Entropy

Princeton University Press

The concept of entropy arose in the physical sciences during the nineteenth century, particularly in thermodynamics and statistical physics, as a measure of the equilibria and evolution of thermodynamic systems. Two main views developed: the macroscopic view formulated originally by Carnot, Clausius, Gibbs, Planck, and Caratheodory and the microscopic approach associated with Boltzmann and Maxwell. Since then both approaches have made possible deep insights into the nature and behavior of thermodynamic and other microscopically unpredictable processes. However, the mathematical tools used have later developed independently of their original physical background and have led to a plethora of methods and differing conventions.

The aim of this book is to identify the unifying threads by providing surveys of the uses and concepts of entropy in diverse areas of mathematics and the physical sciences. Two major threads, emphasized throughout the book, are variational principles and Ljapunov functionals. The book starts by providing basic concepts and terminology, illustrated by examples from both the macroscopic and microscopic lines of thought. In-depth surveys covering the macroscopic, microscopic and probabilistic approaches follow. Part I gives a basic introduction from the views of thermodynamics and probability theory. Part II collects surveys that look at the macroscopic approach of continuum mechanics and physics. Part III deals with the microscopic approach exposing the role of entropy as a concept in probability theory, namely in the analysis of the large time behavior of stochastic processes and in the study of qualitative properties of models in statistical physics. Finally in Part IV applications in dynamical systems, ergodic and information theory are presented.


The chapters were written to provide as cohesive an account as possible, making the book accessible to a wide range of graduate students and researchers. Any scientist dealing with systems that exhibit entropy will find the book an invaluable aid to their understanding.

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About the author

Andreas Greven and Gerhard Kellerare Professors of Mathematics at the University of Erlangen. Gerald Warnecke is Professor of Numerical Mathematics at the University of Magdeburg.
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Additional Information

