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

(Professor, University Paris Ouest Nanterre la Defense, Laboratory of Energetics, Mechanics and Electromagnetism), (Professor Emeritus, University of Lorraine, France)
  • Formāts: EPUB+DRM
  • Izdošanas datums: 16-Nov-2018
  • Izdevniecība: ISTE Press Ltd - Elsevier Inc
  • Valoda: eng
  • ISBN-13: 9780081017906
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Fractal and Trans-scale Nature of Entropy: Towards a Geometrization of ThermodynamicsPalielināt
  • Formāts: EPUB+DRM
  • Izdošanas datums: 16-Nov-2018
  • Izdevniecība: ISTE Press Ltd - Elsevier Inc
  • Valoda: eng
  • ISBN-13: 9780081017906
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Fractal and Trans-scale Nature of Entropy: Towards a Geometrization of Thermodynamics develops a new vision for entropy in thermodynamics by proposing a new way to geometrize the concept of entropy. It is based on a specific geometry which would be the spatial support for entropy, a geometry called entropic skins geometry, where the fundamental components of a general thermodynamical process are surfaces or lines having some specific multi-scale geometries. The authors investigate how this approach can accommodate a large number of very different physical systems, going from combustion and turbulence towards cosmology.

The authors propose, for instance, a simple interpretation of the Hawking entropy in black-hole physics. The geometrical concepts developed can also provide a new way to consider ecosystems, urban dynamics and human cognition. The authors show that this leads to an original and positive interpretation of entropy which today is lacking. In life sciences, entropy appears as the driving element for the organization of systems; this book aims to demonstrate this fact via simple pedagogical tools. We will show that entropy cannot be interpreted as a basic measure of disorder anymore but rather as the measure of the dispersion of energy and of its capacity to form structure and organization.

  • Explores the strong link between geometry and thermodynamics
  • Offers an original understanding of entropy
  • Explains how to visualize phenomena where thermodynamics and entropy are important in the optimization of thermal machines but also in other fields such as nanophysics, life sciences or astrophysics
  • Explains entropy in thermodynamics through fractal geometry and more generally with trans-scale geometries

