Positive Entropy Production Research Articles

The Entropy Production of Diffusion Processes on Manifolds and Its Circulation Decompositions

In non-equilibrium statistical mechanics, the entropy production is used to describe flowing in or pumping out of the entropy of a time-dependent system. Even if a system is in a steady state (invariant in time), Prigogine suggested that there should be a positive entropy production if it is open. In 1979, the first author of this paper and Qian Min-Ping discovered that the entropy production describes the irreversibility of stationary Markov chains, and proved the circulation decomposition formula of the entropy production. They also obtained the entropy production formula for drifted Brownian motions on Euclidean space R n (see a report without proof in the Proc. 1st World Congr. Bernoulli Soc.). By the topological triviality of R n , there is no discrete circulation associated to the diffusion processes on $R^n$. In this paper, the entropy production formula for stationary drifted Brownian motions on a compact Riemannian manifold M is proved. Furthermore, the entropy production is decomposed into two parts – in addition to the first part analogous to that of a diffusion process on R n , some discrete circulations intrinsic to the topology of M appear! The first part is called the hidden circulation and is then explained as the circulation of a lifted process on M×S 1 around the circle S 1. The main result of this paper is the circulation decomposition formula which states that the entropy production of a stationary drifted Brownian motion on M is a linear sum of its circulations around the generators of the fundamental group of M and the hidden circulation.

Constructing an Entropy-Stable Upwind Scheme for Compressible Fluid Flow Computations

A relationship between entropy processes and numerical upwinding is examined in the context of computational gasdynamics. A discretized form of the entropy inequality is constructed at the integration point where convection-diffusion modeling occurs in finite volume methods. Conventional upwinding schemes may violate the local form of the second law of thermodynamics, but a modified upwinding scheme uses additional momentum constraints and pressure terms to provide a positive entropy production rate. The second law is seen as an important quantitative measure of nonphysical numerical results, as well as a sound basis for error analysis. Applications to converging-diverging nozzle and blunt-body flow problems demonstrate the promising performance of the overall numerical algorithm

Phase-field model of solidification of a binary alloy

We develop a rather general thermodynamically consistent phase-field model for solidification of a binary alloy, based on an entropy functional that contains squared gradient terms in the energy density, the composition and the phase-field variable. By assuming positive local entropy production, we derive generalized phase-field equations for an alloy, including cross terms that connect thermal and compositional driving forces to energy and solute fluxes. We explore this model in detail for a regular solution and show that four existing models can be recovered as special cases. We also use it to develop a new phase-field model for an alloy in which an explicit phase-field variable is absent.

The kinetic chemical equilibrium regime

We investigate reactive gas mixtures in the kinetic chemical equilibrium regime. Our starting point is a generalized Boltzmann equation with a chemical source term valid for arbitrary reaction mechanisms and yielding a positive entropy production. We first study the Enskog expansion in the kinetic chemical equilibrium regime. We derive a new set of macroscopic equations in the zeroth- and first-order regimes, expressing conservation of element densities, momentum and energy. The transport fluxes arising in the Navier–Stokes equilibrium regime are the element diffusion velocities, the heat flux vector and the pressure tensor and are written in terms of transport coefficients. Upon introducing species diffusion velocities, the kinetic equilibrium regime appears to be formally equivalent to the one obtained for gas mixtures in chemical nonequilibrium and then letting the chemical reactions approach equilibrium. The actual values of the transport coefficients are, however, different. Finally, we derive the entropy conservation equation in the Navier–Stokes equilibrium regime and show that the source term is positive and that it is compatible with Onsager’s reciprocal relations.

On Entropy Production in Informational Statistical Thermodynamics

We consider the question of the existence of a generalized H-theorem in the context of the variational method in the information-theoretical approach that generates the nonequilibrium statistical operator formalism. After briefly reviewing how the latter provides mechano-statistical foundations for phenomenological irreversible thermodynamics, the so-called Informational Statistical Thermodynamics, we discuss how dissipative phenomena are accounted for by the procedure. Such effects are related to a generalized H-theorem and a weak criterion of positive entropy production. These results are a consequence of the definition of a coarse-grained statistical entropy, resulting from the projection of the full nonequilibrium distribution on the subspace of the slow relaxing dynamical quantities, that are appropriate for the description of the irreversible evolution of the system from the state of initial preparation.

