Ilya Prigogine

Symmetry Breaking Instabilities in Dissipative Systems. II

The thermodynamic theory of symmetry breaking instabilities in dissipative systems is presented. Several kinetic schemes which lead to an unstable behavior are indicated. The role of diffusion is studied in a more detailed way. Moreover we devote some attention to the problem of occurrence of time order in dissipative systems. It is concluded that there exists now a firm theoretical basis for the understanding of chemical dissipative structures. It may therefore be stated that a theoretical basis also exists for the understanding of structural and functional order in chemical open systems.

Thermodynamics of evolution

The physicochemical basis of biological order is a puzzling problem that has occupied whole generations of biologists and physicists and has given rise, in the it, to passionate discussions. Biological systems are highly complex and ordered objects. It is generally accepted that the present order reflects structures acquired during a long evolution. Moreover, the maintenance of order in actual living systems requires a great number of metabolic and synthetic reactions as well as the existence of complex mechanisms controlling the rate and the timing of the various processes. All these features bring the scientist a wealth of new problems. In the first place one has systems that have evolved spontaneously to extremely organized and complex forms. On the other hand metabolism, synthesis and regulation imply a highly heterogeneous distribution of matter inside the cell through chemical reactions and active transport. Coherent behavior is really the characteristic feature of biological systems (see the box on...

On Symmetry‐Breaking Instabilities in Dissipative Systems

The theory of hydrodynamic instability has always been an important part of fluid dynamics [see, e.g., Chandrasekhar, in Hydrodynamic and Hydromagnetic Stability (Clarendon Press, Oxford, England, 1961) and Non‐Equilibrium Thermodynamics, Variation Techniques, and Stability, R. J. Donnelly, R. Herman, and I. Prigogine, Eds. (University of Chicago Press, Chicago, Ill., 1966)]. Such instabilities involve both convective processes (such as mechanical flow) and dissipative processes (such as viscous dissipation). We investigate the possibility of an instability in purely dissipative systems involving chemical reactions and transport processes such as diffusion, but no hydrodynamic motion. We demonstrate that for well‐defined values of the constraints such as the chemical affinities of the over‐all reactions and the constants involved, such systems can indeed become unstable. Such an instability is investigated following an example of autocatalytic reactions first proposed by Turing. The major feature of this ...

Biologie et thermodynamique des phénomènes irréversibles

The thermodynamic study of systems in which stationary (non equilibrium) states were possible, led one of us (I. P.) to a number of general conclusions. In the present paper these conclusions are summarized and briefly discussed from a biological standpoint. It appears that the evolution of such systems is towards states with the least production of entropy (per mass unit) compatible with the conditions imposed. In the case of living matter this corresponds approximately to states of minimum metabolism. During this evolution the entropy contained in the system may decrease whilst the heterogenity increases. But this increase in heterogenity can only take place when there is a decrease in the entropy production, that is an evolution of the metabolism. We are thus led to suggest a physicochemical interpretation of Lamarchism. Finally we call attention to the fact that the moderation principle ofLe Chatelier-Braun is not limited to equilibrium states.

Sur les propriétés différentielles de la production d'entropie

Synopsis The time variation of the entropy production (per unit volume and unit time) is split into two parts. The first is related to the changes of the generalized forces with time and the second to the changes of the rates of the irreversible processes. If some restrictive conditions are satisfied (linear phenomenological laws, Onsagers reciprocity relations, constancy of the phenomenological coefficients) both parts are equal and decrease in time. This does no more hold if these conditions are not satisfied. Postulating only mechanical equilibrium and time independent boundary conditions, the authors show that the first part, related to the change of the generalized forces is always negative or zero. The change of the generalized forces with time is therefore such that it lowers the value of the entropy production. Some consequences of this theorem are briefly discussed.

