Derivation of a two-generator framework of nonequilibrium thermodynamics for quantum systems

Formulation of moment equations for rarefied gases within two frameworks of non-equilibrium thermodynamics: RET and GENERIC

In this work, we make a further step in bringing together different approaches to non-equilibrium thermodynamics. The structure of the moment hierarchy derived from the Boltzmann equation is at the heart of rational extended thermodynamics (RET, developed by Ingo Müller and Tommaso Ruggeri). Whereas the full moment hierarchy has the structure expressed in the general equation for the nonequilibrium reversible-irreversible coup- ling (GENERIC), the Poisson bracket structure of reversible dynamics postulated in that approach is a major obstacle for truncating moment hierarchies, which seems to work only in exceptional cases (most importantly, for the five moments associated with conservation laws). The practical importance of truncated moment hierarchies in rarefied gas dynamics and microfluidics motivates us to develop a new strategy for establishing the full GENERIC structure of truncated moment equations, based on non-entropy-producing irreversible processes associated with Casimir symmetry. Detailed results are given for the special case of 10 moments. This article is part of the theme issue 'Fundamental aspects of nonequilibrium thermodynamics'.

The way from microscopic many-particle theory to macroscopic hydrodynamics

Starting from the microscopic description of a normal fluid in terms of any kind of local interacting many-particle theory we present a well defined step by step procedure to derive the hydrodynamic equations for the macroscopic phenomena. We specify the densities of the conserved quantities as the relevant hydrodynamic variables and apply the methods of non-equilibrium statistical mechanics with projection operator techniques. As a result we obtain time-evolution equations for the hydrodynamic variables with three kinds of terms on the right-hand sides: reversible, dissipative and fluctuating terms. In their original form these equations are completely exact and contain nonlocal terms in space and time which describe nonlocal memory effects. Applying a few approximations the nonlocal properties and the memory effects are removed. As a result we find the well known hydrodynamic equations of a normal fluid with Gaussian fluctuating forces. In the following we investigate if and how the time-inversion invariance is broken and how the second law of thermodynamics comes about. Furthermore, we show that the hydrodynamic equations with fluctuating forces are equivalent to stochastic Langevin equations and the related Fokker-Planck equation. Finally, we investigate the fluctuation theorem and find a modification by an additional term.

Preservation of thermodynamic structure in model reduction

Based on the availability of an invariant manifold, we develop a model-reduction procedure that preserves thermodynamic structure. More concretely, we construct the Poisson and irreversible brackets of the general equation for the nonequilibrium reversible-irreversible coupling of nonequilibrium thermodynamics by means of the ideas originally introduced for handling constraints. The general ideas are then applied to the Kramers problem, that is, the description of transitions between two potential wells separated by a high barrier. This example reveals how a fortuitous cancellation mechanism that allows a logarithmic entropy to generate a linear diffusion equation is inherited by a master equation resulting from model reduction.

Communication: quantum dynamics in classical spin baths

A formalism for studying the dynamics of quantum systems embedded in classical spin baths is introduced. The theory is based on generalized antisymmetric brackets and predicts the presence of open-path off-diagonal geometric phases in the evolution of the density matrix. The weak coupling limit of the equation can be integrated by standard algorithms and provides a non-Markovian approach to the computer simulation of quantum systems in classical spin environments. It is expected that the theory and numerical schemes presented here have a wide applicability.

Dynamic renormalization in the framework of nonequilibrium thermodynamics

We show how the dynamic renormalization of nonequilibrium systems can be carried out within the general framework of nonequilibrium thermodynamics. Whereas the renormalization of Hamiltonians is well known from equilibrium thermodynamics, the renormalization of dissipative brackets, or friction matrices, is the main new feature for nonequilibrium systems. Renormalization is a reduction rather than a coarse-graining technique; that is, no new dissipative processes arise in the dynamic renormalization procedure. The general ideas are illustrated for dilute polymer solutions where, in renormalizing bead-spring chain models, dissipative hydrodynamic interactions between different smaller beads contribute to the friction coefficient of a single larger bead.

Path-integral centroid dynamics for general initial conditions: A nonequilibrium projection operator formulation

The formulation of path-integral centroid dynamics is extended to the quantum dynamics of density operators evolving from general initial states by means of the nonequilibrium projection operator technique. It is shown that the new formulation provides a basis for applying the method of centroid dynamics to nonequilibrium situations and that it allows the derivation of new formal relations, which can be useful in improving current equilibrium centroid dynamics methods. A simple approximation of uniform relaxation for the unprojected portion of the Liouville space propagator leads to a class of practically solvable equations of motion for the centroid variables, but with an undetermined parameter of relaxation. This new class of equations encompasses the centroid molecular-dynamics (CMD) method as a limiting case, and can be applied to both equilibrium and nonequilibrium situations. Tests for the equilibrium dynamics of one-dimensional model systems demonstrate that the new equations with appropriate choice of the relaxation parameter are comparable to the CMD method.

Duality in nonextensive statistical mechanics

We revisit recent derivations of kinetic equations based on Tsallis' entropy concept. The method of kinetic functions is introduced as a standard tool for extensions of classical kinetic equations in the framework of Tsallis' statistical mechanics. Our analysis of the Boltzmann equation demonstrates a remarkable relation between thermodynamics and kinetics caused by the deformation of macroscopic observables.

Macroscopic dynamics through coarse-graining: a solvable example

The recently derived fluctuation-dissipation formula [A. N. Gorban et al., Phys. Rev. E 63, 066124 (2001)] is illustrated by the explicit computation for McKean's kinetic model [H. P. McKean, J. Math. Phys. 8, 547 (1967)]. It is demonstrated that the result is identical, on the one hand, to the sum of the Chapman-Enskog expansion, and, on the other hand, to the exact solution of the invariance equation. The equality between all three results holds up to the crossover from the hydrodynamic to the kinetic domain.

Thermodynamically consistent incorporation of the Schneider rate equations into two-phase models

We formulate a solid-liquid two-phase model including viscous stresses, heat conduction in the two phases, as well as heat exchange through the interface, and a phase change in the structure of nonequilibrium thermodynamics described by a general equation for the nonequilibrium reversible-irreversible coupling (GENERIC). The evolution of the microstructure is studied in terms of the Schneider rate equations introducing the nucleation rate and the radial growth rate of the solid phase. The application of the GENERIC structure shows that this radial growth factor is not an additional, independent material function but is to be expressed in terms of the difference in the chemical potentials, in the temperatures, and in the pressures between the two phases. The contribution due to the pressure difference appears in conjunction with the surface tension in such a way, that a driving force results only if deviations from a generalized version of the Laplace equation occur. Furthermore, it is found that for conditions under which the radial growth rate is zero, the nucleation rate must vanish.

Ehrenfest's argument extended to a formalism of nonequilibrium thermodynamics

A general method of constructing dissipative equations is developed, following Ehrenfest's idea of coarse graining. The approach resolves the major issue of discrete time coarse graining versus continuous time macroscopic equations. Proof of the H theorem for macroscopic equations is given, several examples supporting the construction are presented, and generalizations are suggested.