Papildus informācija

A new vision for entropy in thermodynamics based on a fresh geometrical interpretation, with implications for many physical systems
Introduction xi
Chapter 1 The Thermal Worm Model to Represent Entropy-Exergy Duality
1(48)
1.1 A fractal and diffusive approach to entropy and exergy
1(6)
1.1.1 The filamentary thermal worm model of exergy
1(2)
1.1.2 The geometrical Carnot factor and the temperature of a filamentary worm
3(3)
1.1.3 The T-like filamentary worm
6(1)
1.2 A granular model of energy: toward the entropy and the exergy of a curve
7(12)
1.2.1 The model of granular energy: the concept of ergon
7(1)
1.2.2 Exergy and anergy of an ergon: the entropic angle of an energy
7(2)
1.2.3 An elementary ergon discharged in a field of temperature T: the thermal puff time of energy
9(2)
1.2.4 The granular energy model applied to the hydraulic analogy of Lazare and Sadi Carnot: entropy and action of energy
11(4)
1.2.5 Exergy and anergy of a curve
15(3)
1.2.6 Examples of Carnot factor of some curves
18(1)
1.3 The thermal worm model of entropy--exergy duality
19(7)
1.3.1 Entropic skins to describe irreversibilities and entropy-exergy duality
19(2)
1.3.2 The 2D worm model
21(1)
1.3.3 The 3D worm model: the coefficient of entropic dispersion
22(1)
1.3.4 Entropic structure of a steady-state heat flux
23(1)
1.3.5 2D worm entropic dispersion of a steady heat flux
24(1)
1.3.6 3D worm entropic dispersion of a steady heat flux
25(1)
1.4 The 2D worm model
26(10)
1.4.1 The isothermal 2D worm
26(1)
1.4.2 Exergy destruction between two 2D worms
27(1)
1.4.3 The non-isothermal 2D worm
28(3)
1.4.4 The 2D worm of a linear profile of temperature: method 1
31(4)
1.4.5 2D worm displaying a linear temperature profile: method 2
35(1)
1.5 The 3D thermal worm-like model
36(13)
1.5.1 The isothermal 3D worm model
36(3)
1.5.2 The entropic angle of energy
39(1)
1.5.3 The non-isothermal 3D worm model
40(3)
1.5.4 Table to recapitulate
43(1)
1.5.5 A link with the phenomenon of intermittency in fully developed turbulence
44(2)
1.5.6 Longitudinal diffusion and lateral diffusion in the worm-like model
46(3)
Chapter 2 Black Hole Entropy and the Thermal Worm Model
49(22)
2.1 Entropy of a black hole: the Bekenstein--Hawking temperature
49(7)
2.1.1 Introduction: a geometric entropy for black holes
49(3)
2.1.2 Gravitational and quantum diffusivities: longitudinal and lateral diffusivities of a black hole?
52(2)
2.1.3 The existence of an absolute minimum temperature which is not 0
54(2)
2.2 The thermal worm model of black holes
56(6)
2.2.1 Entropic thermal worm representation of a black hole
56(2)
2.2.2 Temperature of the worm and temperature of the black hole
58(1)
2.2.3 The thickness of the horizon of a black hole
59(1)
2.2.4 The quantum variation of the temperature in a black hole
60(2)
2.3 Carnot representation of black holes
62(9)
2.3.1 Black hole as a reversible power cycle
62(1)
2.3.2 Black hole as a reversible refrigeration cycle
62(1)
2.3.3 The quantum interaction velocity
63(2)
2.3.4 Relaxation time of thermal inhomogeneities: the "thermal puff time" of energy in a black hole
65(2)
2.3.5 From Planck's constant and Boltzmann's constant toward Brillouin's constant b = h/k
67(1)
2.3.6 Black hole physics: a deep and fascinating representation of the finiteness of our world
68(3)
Chapter 3 The Entropic Skins of Black-Body Radiation: a Geometrical Theory of Radiation
71(64)
3.1 Intermittency of black-body radiation
71(11)
3.1.1 Black-body radiation: Wien's law, Rayleigh-Jeans' law and Planck's law
71(2)
3.1.2 The spectral volume fraction and the intermittency of radiation
73(2)
3.1.3 A simple derivation of u(v)- 8πv2c3Ev using an analogy with fully developed turbulence
75(1)
3.1.4 A simple understanding of the importance of the ratio v/T in the black-body problem
76(1)
3.1.5 The interacting length and interaction time of a photon and its scale-entropy
77(4)
3.1.6 The interacting length of a fermion and its scale-entropy
81(1)
3.1.7 Summary of the chapter
82(1)
3.2 Generalized RJ law based on a scale-dependent fractal geometry
82(6)
3.2.1 Radiation as stationary waves in a box: the Rayleigh description
83(1)
3.2.2 First case: uniform distribution of modes in phase-space -- the RJ theory
84(2)
3.2.3 Second case: fractal distribution of modes in phase space
86(1)
3.2.4 General case: the fractal dimension is mode dependent
87(1)
3.2.5 Conclusion
88(1)
3.3 Fluctuations and energy dispersion in black-body radiation
88(26)
3.3.1 Planck's derivation using the second-order derivative of entropy: the "lucky guess"
88(3)
3.3.2 Planck's entropy
91(2)
3.3.3 Planck's entropy, Planck's counting of complexions and Boltzmann's theory
93(4)
3.3.4 Planck's entropy and the scale-entropy of a radiative field
97(2)
3.3.5 Equivalent chemical potential of a Boltzmann distribution
99(1)
3.3.6 Graphical method to represent Boltzmann's counting
100(3)
3.3.7 What is the fundamental status of the second order derivative of entropy relative to energy? Variance and the dispersion factor of radiative energy
103(5)
3.3.8 The fluctuations and the dispersion of energy
108(3)
3.3.9 The dispersion factor interpreted by the worm model
111(1)
3.3.10 An heuristic thermodynamical uncertainty relation from the thermal worm model
112(2)
3.4 A scale-entropy diffusion equation for black-body radiation
114(8)
3.4.1 Spacions and scale-entropy
114(2)
