Glansdorff-Prigogine Criterion Research Articles

Information geometry, trade-off relations, and generalized Glansdorff–Prigogine criterion for stability

We discuss a relationship between information geometry and the Glansdorff–Prigogine criterion for stability. For the linear master equation, we found a relation between the line element and the excess entropy production rate. This relation leads to a new perspective of stability in a nonequilibrium steady-state. We also generalize the Glansdorff–Prigogine criterion for stability based on information geometry. Our information-geometric criterion for stability works well for the nonlinear master equation, where the Glansdorff–Prigogine criterion for stability does not work well. We derive a trade-off relation among the fluctuation of the observable, the mean change of the observable, and the intrinsic speed. We also derive a novel thermodynamic trade-off relation between the excess entropy production rate and the intrinsic speed. These trade-off relations provide a physical interpretation of our information-geometric criterion for stability. We illustrate our information-geometric criterion for stability by an autocatalytic reaction model, where dynamics are driven by a nonlinear master equation.

Nonequilibrium time dynamics of genetic evolution.

Biological systems are typically highly open, nonequilibrium systems that are very challenging to understand from a statistical mechanics perspective. While statistical treatments of evolutionary biological systems have a long and rich history, examination of the time-dependent nonequilibrium dynamics has been less studied. In this paper we first derive a generalized master equation in the genotype space for diploid organisms incorporating the processes of selection, mutation, recombination, and reproduction. The master equation is defined in terms of continuous time and can handle an arbitrary number of gene loci and alleles and can be defined in terms of an absolute population or probabilities. We examine and analytically solve several prototypical cases which illustrate the interplay of the various processes and discuss the timescales of their evolution. The entropy production during the evolution towards steady state is calculated and we find that it agrees with predictions from nonequilibrium statistical mechanics where it is large when the population distribution evolves towards a more viable genotype. The stability of the nonequilibrium steady state is confirmed using the Glansdorff-Prigogine criterion.

Electric field-induced rearrangement of charged species in metal oxide devices with resistive change: thermodynamic limitations

Hypotheses used to interpret results of electrical measurements on metal oxides which are employed for gas sensors and resistive switching memristors are often based on consideration of drift migration of species (such as oxygen vacancies, chemisorbed ions) under electric field. We offer simple conceptual arguments restricting this approach from the view points of potential minimum principle, ambipolar diffusion and non-equilibrium thermodynamics (Glansdorff-Prigogine criterion).

The Glansdorff–Prigogine stability criterion for biochemical reaction networks

Local stability analysis of steady states in mathematical models of biochemical reaction networks is an important tool for systems biology. The second variation of the Gibbs energy around a steady state is a positive definite function and a candidate for a Lyapunov function. A sufficient condition for the local stability is the local negative definiteness of the time derivative of this function. This is expressed by the Glansdorff–Prigogine stability criterion. Previously, the criterion was criticized to be overly conservative and difficult to check. Here, we derive an easily testable form of the criterion for models of biochemical networks. The criterion can be evaluated with incomplete knowledge of the parameters. For ideal mass-action kinetics, it depends only on the steady state fluxes. For reaction systems in ideal solutions, the Glansdorff–Prigogine criterion is overly conservative and we give a tighter criterion that depends on the same subset of the parameters as the Glansdorff–Prigogine criterion. Whenever these criteria are indefinite, there exist parameter values such that the steady state is unstable. By means of simple example systems we explore these aspects and discuss the possible uses of the criteria for systems biology.

Study on self-excited oscillation in a thermoacoustic engine with GlansdorffPrigogine stability criterion

The onset process of self-excited oscillation in a thermoacoustic engine is a nonlinear self-organization process far from equilibrium, and is of great importance for the operation of a thermoacoustic engine. From the viewpoint of thermodynamic stability analysis, the GlansdorffPrigogine stability criterion of non-equilibrium thermodynamics is used to study the onset process. The excess entropy for a single-component ideal-gas thermoacoustic system is formulated, and the procedure of introducing the GlansdorffPrigogine criterion into numerical computation is discussed. A modification of the GlansdorffPrigogine criterion is made to treat the errors deriving from the numerical computation. The computation is conducted on a standing-wave thermoacoustic engine to achieve the time evolutions of the excess entropy and the excess entropy production for different steady states corresponding to different given hot end temperatures of the stack. The modified GlansdorffPrigogine criterion is successfully used to judge the stability of steady state according to the computed time evolution of the excess entropy production, and then the onset temperature (i.e., the critical hot end temperature of the stack for the onset process) can be predicted with a resolution of 1 K.

Generalization of the evolution criterion in extended irreversible thermodynamics

A postulate of extended irreversible thermodynamics is considered, according to which the entropy density is a function of ordinary thermodynamic variables and their material time derivatives. Within the thermodynamic formalism proposed, an entropy balance equation is derived and it is shown that a system in a mechanical stationary state meets an extended evolution criterion, which is a generalization of the known Glansdorff–Prigogine criterion.

A new version of extended irreversible thermodynamics and dual-phase-lag model in heat transfer

For a system involving heat transfer, a new postulate of extended irreversible thermodynamics is suggested, which, in case of a stationary states, reduces to the local equilibrium hypothesis. On the basis of this postulate, a transport equation of the dual-phase-lag model of heat transfer is derived. In the proposed formalism, an extended evolution criterion generalizing the known Glansdorff–Prigogine criterion takes place.

What can be stated by the Glansdorff–Prigogine criterion concerning the stability of mass-action kinetic systems?

