Prigogine Theorem Research Articles

Effective Boltzmann law and Prigogine theorem of minimum entropy production in highly charged ion plasmas

Thermodynamics of irreversible processes is applied to study the interaction of matter and radiation field out of thermodynamic equilibrium using a collisional-radiative equilibrium model. We propose to approximate the electronic configuration populations by an effective Boltzmann law. The corresponding effective temperature is obtained by minimizing the rate of entropy production. Numerical calculations for iron plasmas are presented and discussed for up to 100% changes in photon temperature with respect to electron temperature. We found that the notion of effective Boltzmann law combined with the Prigogine theorem of minimum entropy production is very efficient to describe steady-state plasmas far from local thermodynamic equilibrium.

Diffusion modelling of metamorphic layered coronas with stability criterion and consideration of affinity

Layered coronas between two reactant minerals can, in many cases, be attributed to diffusion-controlled growth with local equilibrium. This paper clarifies and unifies the previous approaches of various authors to the simplest form of modelling, which uses no assumed values for thermochemical quantities. A realistic overall reaction must be estimated from measured overall proportions of minerals and their major element compositions. Modelling is not restricted to a particular number of components S, relative to the number of phases Φ. IfΦ > S + 1, the overall reaction is a combination of simultaneous reactions. The stepwise method, solving for the local reaction at each boundary in turn, is extended to allow for recurrence of a mineral (its presence in two parts of the layer structure separated by a gap). The equations are also given in matrix form. A thermodynamic stability criterion is derived, determining which layer sequence is truly stable if several are computable from the same inputs. A layer structure satisfying the stability criterion has greater growth rate (and greater rate of entropy production) than the other computable layer sequences. This criterion of greatest entropy production is distinct from Prigogine's theorem of minimum entropy production, which distinguishes the stationary or quasi-stationary state from other states of the same layer sequence. The criterion leads to modification of previous results for coronas comprising hornblende, spinel, and orthopyroxene between olivine (Ol) and plagioclase (Pl). The outcome supports the previous inference that Si, and particularly Al, commonly behave as immobile relative to other cation-forming major elements. The affinity (−ΔG) of a corona-forming reaction is estimated, using previous estimates of diffusion coefficient and the duration t of reaction, together with a new model quantity (−ΔG) *. For an example of the Ol + Pl reaction, a rough calculation gives (−ΔG) > 1.7RT (per mole of P1 consumed, based on a 24-oxygen formula for Pl). At 600–700°C, this represents (−ΔG) > 10kJ mol −1 and departure from equilibrium temperature by at least ∼ 100°C. The lower end of this range is petrologically reasonable and, for t < 100Ma, corresponds to a Fick's-law diffusion coefficient for Al, D Al > 10 −25m 2s −1, larger than expected for lattice diffusion but consistent with fluid-absent grain-boundary diffusion and small concentration gradients.

On the statistical thermodynamics of a model for non-equilibrium semiconductors

We describe the non-equilibrium thermodynamics of a model for semiconductors under high levels of excitation using the non-equilibrium statistical operator method. We obtain thermodynamic functions in terms of thermodynamic variables that are accessible to experimental measurements via ultrafast laser spectroscopy. Calculations of entropy production and the rate of entropy production are performed. Kinetic equations for the relaxation processes are derived and Onsager-like coefficients are defined. It is shown that for the system we consider Prigogine's theorem of minimum entropy production holds even in the non-linear regime, and the Glansdorff-Prigogine universal evolution criterion is verified.

The Gibbs-Duhem equation in membrane transport.

It is argued that the Gibbs-Duhem equation alone cannot be used for deriving conclusions about the pressure gradient in membrane permeation. Statements regarding spatial variation of pressure in conjunction with chemical potential gradients of the components can legitimately be drawn from an equation that results from a combination of the G-D equation and the mechanical equilibrium equation. The derived equation has been applied here for explaining the mechanics of osmosis. In a further application, the frictional model has been improved here because the driving force also includes the membrane-solute potential interaction, thus allowing the solute partition coefficient to appear in the calculations naturally. By recognizing that because of the membrane-solution interaction, external forces of both potential and frictional character are present, the dissipation function is shown to depend explicitly on the centre-of-mass velocity. Thus the reference velocity for diffusive fluxes cannot be chosen arbitrarily, making Prigogine's theorem invalid in this approach to describing membrane permeation.