On the inverse problem of statistical physics: from irreversible semigroups to chaotic dynamics

We show that all measure preserving stationary Markov processes arise as projections of Kolmogorov dynamical systems which have positive entropy production and are prototypes of chaos. This result not only contributes to the clarification of the relation of dynamics with stochastic processes but also shows the physical significance of the Misra–Prigogine–Courbage theory of irreversibility in the more general context of the inverse problem of statistical physics. Because we want positivity preserving transformations, our procedure although analogous to the Sz–Nagy–Foias Dilation theory has a different viewpoint, that of positive dilations.

Entropy production in open volume-preserving systems

We describe a mechanism leading to positive entropy production in volume-preserving systems under nonequilibrium conditions. We consider volume-preserving systems sustaining a diffusion process like the multibaker map or the Lorentz gas. A continuous flux of particles is imposed across the system resulting in a steady gradient of concentration. In the limit where such flux boundary conditions are imposed at arbitrarily separated boundaries for a fixed gradient, the invariant measure becomes singular. For instance, in the multibaker map, the limit invariant measure has a cumulative function given in terms of the nondifferentiable Takagi function. Because of this singularity of the invariant measure, the entropy must be defined as a coarse-grained entropy instead of the fined-grained Gibbs entropy, which would require the existence of a regular measure with a density. The coarse-grained entropy production is then shown to be asymptotically positive and, moreover, given by the entropy production expected from irreversible thermodynamics.

Positivity of entropy production in the presence of a random thermostat

We study nonequilibrium statistical mechanics in the presence of a thermostat acting by random forces, and propose a formula for the rate of entropy productione(μ) in a state μ. When μ is a natural nonequilibrium steady state we show thate(μ)≥0, and sometimes we can provee(μ)>0.

Positivity of entropy production in nonequilibrium statistical mechanics

We analyze different mechanisms of entropy production in statistical mechanics, and propose formulas for the entropy production ratee(μ) in a state μ. When μ is steady state describing the long term behavior of a system we show thate(μ)≥0, and sometimes we can provee(μ)>0.

Process systems, passivity and the second law of thermodynamics

In this paper we use the first and second laws of thermodynamics to motivate a theory for nonlinear process control. Our main tenets are: positive entropy production, boundedness of entropy in terms of energy, concavity of the entropy density and Helmholtz free energy as a storage function. These give the process system a causal input-output description, zero state detectability and stabilizability. To make the theory apply to practical systems we follow ideas from classical irreversible thermodynamics and extend the concept of entropy of the non-equilibrium by assuming local equilibrium.

Thermodynamically-consistent phase-field models for solidification

In an effort to unify the various phase-field models that have been used to study solidification, we have developed a class of phase-field models for crystallization of a pure substance from its melt. These models are based on an entropy functional, as in the treatment of Penrose and Fife, and are therefore thermodynamically consistent inasmuch as they guarantee spatially local positive entropy production. General conditions are developed to ensure that the phase field takes on constant values in the bulk phases. Specific forms of a phase-field function are chosen to produce two models that bear strong resemblances to the models proposed by Langer and Kobayashi. Our models contain additional nonlinear functions of the phase field that are necessary to guarantee thermodynamic consistency.

Entropy budget of conifer branches

From energy budget data for a branch of ponderosa pine given by Gates, Tibbals and Kreith, entropy fluxes into or out of the branch due to solar radiation, infrared radiation, transpiration and convection are calculated. Net entropy flow into the branch is negative. Assuming that the entropy in the branch is at steady state, the entropy production in the branch of ponderosa pine is calculated and shown to be positive. A positive entropy production indicates that the Second Law of Thermodynamics is certainly valid in the branch. Entropy productions in other conifers, blue spruce and white fir, and in a single pine needle in a horizontal position are also calculated. The entropy production (Sprod) increases linearly with the solar energy absorbed by the plant surface (Esolar); Sprod≈(30.6 Esolar)×10−4. The ratio (Sprod/Esolar) does not differ between deciduous leaves reported earlier and conifer branches. The theorem of oscillating entropy production proposed earlier holds also for conifer branches and will be of universal nature, applicable to all plant leaves.

OnH-theorem and entropy production

The paper deals with the study of aH-theorem for an isolated system and the positivity of entropy production of a closed system on the basis of the process of removing correlation.

Entropy Production in Oscillating Chemical Systems

Abstract The entropy production in oscillating homogenous chemical systems is investigated by analyzing the difference between the average entropy production rate in a stable periodic oscillatory mode and in the corresponding unstable stationary state. A general analytical expression for this difference in the neighborhood of a Hopf bifurcation is derived. The entropy production in two typical models of chemical systems with unstable stationary states and stable periodic oscillations is investigated, using fixed concentrations as control parameters. The models exemplify both positive and negative entropy production rate differences. One of the investigated models has four free concentrations, the other three. The rate expressions are given by second order mass action kinetics with reverse reactions taken into account. The flows of reactants and products are controlled so that only the free concentrations vary, and the entropy of mixing associated with these flows is discussed.

