The necessity of defining entropy measures

An a–disciplinary concept

The word entropy comes from a Greek word meaning evolution. The physical meaning of the concept of entropy is much disputed; it is still considered to be not very clear. According to Poincaré, it is a “prodigiously abstract concept.” There does not exist a completely rigorous mathematical formulation of thermodynamics. Wiener (1948) indicated the need to extend the notion of physical entropy when he stated that information is negative entropy. For Brillouin (1959) information and physical entropy are of the same nature, the increase in entropy corresponding to a loss of information.

The literature – in physics; in statistical theory of communications and information theory; in social sciences and the life sciences; in probability theory, graph theory, and Lebesgue's integration theory – contains many concepts and formulas for entropy and for quantity of information. Among them we find the topological and structural information content of Rashevsky (1955) and Trucco (1956a,b) defined on graphs as a measure of their complexity; the more well–known Shannon–Weaver entropy of a set of probabilities given by I = - Σ pi log pi,; the Hartley – Nyquist formula I = – log p where p is the probability of drawing an n letters message in an urn containing all messages; the chromatic information content of Mowshowitz (1968); the entropy of measurable functions and the epsilon–entropy; and the absolute S-entropy and the weighted entropy.

Many authors consider that the mechanisms of pre–pattern formation constitute one of the most important aspects of morphogenetic phenomena. For example, Edwards (1982) and Hermant (Section 3.4.2) produced phyllotactic pre–patterns using projective geometry and topology, respectively. Many theories propose that the site of organ formation is determined by a chemical pre–pattern, generated by a diffusion–reaction mechanism. By pre–pattern is meant in this appendix the establishment of a concentration gradient of substances from an almost equal initial distribution. In this context the patterns observed in organisms are considered to be the results of the interaction of their constituents such as cells, genes, or molecules. One of the models that has received the utmost attention is the one by Gierer and Meinhardt (see these authors in the Bibliography). The model is based on a principle of formation of patterns by a process of molecular activation and inhibition. Activation is an autocatalytic process with short spatial range, whereas inhibition is considered to be a comparatively long–range interaction in order to ensure stability of the developing structures. We have seen in Section 8.6.2 that Richter and Schranner (1978) used this model. Berding, Harbich and Haken (1983) used it to generate pre–patterns of capituli and leafy stems.

In order to better understand the principle, Meinhardt (1984) proposes the following analogy. A river can be formed from a minor depression in the landscape. Raindrops accumulate, erosion is accelerated, and a larger quantity of water rushes towards the newly created valley.

In this appendix we shall show how the methods studied in the main part of the book can be used to resolve far-from-trivial engineering or physical problems. All purely technical details will be omitted, although references to the corresponding sections of the main text will be provided.

Heat loss in injection of heat into oil stratum [67]

The injection of heat into oil strata is one of the tertiary methods of oil recovery, and has been extensively discussed in the technological literature. Although the most effective method of thermally influencing oil production seems to be the injection of superheated steam into production wells, the injection of a hot incompressible liquid is discussed in the literature primarily because it is much more amenable to analysis than steam injection, which involves consideration of the very complicated phenomenon of phase transition in porous media. Analysis of the injection of a hot incompressible liquid is incomparably easier, and provides useful information from the engineering point of view.

One of the basic problems in analyzing the process of heat injection is determining the ratio of the amount of heat used efficiently to improve oil recovery to the heat lost due to the unavoidable heat exchange between the productive stratum and surrounding unproductive rocks. In order to calculate this ratio, one need not know the spatial distribution of temperature within the productive stratum but only the overall effect of the temperature distribution.

Historical meeting again

Past chapters in this book very often mention Bravais and Bravais who successfully worked on the problem of phyllotaxis in the 1830s (Bravais– Bravais theorem, lattice, formula, etc.). One of the reasons for this success might be that these brothers were a botanist and a crystallographer. In those times living organisms were generally perceived as living crystals. That the crystallographic paradigm was shortly abandoned after the Bravais brothers may explain the relatively poor development of the subject of phyllotaxis in the second half of the nineteenth century. Van Iterson (1907) introduced geometrical methods of analysis of phyllotactic patterns based on symmetry theories of crystal structures. His work received very little attention and was revived to some extent in the second part of the century by botanists and mathematicians dealing with phyllotaxis in terms of contact circles on cylinders. The fact that crystallographers have rejoined the effort very recently might be an unconscious collective historical recognition that the discipline was from its very beginning under a good omen, and that the intuition of the initiators of the cylindrical treatment of phyllotaxis put the subject on the right track.