Some properties of generalized irreversible thermodynamics founded by Bearman–Kirkwood equations

Generalized thermodynamics founded by Bearman–Kirkwood equations is considered. It is shown that the Prigogine theorem on the entropy production invariance also holds when inertia and local stresses contribute to the diffusion-dependent part of the dissipation function. A comparison with other approaches is made and conditions which reduce the generalized theory to the classical phenomenological approach are obtained for the case of mechanical equilibrium. The role of coupling between diffusion and viscous flow in barodiffusion processes is discussed. Bearman–Kirkwood partial salt viscosities are estimated for the low concentration region. The generalized theory predicts that in the presence of viscous effects and in the low concentration region, electrolyte barodiffusion depends mainly on local stresses. As a consequence, under the mentioned conditions, electrolyte barodiffusion is directed ‘‘against’’ the motion of the mass center. Thus, for some porous membranes the ‘‘practical’’coefficient LπP may be interpreted by the diffusion–viscous flow coupling. The possibility of a measurement of the Bearman–Kirkwood ionic partial viscosities is shortly discussed.

The kinetic phenomena in nonisothermal motion of a binary gas mixture through a plane channel

Based on a linearized kinetic equation (the McCormack model), the theory of motion of binary gas mixtures in a plane channel owing to the temperature, concentration and pressure gradients is established. As a result, the relationships are obtained showing the dependence of the thermal, diffusional and numerical mean fluxes on the Knudsen number and concentration of the components. The calculations are carried out for the mixtures He-Ar, He-Ne, Ne-Ar with the use of three models of intermolecular potential: ‘solid spheres’, ‘Maxwell molecules’, Lennard-Jones (6–12)-potential. Based on the Prigogine theorem, the effects characteristic of gas mixtures (thermal diffusion and pressure diffusion separation of mixtures, diffusional baroeffect, thermomolecular pressure difference) are considered. The theory is compared with experiment.

Principles of minimum entropy production in transport theory

Formulates in microscopic terms a principle of minimum entropy production which may alternatively be regarded as a full generalisation of Kohler's principle for the solution of the semiclassical Boltzmann equation. The principle is thus in fact a general variational principle for the expressions of Kubo type for the transport coefficients of a linear system. It is also shown that the usual principle of minimum entropy production in macroscopic terms (Prigogine's theorem, 1947) follows from the general principle. When a magnetic field is present there is a sense in which the principle is still valid, but one must discuss together the original system and one identical to it except that the magnetic field is reversed.

Free energy flow in the process of collecting material for body build up of aquatic organisms

Abstract A general formula defining the free energy expenditures in the processes of algal nutrition and zooplankton feeding has been derived. It appears to be symmetrical in respect of space and time scales. In conditions of optimal control the theoretical relationship between effective free energy expenditures and losses appears to be the ‘golden proportion’. The general trends of the system due to the first and second laws of thermodynamics and Prigogine's theorem form a theoretical basis for phenomenological rules of ecosystem's development.

Electromechanical equilibrium in osmosis

To apply the flux equations originating from Onsager's principle of least dissipation of energy to osmosis in permeable membranes, the membrane must be taken as the system of reference. Postulating that in an isobaric system the permeation of any particle through the membrane exerts no force on the membrane itself, a fundamental property of such a system of reference is ensured. This leads to an electromechanical equilibrium between the frictional and the electric forces in the membrane. According to Prigogine's theorem of mechanical equilibrium, chemical forces (potentials) are absent in such a relation.

Validity of the prigogine theorem in establishing a significantly nonequilibrium steady state of a thermodynamic system

It is concluded that there exists a certain class of thermodynamic systems for which the Prigogine theorem is satisfied in setting up a steady state significantly different from a equilibrium state.

