Fractal and Trans-scale Nature of Entropy: Towards a Geometrization of Thermodynamics

By Diogo Queiros Conde and Michel Feidt

Fractal and Trans-scale Nature of Entropy: Towards a Geometrization of Thermodynamics develops a new vision for entropy in thermodynamics by proposing a new method to geometrize. It investigates how this approach can accommodate a large number of very different physical systems, going from combustion and turbulence towards cosmology. As an example, a simple interpretation of the Hawking entropy in black-hole physics is provided. In the life sciences, entropy appears as the driving element for the organization of systems. This book demonstrates this fact using simple pedagogical tools, thus showing that entropy cannot be interpreted as a basic measure of disorder.

• Develop a new vision of entropy in thermodynamics
• Study the concept of entropy
• Propose a simple interpretation the entropy of Hawking
• Demonstrate entropy as a measure of energy dispersal

• Language: English
• Publisher: Elsevier Science
• Release date: Nov 16, 2018
• ISBN: 9780081017906

Diogo Queiros Conde

Diogo Queiros-Condé is a professor at University of Paris Ouest Nanterre La Défense since 2009. Before he was a research professor at ENSTA ParisTech from 2005 to 2009. He has a PhD in Physic and Science (1995)..

Book preview

Fractal and Trans-scale Nature of Entropy

Towards a Geometrization of Thermodynamics

Diogo Queiros-Condé

Michel Feidt

Thermodynamics – Energy, Environment, Economy Set

coordinated by

Michel Feidt

Table of Contents

Cover image

Title page

Dedication

Copyright

Introduction

1: The Thermal Worm Model to Represent Entropy–Exergy Duality

Abstract

1.1 A fractal and diffusive approach to entropy and exergy

1.2 A granular model of energy: toward the entropy and the exergy of a curve

1.3 The thermal worm model of entropy–exergy duality

1.4 The 2D worm model

1.5 The 3D thermal worm-like model

2: Black Hole Entropy and the Thermal Worm Model

Abstract

2.1 Entropy of a black hole: the Bekenstein–Hawking temperature

2.2 The thermal worm model of black holes

2.3 Carnot representation of black holes

3: The Entropic Skins of Black-Body Radiation: a Geometrical Theory of Radiation

Abstract

3.1 Intermittency of black-body radiation

3.2 Generalized RJ law based on a scale-dependent fractal geometry

3.3 Fluctuations and energy dispersion in black-body radiation

3.4 A scale-entropy diffusion equation for black-body radiation

3.5 Spectral fractal dimensions and scale-entropy of black-body radiation

3.6 Conclusion

4: Non-extensive Thermodynamics, Fractal Geometry and Scale-entropy

Abstract

4.1 Tsallis entropy in non-extensive thermostatistics

4.2 Two physical systems leading to Tsallis entropy: a simple interpretation of the entropic index

4.3 Non-extensive thermostatistics, scale-dependent fractality and Kaniadakis entropy

5: Finite Physical Dimensions Thermodynamics

Abstract

5.1 A brief history of finite physical dimensions thermodynamics

5.2 Transfer phenomena by FPDT

5.3 Energy conversion by FPDT

5.4 Extension to complex systems: cascades of endoreversible Carnot engines

5.5 Time dynamics of Carnot engines

5.6 Conclusions on FPDT

6: A Scale-Dependent Fractal and Intermittent Structure to Describe Chemical Potential and Matter Diffusion

Abstract

6.1 Defining and quantifying the diffusion of matter through chemical potential

6.2 Topic scales and scale-entropy of a set of particles

6.3 Entropy and chemical potential of an ideal gas by Sackur–Tetrode theory

6.4 Entropy of a set of particles described through topic scales and scale-entropy

6.5 Fractal and scale-dependent fractal geometries to interpret and calculate the chemical potential

6.6 The intermittency parameter and clustering entropy of particles in the fractal case

6.7 The clustering entropy and chemical potential in the parabolic fractal case

6.8 Summing up formulas and conclusion

Conclusion

References

Index

Dedication

This book is dedicated to Michel Ballereau (1948–2016)

In memory of a great and generous person

Copyright

First published 2018 in Great Britain and the United States by ISTE Press Ltd and Elsevier Ltd

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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Notices

Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary.

Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility.

To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein.

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The rights of Diogo Queiros-Condé and Michel Feidt to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

Library of Congress Cataloging in Publication Data

A catalog record for this book is available from the Library of Congress

ISBN 978-1-78548-193-2

Printed and bound in the UK and US

Introduction

In 2005, the physicist J.-P. Badiali wrote:

It is possible that the general definition of entropy is connected with a definition of order in space-time rather than associated with a counting of microstates.