For crystallographers all aspects of the challenge of phyllotaxis are excluded except the geometrical aspect, and phyllotaxis is identified with the study of spiral lattices.

This book represents an attempt to implement a general approach that in essence views the theory of partial differential equations (PDEs) of mathematical physics as the language of continuous processes, that is, an interdisciplinary science that considers the hierarchy of mathematical phenomena as a reflection of their physical counterparts. A comprehensive, mathematically rigorous account of the classical theory of PDEs in mathematical physics is thus inseparably bound with the features of the corresponding natural continuum objects. We shall therefore endeavor to trace the simultaneous origins of some basic mathematical objects in different natural contexts (continuum mechanics, electrodynamics, transport phenomena, thermodynamics, and chemical kinetics). In parallel, we shall trace the interrelation between different types of problems (elliptic, parabolic, and hyperbolic) as mathematical counterparts of their natural prototypes: steady-state and evolutionary processes (dissipative and conservative). This will be done by an asymptotic analysis of the behavior of these processes in time and their dependence on the relevant governing parameters.

In view of the almost complete absence of a physics background in undergraduate and graduate curricula of mathematics and applied mathematics, it seems important, in a course of mathematical physics, to provide an introduction to the basic concepts of different natural sciences and their relation with PDEs in terms of certain typical boundary-value problems that recur in different scientific contexts. Chapters 1 and 2 are therefore addressed primarily to students of mathematics.

Nonlinear systems, bifurcations and symmetry breaking

A nonlinear system is a set of nonlinear equations, which may be algebraic, functional, ordinary differential, partial differential, integral or a combination of these. The system may depend on given parameters. Dynamical system is now used as a synonym of nonlinear system when the nonlinear equations represent evolution of a solution with time or some variable like time; the name dynamical system arose, by extension, after the name of the equations governing the motion of a system of particles, even though the nonlinear system may have no application to mechanics. We may also regard a nonlinear system as representing a feedback loop in which the output of an element is not proportional to its input. Nonlinear systems are used to describe a great variety of phenomena, in the social and life sciences as well as the physical sciences, earth sciences and engineering. The theory of nonlinear systems has applications to problems of economics, population growth, the propagation of genes, the physiology of nerves, the regulation of heart-beats, chemical reactions, phase transitions, elastic buckling, the onset of turbulence, celestial mechanics, electronic circuits and many other phenomena. This introduction to nonlinear systems, then, is an introduction to a great variety of mathematics and to diverse and numerous applications.

We have seen that bifurcations and chaos for a system of difference or ordinary differential equations often occur in lower dimensions than the dimension of the system. Similarly, although a partial differential system has an infinite dimension, its bifurcations and chaos often occur in a manifold of low finite dimension. Indeed, turning points, transcritical bifurcations, pitchfork bifurcations, Hopf bifurcations, limit cycles etc. arise for partial differential systems. This can be demonstrated in many cases by use of one of a few perturbation techniques, for example the Liapounov- Schmidt reduction or centre manifold theory. The essence of these techniques is to consider perturbations of marginal stability in which the values of both the parameters and the state variables are close to those corresponding to marginal stability, and in which the effects of these two kinds of perturbations are balanced asymptotically. At the margin of stability, the number of eigenvalues whose real parts are zero is usually small, so that their eigenfunctions span a low-dimensional space; all components of an initial disturbance not in this space being strongly damped. The centre manifold of a weakly nonlinear system is tangential to this space as the margin of stability is approached.

The fact that phenomena of interest occur in a low-dimensional manifold makes the dynamics much easier to understand, but it seems that some phenomena, for example turbulent motion of a fluid, cannot be represented in a low-dimensional manifold.