Operation of the simplest gas ejector from the point of view of the thermodynamics of irreversible processes

It is shown that the existence of critical modes of operation of a sonic gas ejector with a cylindrical mixing chamber can be determined by one of the theorems of the thermodynamics of irreversible processes, the Prigogine theorem.

Theorie phenomenologique de la diffusion chimique

The phenomenological theory of chemical diffusion consists in applying Gibbs' formula in a barycentric referential. In this barycentric referential, whether mass or atomic (in the latter case there must be no chemical reaction), the sum of the fluxes of matter is zero if working with a constant number of sites per unit volume. It is then identical with the laboratory referential (Matano referential). The entropy source can then be formulated ( σ M ). The discovery of a displacement of crystal planes during chemical diffusion leads us to consider a second referential (Kirkendall referential) moving at a velocity V with respect to the Matano referential. In the Kirkendall referential, the flux of vacancies J 1 exactly compensates the difference in the atomic fluxes J A and J B (unidirectional diffusion) and the entropy source σ k therein must be equal to σ M according to * * n i is the number of atoms i per unit volume, n 1 has the same meaning for the vacancies. Prigogine's theorem. It can be demonstrated, subject to certain hypotheses and approximations, viz., that: The number of sites per unit volume, n, is conservative The concentration of vacancies is always negligible compared with that of the species A and B The displacement of the crystal planes is due to a first order reaction: l a c u n e + ⇄ d i s l o c a t i o n 0 (it is this reaction which introduces a source term σ i in the variation of the number of vacancies per unit volume and time) The number of dislocations is high enough for their density not to be affected by this reaction and for vacancy supersaturation to remain slight. The chemical potential gradient of the vacancies can always be neglected compared with that of the mass species. The result is a formulation of the entropy source σ k (De Groot's and Mazur's identification criteria being respected) in which all the terms linked directly with the vacancies disappear (potential, source, etc.). Tσ κ =-J A· grad(µ A)-J B· grad(µ B) with: T=absolute temperature N i =chemical potential of the species i The identification: σ k=σ M can then be made, from which it can be shown that the velocity V of the crystal planes is the mean atomic velocity: V = n A V A + n B V B + n 1 V 1 n which can be taken as equivalent to an atomic barycentric velocity weighted to cover those of all species (mass and vacancy) with V i being the velocity of the species i in the laboratory referential, and V t being the velocity of the vacancies. * In our analysis we drew widely from the work of Bailly (Thermodynamic study of the Kirkendall effect in interdiffusion, paper presented to the Autumn Meeting of the Société Française de Métallurgie in October 1970), and of Adda and Philibert (Diffusion in Solids, Presses Universitaires de France, Paris 1966).

General validity of Prigogine theorem in irreversible thermodynamics

General validity of Prigogine theorem in irreversible thermodynamics

The second law of thermodynamics for heterogenous flow systems — III Effect of the conditions of mechanical equilibrium and electroneutrality on simultaneous heat and mass transfer and the Prigogine theorem

An integrated form of the Gibbs-Duhem equation across the phase boundary is developed for the various forms of mass and thermal driving forces previously introduced [4] and is shown to provide a linear dependence among these forces. The simplification arising for the special case of surface mechanical equilibrium and electroneutrality is pointed out when it is found that the number of independent driving forces is now one less than the number of fluxes, and it is also shown that the Prigogine theorem is not valid on the phase boundary for this case. Finally the relation of the Gibbs-Duhem condition to the simple constituent mole fraction sum relation is illustrated for the further restriction to isothermal mass transfer in ideal solutions.

Variational Principles in Irreversible Thermodynamics with Application to Viscoelasticity

General differential equations are derived for the time history of a thermodynamic system undergoing irreversible transformations. This is done by using Onsager's principle, and introducing generalized concepts of free energy and thermodynamic potentials. From these equations it is shown that the instantaneous evolution of the system satisfies a principle of minimum rate of entropy production. It is also shown how Prigogine's theorem for the stationary state fits into the present theory. Another variational principle is established for the case where certain variables are ignored in analogy with the methods of virtual work in mechanics. This principle which applies to complex physical-chemical systems is developed more specifically for viscoelastic phenomena, and as an example the differential equations for the deflection of a viscoelastic plate is derived.