This sentence expresses the idea that the concept of entropy is not completely explained by Boltzmann’s theory, which is based on a counting of microstates without any consideration of the structure and texture of space-time. The main objective of our book is to propose a geometrical understanding of entropy through a multiscale geometrical structure of the space-time in which entropy is considered: we converged to the idea that entropy characterizes a dispersion of energy by a change on the scales of the system. If we understand this dynamic in scale-space, we may understand entropy more deeply. By defining adequate scales, we show that it becomes possible to give a geometrical and direct interpretation of what entropy is. We follow here a modern interpretation of entropy as being the measure of the dispersal of energy. This dispersive nature of energy quantified by entropy is finite and limited by two specific scales: an inner cut-off scale, which defines the smallest scale of dispersion, and an integral scale, which defines the largest scale of dispersion.

In a first model called the thermal worm model, entropy is explained by a lateral diffusion, which has no direct mechanical effect in the longitudinal direction taken to catch the working capacity of the process. Two characteristic scales are introduced: the energetic thickness and the exergetic thickness. This allows a definition of the entropic thickness. The entropic evolution of a system is therefore the relative variations of exergetic and entropic thicknesses. In a second model based on the scale-entropy concept developed to describe intermittency in turbulence and which found applications in physics and engineering (Queiros-Condé 2003; Queiros-Condé et al. 2015), dispersion is quantified by a scale range and a fractal dimension or a scale fractal-dependent behavior. There are already existing links between fractal geometry and thermodynamics but the dynamical arrows of these links are often from thermodynamics toward geometry (thermodynamics → geometry) and, more particularly, toward fractal geometry: it means that thermodynamics is used to understand fractal structures, curves or chaotic processes; Mendès-France for example, introduced, the temperature of a curve (Mendès-France 1983) and also defined its entropy and even its action. The theory of multifractals used Legendre transformations and thermodynamical formalism to characterize multifractality (Parisi and Frisch 1985). Renyi (1960) and Tsallis (1988) introduced a new statistical definition of entropy that quantifies departure from Boltzmannn–Gibbs entropy. Thermodynamics of chaotic processes have been proposed (Beck and Schlögl 1993).

Let us recall that Mandelbrot’s scientific career began with fundamental papers on thermodynamics before converging at IBM on fractals. This is something that has been forgotten but in the context of our book it really deserves to be recalled: the father of fractals as he is often called was fascinated by thermodynamics and more precisely by the thermal uncertainty behavior between energy and temperature, an uncertainty mixed with a sort of complementarity proposed by Bohr (1932) by analogy with the Copenhagen interpretation of quantum mechanics. In his memories, Mandelbrot (2012) wrote that when he discovered fractal structure in the Zipf law, he thought he found the way to understand thermodynamics. We may think that Mandelbrot came to fractals by trying to understand some paradoxical aspects of the canonical classical definition of temperature. In fact, he proposed in 1956 a thermodynamical uncertainty relation in order to solve this question. Following Uffink and Van Lith-van Dis (1998), Mandelbrot’s thermodynamical uncertainty relation expresses that the efficiency with which temperature can be estimated is bounded by the spread in energy. Let us also remark that Schlögl (1988), the co-author (with C. Beck) of Thermodynamics of Chaotic Processes (1995), derived uncertainty thermal relations. The fractal geometry seems thus to be – at least through Mandelbrot’s and Schlögl’s works – deeply linked to the quest for an uncertainty thermal structure. We will see in our book that the thermal worm model we developed here leads to a similar thermal uncertainty relation.

Geometry in thermodynamics. The reverse arrow fractal geometry → thermodynamics has not been so developed. Applications of fractal geometry to understand thermodynamics and especially the concept of entropy are rare. Nevertheless, we can quote the work of A. Le Méhauté et al. (1990). The book we propose here defends a geometrization of thermo-dynamics by using scale-dependent fractals and other geometrical tools. How did we come to this idea that entropy displays a fractal nature? This comes naturally from a modern interpretation of entropy as being the measure of the tendency of energy to be dispersed with time. The entropic arrow is mainly an arrow of dispersion. To characterize dispersion, statistical tools are necessary, such as the moments of a quantity, but we can also use fractal geometry. We must emphasize again that we do not work with mathematical fractals, which do not have an inner cut-off scale, but with truncated fractal objects exhibiting their fractality in a finite scale-range [lc; l0] where lc is the inner cut-off and the integral scale l0. Strict fractality is defined by a unique fractal dimension in the scale range. Nevertheless, it appears that this can only be an ideality and that a fractal dimension is in fact a scale-dependent quantity. Fractal dimensions can also be time dependent. To describe these features and to generalize the idea of fractals, we introduce a general scale-entropy diffusion equation (Queiros-Condé 2003; Queiros-Condé et al. 2015).

Because there is always entropy in any process, even an elementary quantity of energy cannot be a homogeneous object in terms of its capacity to furnish work: it necessarily displays a dichotomic nature. Taking the production of work as a reference, it became clear since Carnot (1824) and Clausius (1851) that energy has to be discriminated into two parts. A part of it can be transformed into work and another part linked to entropy cannot. Entropy appears to be the result of a friction that degrades the working capacity of a fluid. Nowadays, after almost two centuries since Carnot’s pioneering work, the mechanical part of energy is called exergy (or free energy) while the non-mechanical one is called anergy. We emphasize the fact that this does not mean that there is energy, which is useful and energy that is not useful. It depends on what is done with this energy.