Publisher
Princeton University Press
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Published on
Dec 31, 2003
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Pages
358
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ISBN
9780691113388
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Language
English
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Genres
Mathematics / Applied
Science / Mechanics / Thermodynamics
Science / Physics / Mathematical & Computational
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Content Protection
This content is DRM protected.
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Whatdoasupernovaexplosioninouterspace,?owaroundanairfoil and knocking in combustion engines have in common? The physical and chemical mechanisms as well as the sizes of these processes are quite di?erent. So are the motivations for studying them scienti?cally. The super- 8 nova is a thermo-nuclear explosion on a scale of 10 cm. Astrophysicists try to understand them in order to get insight into fundamental properties of the universe. In ?ows around airfoils of commercial airliners at the scale of 3 10 cm shock waves occur that in?uence the stability of the wings as well as fuel consumption in ?ight. This requires appropriate design of the shape and structure of airfoils by engineers. Knocking occurs in combustion, a chemical 1 process, and must be avoided since it damages motors. The scale is 10 cm and these processes must be optimized for e?ciency and environmental conside- tions. The common thread is that the underlying ?uid ?ows may at a certain scale of observation be described by basically the same type of hyperbolic s- tems of partial di?erential equations in divergence form, called conservation laws. Astrophysicists, engineers and mathematicians share a common interest in scienti?c progress on theory for these equations and the development of computational methods for solutions of the equations. Due to their wide applicability in modeling of continua, partial di?erential equationsareamajor?eldofresearchinmathematics. Asubstantialportionof mathematical research is related to the analysis and numerical approximation of solutions to such equations. Hyperbolic conservation laws in two or more spacedimensionsstillposeoneofthemainchallengestomodernmathematics.
We study features of the longtime behavior and the spatial continuum limit for the diffusion limit of the following particle model. Consider populations consisting of two types of particles located on sites labeled by a countable group. The populations of each of the types evolve as follows: each particle performs a random walk and dies or splits in two with probability $\frac{1} {2}$ and the branching rates of a particle of each type at a site $x$ at time $t$ is proportional to the size of the population at $x$ at time $t$ of the other type. The diffusion limit of ''small mass, large number of initial particles'' is a pair of two coupled countable collections of interacting diffusions, the mutually catalytic super branching random walk.Consider now increasing sequences of finite subsets of sites and define the corresponding finite versions of the process. We study the evolution of these large finite spatial systems in size-dependent time scales and compare them with the behavior of the infinite systems, which amounts to establishing the so-called finite system scheme. A dichotomy is known between transient and recurrent symmetrized migrations for the infinite system, namely, between convergence to equilibria allowing for coexistence in the first case and concentration on monotype configurations in the second case.Correspondingly we show in the recurrent case both large finite and infinite systems behave similar in all time scales, in the transient case we see for small time scales a behavior resembling the one of the infinite system, whereas for large time scales the system behaves as in the finite case with fixed size and finally in intermediate scales interesting behavior is exhibited, the system diffuses through the equilibria of the infinite system which are indexed by the pair of intensities and this diffusion process can be described as mutually catalytic diffusion on $(\R^ )^2$. At the same time, the above finite system asymptotics can be applied to mean-field systems of $N$ exchangeable mutually catalytic diffusions. This is the building block for a renormalization analysis of the spatially infinite hierarchical model and leads to an association of this system with the so-called interaction chain, which reflects the behavior of the process on large space-time scales.Similarly we introduce the concept of a continuum limit in the hierarchical mean field limit and show that this limit always exists and that the small-scale properties are described by another Markov chain called small scale characteristics. Both chains are analyzed in detail and exhibit the following interesting effects. The small scale properties of the continuum limit exhibit the dichotomy, overlap or segregation of densities of the two populations, as a function of the underlying random walk kernel. A corresponding concept to study hot spots is presented. Next we look in the transient regime for global equilibria and their equilibrium fluctuations and in the recurrent regime on the formation of monotype regions.For particular migration kernels in the recurrent regime we exhibit diffusive clustering, which means that the sizes (suitable defined) of monotype regions have a random order of magnitude as time proceeds and its distribution is explicitly identifiable. On the other hand in the regime of very large clusters we identify the deterministic order of magnitude of monotype regions and determine the law of the random size. These two regimes occur for different migration kernels than for the cases of ordinary branching or Fisher-Wright diffusion. Finally we find a third regime of very rapid deterministic spatial cluster growth which is not present in other models just mentioned. A further consequence of the analysis is that mutually catalytic branching has a fixed point property under renormalization and gives a natural example different from the trivial case of multitype models consisting of two independent versions of the fixed points for the one type case.
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Whatdoasupernovaexplosioninouterspace,?owaroundanairfoil and knocking in combustion engines have in common? The physical and chemical mechanisms as well as the sizes of these processes are quite di?erent. So are the motivations for studying them scienti?cally. The super- 8 nova is a thermo-nuclear explosion on a scale of 10 cm. Astrophysicists try to understand them in order to get insight into fundamental properties of the universe. In ?ows around airfoils of commercial airliners at the scale of 3 10 cm shock waves occur that in?uence the stability of the wings as well as fuel consumption in ?ight. This requires appropriate design of the shape and structure of airfoils by engineers. Knocking occurs in combustion, a chemical 1 process, and must be avoided since it damages motors. The scale is 10 cm and these processes must be optimized for e?ciency and environmental conside- tions. The common thread is that the underlying ?uid ?ows may at a certain scale of observation be described by basically the same type of hyperbolic s- tems of partial di?erential equations in divergence form, called conservation laws. Astrophysicists, engineers and mathematicians share a common interest in scienti?c progress on theory for these equations and the development of computational methods for solutions of the equations. Due to their wide applicability in modeling of continua, partial di?erential equationsareamajor?eldofresearchinmathematics. Asubstantialportionof mathematical research is related to the analysis and numerical approximation of solutions to such equations. Hyperbolic conservation laws in two or more spacedimensionsstillposeoneofthemainchallengestomodernmathematics.
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