Michal Pavelka

Reductions and extensions in mesoscopic dynamics.

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 extend. Reductions, extensions, and compatibility relations among them are formulated first in an abstract setting and then illustrated in specific mesoscopic theories.

A hierarchy of Poisson brackets in non-equilibrium thermodynamics

Abstract Reversible evolution of macroscopic and mesoscopic systems can be conveniently 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 reversible evolution in one-particle kinetic theory, kinetic theory of binary mixtures, binary fluid mixtures, classical irreversible thermodynamics and classical hydrodynamics are derived from Liouville equation. Although the construction is quite natural, a few examples where it does not work are included (e.g. the BBGKY hierarchy). Finally, a new infinite grand-canonical hierarchy of Poisson brackets is proposed, which leads to Poisson brackets expressing non-local phenomena such as turbulent motion or evolution of polymeric fluids. Eventually, Lie–Poisson structures standing behind some of the brackets are identified.

Extra Mass Flux in Fluid Mechanics

Abstract The conditions of existence of extra mass flux in single-component dissipative nonrelativistic 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 fulfilled also by other means. We show an example of weakly nonlocal hydrodynamics where the conservation laws are satisfied as well although the total mass flux is different from momentum density.

Gradient Dynamics and Entropy Production Maximization

Abstract 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 satisfies Onsager reciprocal relations as well as their nonlinear generalization (Maxwell–Onsager relations), and it has statistical interpretation. Entropy production maximization is based on knowledge of free energy (or another thermodynamic potential) and entropy production. It also leads to the linear Onsager reciprocal relations and it has proven successful in thermodynamics of complex materials. Both methods are thermodynamically sound as they ensure approach to equilibrium, and we compare them and discuss their advantages and shortcomings. In particular, conditions under which the two approaches coincide and are capable of providing the same constitutive relations are identified. Besides, a commonly used but not often mentioned step in the entropy production maximization is pinpointed and the condition of incompressibility is incorporated into gradient dynamics.

Functional constraints on phenomenological coefficients

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. Onsagers relations imply that the matrix is symmetric, which reduces the number of unknown coefficients is reduced. In this article we demonstrate that for a class of nonequilibrium thermodynamic models in addition to Onsagers relations the phenomenological coefficients must share the same functional dependence on the local thermodynamic state variables. Thermodynamic models and experimental data should be validated through consistency with the functional constraint. We present examples of coupled heat and mass transport (thermodiffusion) and coupled charge and mass transport (electro-osmotic drag). Additionally, these newly identified constraints further reduce the number of experiments needed to describe the phenomenological coefficient.

Hamiltonian and Godunov structures of the Grad hierarchy

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 three structures to the Grad reformulation of the Boltzmann equation (to the Grad hierarchy). First, we recognize the structures in the infinite Grad hierarchy and then in several examples of finite hierarchies representing extended hydrodynamic equations. In the context of Grads hierarchies, we also investigate relations between Hamiltonian and Godunov structures.

Pitfalls of Exergy Analysis

Abstract 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 also given by entropy production inside the device. Therefore, assessing efficiency of a device by exergy analysis is also based on the Gouy–Stodola theorem. However, assumptions that had led to the Gouy–Stodola theorem are not satisfied in several optimization scenarios, e.g. non-isothermal steady-state fuel cells, where both entropy production minimization and exergy analysis should be used with caution. We demonstrate, using non-equilibrium thermodynamics, a few cases where entropy production minimization and exergy analysis should not be applied.

Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations

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 reversible part of the governing equations is mathematically expressed in the former formulation as degeneracy of the Hamiltonian structure and in the latter formulation as the existence of a companion conservation law. In both formulations the time irreversible part represents gradient dynamics. The Godunov-type formulation brings the mathematical rigor (the local well posedness of the Cauchy initial value problem) and the possibility to discretize while keeping the physical content of the governing equations (the Godunov finite volume discretization).

Thermodynamic Explanation of Landau Damping by Reduction to Hydrodynamics

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 of the distribution function, approach spatially homogeneous states strongly, which is accompanied by growth of the hydrodynamic entropy. Such a behavior can be seen when the Vlasov equation is reduced to the evolution equations for density and kinetic energy density by means of the Ehrenfest reduction.

Non-convex dissipation potentials in multiscale non-equilibrium thermodynamics

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 dissipation potential, which makes the Legendre-conjugate dissipation potential multivalued and causes dissipative phase transitions that are not induced by non-convexity of free energy, but by non-convexity of the dissipation potential. CR-stability of the constitutive relation with respect to perturbations is then manifested by constructing evolution equations for the perturbations in a thermodynamically sound way (CR-extension). As a result, interesting experimental observations of behavior of complex fluids under shear flow and supercritical boiling curve can be explained.