Spatial distribution of entropic events. If we consider that there is an intrinsic exergy–anergy duality of energy, calculating the exergy destruction in a process becomes equivalent to calculating the entropy production. Accepting the exergy–anergy duality, the fundamental question is to know what their relative spatial distribution is. How can we visualize their spatial structure? The interest of working with the pair exergy–anergy instead of the classical pair energy–entropy lies in the simple fact that exergy and anergy are fractions of energy since energy = exergy + anergy. The ratios θ = exergy/energy and 1 − θ = anergy/energy can then be considered as probabilities of two kind of energetic particles. Exergetic and anergetic events can then be seen as correlated particles having some specific geometrical structure but belonging to the same set. We think that this structure displays some fractal and more generally scale-dependent fractal features.

Thanks to numerous works since the 1980s, the interpretation of entropy based on the idea of disorder, which came from the Boltzmannian statistical mechanics, is considered unsatisfactory because it generates paradoxical results and does not give an intuitive understanding of entropy. An interpretation in terms of dispersal of the energy is now accepted (Leff 1996; Lambert 2002) and seems more adequate to experimental observations and intuition. Our book widely develops this idea that entropy a spatial and temporal dispersion of energy as can be the diffusion in a fluid of a set of initially packed and clustered particles. If entropy results from a dispersal, how can we define and quantify this dispersion? The main objective of our book is to propose geometrical ways to take into account this dispersion. We propose two methods to characterize it.

The thermal worm model. This physical model is based on the simple idea that transfer of energy displays two components: one main component is longitudinal, and the other one is lateral and transversal. The longitudinal component of energy represents the mechanical potential of the energy in the sense that this energy can be transformed into work: it is based on a central duct composed of exergetic events all having the same direction. It is thus the exergetic content of energy. The lateral component does not allow the production of mechanical work because the direction is perpendicular to the main direction in which work can be produced. It should thus be associated with entropy. This lateral drift of energy contributes to a lateral diffusion of the worm. The volume occupied by this lateral activity represents the anergetic or entropic content of energy. This is the idea behind the thermal worm model that will be presented in Chapter 1 with an application to black hole entropy in Chapter 2. The thermal worm model will be applied to describe heat transfer by an entropic approach based on the deformation of thermal worms with temperature. The physics of black holes has shown – mainly through the works of Bekenstein (1973) and Hawking (1974) – that black holes have an entropy proportional to the ratio of the area of the black hole horizon relative to the Planck scale. This is reminiscent of the basic idea offered by the thermal worm model. We thus propose a geometrical representation of black hole entropy based on the thermal worm model.

The fractal and scale-dependent fractal model. The second method characterizes dispersion of energy by using fractal and more generally scale-dependent fractal geometries to model the dispersion. If the fractal dimension of the anergetic set is close to the space dimension, the energy is homogeneously dispersed and cannot produce work. It the fractal dimension is much smaller, energy presents a larger heterogeneity between anergy and exergy and is much more clustered. This leads to a larger capacity to produce work.

The dispersive nature of energy: lateral and longitudinal components of energy. In order to give the main interpretation of entropy we will develop in this book, let us consider an imaginary fluid flowing toward a wheel (see Figure I.1), a configuration which is in the same spirit as the hydraulic analogy of Lazare and Sadi Carnot. Through the impact of the flow on blades, the wheel would be able to transform the flowing line into rotation and therefore to produce work. Nevertheless, if the flow displays an increasing dispersion relative to the mean direction, it implies less generation of work on the wheel.

Figure I.1 Sketch of five imaginary flows with the same energy but displaying from (a) to (e) an increasing dispersion, which reduces its capacity to produce work (exergy)

The lines (a) to (e) of Figure I.1 on the right have the same length (i.e. the same energy) but an increasing lateral dispersion, which can be quantified by a mean scale of dispersion or a truncated fractal dimension. The dispersion scale increases from (a) to (e): the parts of the lines that remain straight in the direction of the wheel (able to produce work) decrease and the transversal parts increase. The fractal dimension characterizing space-filling properties increases from (a) to (e) but also the scale range since the line becomes laterally more dispersed. In terms of work production, the line (a) is the most exergetic one since it gives the largest quantity of work while the line (e) delivers the smallest quantity of work due to its high lateral diffusion. Of course, the capacity to produce work is relative to the main direction: it we take the perpendicular as a new mean direction to produce work, the line (e) becomes the more efficient while the line (a) becomes the less efficient.

To quantify the dispersion of a multiscale object, we need to know the scale range in which the object is developed and the behavior of fractal dimensions within this scale range. It is thus necessary to define an inner cutoff scale