Thermodynamical theory of boundary conditions for polyatomic gases

A boundary condition for the distribution operator of a dilute polyatomic gas at a gas-solid interface is presented. The derivation is based on the methods of nonequilibrium thermodynamics in connection with the general reciprocity postulate for the interface. The boundary condition consists of a linear relation between the sum and the difference of the distribution operator and its time-reversal. The connecting wall-collision superoperator is symmetric according to the postulate of universal reciprocity and positive definite in order to guarantee a positive surface entropy production. Finally, the new boundary condition is rewritten as a linear relation between the distribution operator of the molecules reflected from the solid and the distribution operator of the incident molecules. In this form it is compared with the boundary conditions used till now, which are, however, almost entirely restricted to monatomic gases. Finally, in the Knudsen case for a heat conducting polyatomic gas between parallel plates, the distribution function is expressed by an integral over the wall-collision superoperator.

Critical exponent of the heat of transport of water– isobutyric acid mixtures

The Dufour effect (diffusion thermoeffect) has been used to obtain the critical exponent of the heat of transport in binary mixtures of isobutyric acid and water near the liquid–liquid critical point. As the critical temperature is approached experimentally, the heat of transport vanishes with a critical exponent of about 2/3. The i critical exponent for the Onsager coefficient Ω01 is zero. This value is consistent with the semidefinite positive entropy production r requirement and is identical to the best literature value for the critical exponent of the reciprocal Onsager coefficient Ω10. In order to explain the observed critical behavior of the heat of transport a new molecular interpretation based on the random thermal motion model for diffusion is suggested.

Entropy production by anomalous drift wave transport

The macroscopic transport equations describing the evolution of the mean density and temperature of a hot confined plasma subject to drift wave fluctuations are shown to lead a positive definite entropy production.

Gaseous diffusion in a stressed-thermoelastic solid. Part II: Thermodynamic structure and transport theory

In part I of the present work a constitutive theory is established for a gas diffusing, as a dilute solution, in a linear thermoelastic solid. The theory is compatible with the laws of thermomechanics i.e., mass, momentum and energy balance, as well as with the principles of positive entropy production and material objectivity. The assumptions of small concentrations of the gas and the absence of viscous effects simplify the analysis considerably. This allows a thermomechanical model for the solid to be assumed, independent of the presence of the gas. The constitutive model adopted for the solid is the linear uncoupled theory of thermoelasticity. In this level of generality a transport theory is developed for the gas by using the conservation laws of mass, linear momentum and internal energy. Some uncommon identities for the thermodynamic quantities of the gas are also established, based on the derived functional dependence for the internal energy, entropy, free energy and stress tensor of the gas. Thus a Gibb's type equation is obtained. In addition a comparison with classical irreversible thermodynamics, previous intuitive theories and experiments is made.

Transmission electron spin resonance as a probe of the metallic interface

Observation of coupled TESR in bilayers of dissimilar pure metals is reported. Positive entropy production and detailed balance are incorporated in general linear relations between non-equilibrium spin densities and currents at the interface. Spin transport coefficients for samples exhibiting both strong and weak coupling are deduced from the data by comparison with computer generated lineshapes.

On the theory of linear viscoelastic behavior

The after-effect relation between stress and strain in a linear viscoelastic material is examined using general principles only. First of all, it is a linear passive system and all results on linear passive systems can be applied. Secondly, thermodynamic principles are invoked. They are the existence of stress equilibrium for constant strain, of welldefined flow properties or equilibrium of strain for constant stress, the existence of asymptotically harmonic stress or strain if the strain or stress, respectively, is a harmonic function after some transients, and finally the existence of positive entropy production in a real process. Only if one looks more closely into the internal processes, wich are going on under an applied stress or strain history, is it possible to give a more detailed characterisation of the after-effect functions: If only even internal variables are excited, one arrives at the usually adopted description of after-effect functions in terms of retardation and relaxation spectra. It may be that odd internal variables, which are connected with some internal inertia, come into play for relatively fast processes only, which would justify their neglect for slow processes. — A very simple mathematical characterisation of viscoelastic materials with pure relaxation behavior is given. The results are derived for isotropic materials; but they can easily be generalized to anisotropic materials.