Chaos and Entropy

The quantitative laws of chemical combination provide clear pointers to the molecular theory of matter, which increases progressively in vividness and realism with the application of Newton s laws to the motions of the particles. The interpretation of phenomena such as the pressure and viscosity of gases and the Brownian motion, and the assignment of definite magnitudes to molecular speeds, masses, and diameters render it clear that a continual interchange of energies must occur between the molecules of a material system, a circumstance which lies at the basis of temperature equilibrium and determines what in ordinary experience is called the flow of heat. It is responsible indeed for far more than this, and a large part of physical chemistry follows from the conception of the chaotic motion of the molecules. This matter must now be examined more deeply. 

The state of such a molecular system is obviously chaotic in the sense that some molecules move fast and others slowly some are in violent vibration or rotation while others are almost qmescent and the condition of individuals is continually changing. A rough idea of the rapid and irregular variations of motion would be provided by the behaviour of a number of billiard balls propelled in random directions on a table, their translations and spins fluctuating according to the hazards of their mutual encounters. 

For an elastic collision of two smooth spheres the laws of mechanics 

The original state of all the matter in the world is not known nor can we predict what would happen if all molecular speeds were reversed. As far as the laws of mechanics go, we cannot assert that existing conditions are unrelated to an earlier condition of order. Whether, therefore, the complete randomness of all microscopic motion can be logically related to the Newtonian laws has in fact been a subject of controversy and no wholly satisfactory answer emerges. 

What is undoubtedly true is that for practical purposes the chaos can be regarded at least as very nearly complete. That this is so irrespective of the origin of the present order of things is attested by such simple experiences as the shuffling of a pack of cards, which show how rapidly all vestiges of order become undetectable. While forgoing any attempt at a rigid application of the laws of dynamics to the question, we may therefore introduce as a specific postulate the assumption of molecular chaos for ordinary systems endowed with 

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Entropy (S) is energy that cannot be harnessed for work, it is wasted energy associated with chaos and disorder. 

MSN. 198.1. Prigogine and G. Ordonez, Acceleration and entropy A macroscopic analogue of the twin paradox, Chaos and Complexity Lett, (to appear, 2003). 

A negative sign for entropy means more order (or less chaos and less disorder). Raking up leaves is a more orderly state. 

We briefly mention a couple of interesting papers that are directly related to chaos and transport. Ihm et u/.184 demonstrate that the Kolmogorov entropy r KS is related to the self-diffusion coefficient (D) in simple liquids by the relationship... 

Other useful sources of historical information are The Early Development of the Concepts of Temperature and Heat The Rise and Decline of the Caloric Theory by D. Roller in Volume 1 of Harvard Case Histories in Experimental Science edited by J.B. Conant and published by Harvard University Press in 1957 articles in Physics Today, such as A Sketch for a History of Early Thermodynamics by E. Mendoza (February, 1961, p.32), Carnot s Contribution to Thermodynamics by M.J. Klein (August, 1974, p. 23) articles in Scientific American and various books on the history of science. Of special interest is the book The Second Law by P.W. Atkins published by Scientific American Books, W.H. Freeman and Company (New York, 1984) which contains a very extensive discussion of the entropy, the second law of thermodynamics, chaos and symmetry. 

For the adjusted material balance of Example 15.2, as shown in Fig. 15.9 (but using the Chao-Seader correlation for thermodynamic properties), stream temperatures, enthalpies, and entropies are as follows. 

One typically starts with an internal energy of a macroscopic system, expressed as the internal energy of the periodically repeated unit cell. This state function is part of another state function H, the enthalpy, which is a very useful energetic measure for conditions of constant pressure p. For a complete picture, one also needs to know the value of the state function T, the temperature, and that of the state function S, the entropy, a measure of chaos and also probability. These functions may be combined to yield the Gibbs energy (or free enthalpy) G, the true and final measure of stability. In its difference form, the so-called Gibbs-Helmholtz formula reads... 

The essential character of thermal phenomena becomes clear, the conditions of coexistence of solids, liquids, and gases in systems of any number of chemical components are explained, the dependence of equilibria upon concentrations, upon pressure, and upon temperature is defined. The conceptions of entropy and free energy, of statistical equilibrium and energy distribution, provide quantitative laws which describe the perpetual conflict of order and chaos, and which prescribe in a large measure not only the shapes assumed by the material world but also the pattern of its possible changes. 

The interchange of transiently acting multi-solvent-solute systems between non-classical chaos foundations and fundamental complexity goals, waiting for energy input to set all their enthalpy- and entropy-determined functional and informational richness alive . 

Questions regarding the relation between entropy and dynamics have received great attention recently but they are far from simple. Not all dynamical processes require the concept of entropy. The motion of the earth around the sun is an example in which irreversibility (such as friction due to tides) can be ignored and the motion may be described by time-symmetric equations. But recent developments in nonlinear dynamics have shown that such systems are exceptions. Most systems exhibit chaos and irreversible behaviour. We now begin to be able to characterize the dynamical systems for which irreversibility is an essential feature leading to an increase in entropy. 

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