Intrinsic irreversibility and integrability of dynamics

Irreversibility as the emergence of a priviledged direction of time arises in an intrinsic way at the fundamental level for highly unstable dynamical systems, such as Kolmogorov systems or large Poincare systems. The presence of resonances in large Poincare systems causes a breakdown of the conventional perturbation methods analytic in the coupling parameter. These difficulties are manifestations of general limitations to computability for unstable dynamical systems. However, a natural ordering of the dynamical states leads to a well-defined prescription for the regularization of the propagators which lifts the divergence and gives rise to an extension of the eigenvalue problem to the complex plane. The extension acquires meaning in suitable rigged Hilbert spaces which are constructed explicitly for the Friedrichs model. We show that the unitary evolution group, when extended, splits into two semigroups, one decaying in the future and the other in the past. Irreversibility emerges as the selection of the semigroup compatible with our future observations. In this way the problems of integration and irreversibility both enjoy a common solution in the extended space.

Quantum theory of non integrable systems

In 1889 H. Poincare introduced a basic distinction between integrable and non-integrable dynamical systems. This distinction refers to the role of resonances which may lead to divergences. A specially important class of non-integrable systems are «large» Poincare systems (LPS) which have a continuous spectrum and present continuous sets of resonances. As has been shown earlier, LPS play an essential role both in classical and quantum mechanics. Essentially all nontrivial problems of field theory as well as kinetic theory belong this class. We consider here the well known Friedrichs model in which an unstable discrete level is coupled to a continuum. Poincares theorem prevents the existence of solutions of the eigenvalue problem associated to the Hamiltonian which would be analytic in the coupling constant

Sur La Perturbation De La Distribution De Maxwell Par Des Réactions Chimiques En Phase Gazeuse

Abstract This paper is the second of a series devoted to the extension of ChapmanEnskogs method to inelastic collisions1). This note is mainly concerned with the study of the influence of the heat of reaction on the Maxwell distribution. As shown in the first paper this effect is in general small for the activation energy. The heat of reaction, however, can perturb the Maxwell distribution to an appreciable extent. As a result, the reaction rate is increased for exothermal, decreased for endothermal reactions. The effect is especially important for exothermal reactions. At the same time, the order of the reaction is increased. The calculations have been performed for the initial rate of a reaction of the type A0 + B → A1 + B. In this case the result can be expressed by the following equation v = v (0) (1 + 1,2 x A 0 x B r VT ϵ ∗ ) where e ∗ is the activation energy, γVT the heat of reaction and xA0 xB the mole fractions of A0 and B; v(0) is the reaction rate estimated by assuming Maxwell distributions for each constituent.

Le domaine de validité de la thermodynamique des phénomènes irréversibles

Abstract The extension of thermodynamic methods to irreversible processes which has been performed with a high degree of success by several authors, is based on the validity of the fondamental Gibbs formula for the total differential of entropy. This formula permits a straightforward calculation of the entropy flux and also of the entropy creation due to irreversible phenomena. The statistical interpretation of this formula for equilibrium processes has been given by Gibbs himself in his classical paper on statistical mechanics. By an application of the statistical theory of irreversible processes in gases, due mainly to Enskog and Chapman, the author shows in the present paper, that the fundamental Gibbs formula remains valid for a large class of irreversible processes. This class includes the transport phenomena for which the statistical distribution functions are of the form fγ = fγ(0), (1 + φγ(0) with fγ(0) the equilibrium distribution function and fγ(0) φγ(1) the second approximation function calculated by the Chapman-Enskog method. This class includes also chemical reactions slow enough as not to disturb to an appreciable extent the equilibrium form of the distribution functions of each constituent.

The philosophy of instability

Recent discoveries in science have led to the recognition that instability and creativity are inherent to our world. This has major implications for the way we perceive the universe and our place in it. In an unstable world, absolute control and precise forecasting are not possible. In this article, Ilya Prigogine traces the emergence of the new worldview provided by science, and suggests that it offers hope and new responsibility for humankind