3.4.2 Scale-entropy diffusion equation
116(3)
3.4.3 Pure truncated fractal case and parabolic scaling
119(1)
3.4.4 Exponential scaling: the scale-entropy sink is a fraction of the entropy production of a photon
119(3)
3.4.5 Summary
122(1)
3.5 Spectral fractal dimensions and scale-entropy of black-body radiation
122(10)
3.5.1 Fractal skins of black-body radiation
122(2)
3.5.2 Spectral fractal dimensions Dx of black-body radiation
124(1)
3.5.3 Spectral evolution of the scale-entropy sink b(x)
125(2)
3.5.4 Point of minimum scale-entropy sink
127(2)
3.5.5 Minimum scale-entropy sink and CMB
129(1)
3.5.6 Determination of the inner cut-off scale: a scale for which D(x) = 2
129(1)
3.5.7 Average value of the scale-entropy sink
130(2)
3.6 Conclusion
132(3)
Chapter 4 Non-extensive Thermodynamics, Fractal Geometry and Scale-entropy
135(16)
4.1 Tsallis entropy in non-extensive thermostatistics
135(3)
4.2 Two physical systems leading to Tsallis entropy: a simple interpretation of the entropic index
138(6)
4.2.1 Work produced by the isothermal growth of a fractal volume
139(3)
4.2.2 Mass decay of a fractal system
142(2)
4.3 Non-extensive thermostatistics, scale-dependent fractality and Kaniadakis entropy
144(7)
4.3.1 Decaying process of a fractal system submitted to internal cohesion pressure
144(3)
4.3.2 The Kaniadakis K-statistics and Kaniadakis entropy
147(2)
4.3.3 Conclusion on non-extensive thermodynamics and fractal geometry and scale-entropy
149(2)
Chapter 5 Finite Physical Dimensions Thermodynamics
151(30)
5.1 A brief history of finite physical dimensions thermodynamics
151(3)
5.2 Transfer phenomena by FPDT
154(8)
5.2.1 Series model of insulation: thermal resistances in series
154(4)
5.2.2 Parallel model of insulation: thermal resistances in parallel
158(2)
5.2.3 Generalization
160(1)
5.2.4 Partial conclusion
161(1)
5.3 Energy conversion by FPDT
162(7)
5.3.1 Carnot cycle and thermodynamics of equilibrium
163(1)
5.3.2 The non-adiabatic endoreversible Carnot engine
163(1)
5.3.3 The adiabatic endoreversible Carnot engine
164(1)
5.3.4 The adiabatic non-reversible Carnot engine
165(2)
5.3.5 The non-reversible Carnot engine with thermal losses
167(1)
5.3.6 Generalization of previous results
168(1)
5.4 Extension to complex systems: cascades of endoreversible Carnot engines
169(5)
5.4.1 Cascade of power Carnot engines
169(1)
5.4.2 Thermodynamic model in finite dimensions
170(1)
5.4.3 Optimization of the cascade
171(2)
5.4.4 Cascade with N endoreversible machines
173(1)
5.5 Time dynamics of Carnot engines
174(5)
5.5.1 Reversible thermal transfer between source and sink
174(2)
5.5.2 Finite transfer between source and irreversible engine
176(3)
5.6 Conclusions on FPDT
179(2)
Chapter 6 A Scale-Dependent Fractal and Intermittent Structure to Describe Chemical Potential and Matter Diffusion
181(38)
6.1 Defining and quantifying the diffusion of matter through chemical potential
181(5)
6.1.1 The chemical potential
181(2)
6.1.2 Fundamental equations
183(1)
6.1.3 Condition of chemical equilibrium
184(2)
6.2 Topic scales and scale-entropy of a set of particles
186(6)
6.2.1 Topic volume and topic scales of a particle
186(2)
6.2.2 Scale-entropy of a set of particles
188(2)
6.2.3 Scale-entropy of a fractal set
190(2)
6.3 Entropy and chemical potential of an ideal gas by Sackur--Tetrode theory
192(6)
6.3.1 Entropy of monoatomic ideal gases using Sackur--Tetrode theory
192(3)
6.3.2 Chemical potential using Sackur--Tetrode theory
195(1)
6.3.3 Chemical potential of a mixture using Sackur--Tetrode theory
196(2)
6.4 Entropy of a set of particles described through topic scales and scale-entropy
198(6)
6.4.1 Waving and clustering entropies of a set of particles
198(2)
6.4.2 Application to a two-component mixture: the clustering entropy of a mixture
200(3)
6.4.3 Application to heterogeneous solids: clustering entropy of heterogeneity
203(1)
6.5 Fractal and scale-dependent fractal geometries to interpret and calculate the chemical potential
204(2)
6.5.1 Chemical potential interpreted via scale-entropy, expelling potential and clustering entropy
204(1)
6.5.2 Intermittent and multiscale nature of chemical potential: the fractal case
205(1)
6.6 The intermittency parameter and clustering entropy of particles in the fractal case
206(6)
6.6.1 Clustering entropy of a single i-particle belonging to a fractal set of particles
206(2)
6.6.2 Clustering entropy for the whole ith component (fractal case)
208(1)
6.6.3 Clustering entropy of the mixture (fractal case)
209(1)
6.6.4 The case of two-components fractally distributed in a mixture
209(1)
6.6.5 Fractal dimension of the maximum clustering entropy case
210(1)
6.6.6 Equifractality condition of equilibrium
211(1)
6.6.7 Entropy jump by adding a particle in the fractal case
211(1)
6.7 The clustering entropy and chemical potential in the parabolic fractal case
212(2)
6.7.1 Clustering entropy of an individual i-particle in parabolic fractal case
212(1)
6.7.2 Clustering entropy for the ith component in parabolic fractal case
213(1)
6.7.3 Clustering entropy of the mixture in parabolic fractal case
213(1)
6.8 Summing up formulas and conclusion
214(5)
Conclusion 219(6)
References 225(8)
Index 233
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).. Michel Feidt is Professor in the Department of Physics and Mechanics at the University of Lorraine, France.
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