We investigate which general results concerning the local stability of steady states of arbitrary chemical reaction networks can be deduced with the Glansdorff–Prigogine stability criterion. Especially, it is proven that the presence of an autocatalytic reaction is not a necessary condition for a violation of the thermodynamic stability condition. It turns out that every reaction with at least one variable reactant at each side of the reaction equation can potentially destabilize the steady states. An explicit example of a simple reaction system without autocatalytic reactions where the stability of the steady state changes via a supercritical Hopf bifurcation is discussed. Furthermore, in expanding the original concept for proving local stability to global stability analyses, a general way for constructing different Lyapunov functions is given.

A thermodynamical analysis of the stability of electrical systems

In this work a generalization of the Glansdorff-Prigogine criterion for the stability of nonequilibrium states in electrical systems is derived. This generalization is also made in the context of extended irreversible thermodynamics. In both criteria, stability of some electrical conduction situations is determined.

Morphological stability of a planar interface during unidirectional growth of a dilute binary alloy

Considering the solidification front as a discontinuity, the Glansdorff-Prigogine criterion for stability of a non-equilibrium steady state is applied to the unidirectional growth with a planar interface of a dilute binary alloy. Under the assumption of thermodynamic equilibrium at the interface, we recover either the Constitutional Supercooling or the Modified Constitutional Supercooling Criterion, depending on the assumptions made about the diffusion fields. Finally, we propose to reconciliate the various existing criteria instead of opposing them as it is usually done.

Network theory of microscopic and macroscopic behavior of master equation systems

A general microscopic and macroscopic theory is developed for systems which are governed by a (linear) master equation. The theory is based on a network representation of the master equation, and the results are obtained mostly by application of some basic theorems of mathematical graph theory. In the microscopic part of the theory, the construction of a steady state solution of the master equation in terms of graph theoretical elements is described (Kirchhoff's theorem), and it is shown that the master equation satisfies a global asymptotic Liapunov stability criterion with respect to this state. The Glansdorff-Prigogine criterion turns out to be the differential version and thus a special case of the global criterion. In the macroscopic part of the theory, a general prescription is given describing macrostates of the systems arbitrarily far from equilibrium in the language of generalized forces and fluxes of nonlinear irreversible thermodynamics. As a particular result, Onsager's reciprocity relations for the phenomenological coefficients are obtained as coinciding with the reciprocity relations of a near-to-equilibrium network.

Correction: Qualms Regarding the Range of Validity of the Glansdorff-Prigogine Criterion for Stability of Non-Equilibrium States

Correction: Qualms Regarding the Range of Validity of the Glansdorff-Prigogine Criterion for Stability of Non-Equilibrium States

Qualms Regarding the Range of Validity of the Glansdorff-Prigogine Criterion for Stability of Non-Equilibrium States

Doubt is raised concerning the range of validity of a stability criterion for non-equilibrium states which has been proposed by Glansdorff and Prigogine. In the case of a particular autocatalytic reaction, the stability analysis presented by Glansdorff and Prigogine, and by Eigen and by Katchalsky in their reviews of this problem, does not agree with our analysis, which is based upon exact solution of the relevant rate equations. We also find disagreement between the analysis based upon the Glansdorff-Prigogine criterion and our analysis of a second example which involves non-equilibrium steady states. The situation is quite delicate because seemingly innocent approximations (e.g., the use of specialized conditions in the autocatalytic reaction X + Y right arrow over left arrow 2X discussed in the sequel) may lead to the impression that the scope of validity of the criterion is wider than it actually is. By considering the stability of the equilibrium state, we conclude that the second differential of the entropy, which is at the heart of the Glansdorff-Prigogine criterion, is likely to be relevant for stability questions close to equilibrium only.

The Thermodynamic Stability Theory of Non-Equilibrium States

The paper by J. Keizer and R. F. Fox entitled "Qualms Regarding the Range of Validity of the Glansdorff-Prigogine Criterion for Stability of Non-Equilibrium States" [(1974) Proc. Nat. Acad. Sci. USA 71, 192-196.] is shown to be based on a misunderstanding of the work of the Brussels group. In order to avoid further confusion, we summarize the formulation of the stability criterion in accordance with our published work. The differences with the presentation by Keizer and Fox are pointed out and it is shown that, when correctly applied, our approach does not lead to any contradiction with other methods available for studying stability.

On thermodynamics near a steady state

The statistical deduction of a stability criterion for steady states (a generalization of the Glansdorff-Prigogine criterion) given in a previous paper is extended in some respects. Dynamical coefficients for the motion of small deviations from a steady state are expressed by correlation functions. The stability criterion is applied to some models of chemical reactions and to a model for second break-down in semiconductors.

On the statistical foundation of the thermodynamic evolution-criterion of Glansdorff and Prigogine: Friedrich Schogl. Institut fur Theoretische Physik, Lehrestuhl B, Reinisch-Westalische Technische Hoshschule, 51 Aachen, Temlergraben 55, Germany

Local stability analysis of steady states in mathematical models of biochemical reaction networks is an important tool for systems biology. The second variation of the Gibbs energy around a steady state is a positive definite function and a candidate for a Lyapunov function. A sufficient condition for the local stability is the local negative definiteness of the time derivative of this function. This is expressed by the Glansdorff–Prigogine stability criterion. Previously, the criterion was criticized to be overly conservative and difficult to check. Here, we derive an easily testable form of the criterion for models of biochemical networks. The criterion can be evaluated with incomplete knowledge of the parameters. For ideal mass-action kinetics, it depends only on the steady state fluxes. For reaction systems in ideal solutions, the Glansdorff–Prigogine criterion is overly conservative and we give a tighter criterion that depends on the same subset of the parameters as the Glansdorff–Prigogine criterion. Whenever these criteria are indefinite, there exist parameter values such that the steady state is unstable. By means of simple example systems we explore these aspects and discuss the possible uses of the criteria for systems biology.