efficiency is being lost at each point of the fuel cell. It is shown that the losses are not only given b...

Energy conversion in fuel cells is performed by electrochemical and transport processes in the polymer electrolyte membrane, gas diffusion layers, catalyst layers, and in the pipes transporting hydrogen, oxygen, water and air into and out of the fuel cell. All these processes are analyzed from the point of view of phenomenological thermodynamics, w...

Efficiency of hydrogen fuel cells is analyzed using a non-equilibrium theory of mixtures based on classical irreversible ther-modynamics. The efficiency is expressed in terms of processes taking place inside the fuel cells revealing which processes are responsible for efficiency losses. This provides a new method of optimization. It is shown that e...

It is common knowledge that efficiency of fuel cells is highest when no electric current is produced while when the fuel cell is really working, the efficiency is reduced by dissipation. In this paper the relation between efficiency and dissipation inside the fuel cell is formulated within the framework of classical irreversible thermodynamics of m...