Adiabatic Thermodynamics
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ADIABATIC THERMODYNAMICS A monograph Christian Fronsdal Dept. of Physics and Astronomy, University of California Los angeles CA USA Copyright C.Fronsdal 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019. i TABLE TENTS Acknowledgements Page 1 Forword Page 2 I. INTeRODUCTION Page 8 I.1. Elements of hydrodynamics Page 9 The nature of a liquid. Density and pressure. Energy. Equation of continuity. Newtonian fluids. I.2. Hamiltonian and Lagrangian hydrodynamics Page 13 Action principles. Lagrange multipliers. The Lagrangian. Velocity potential. Kinetic energy. The Hamiltonian. I.3. The temperature Page 19 The ideal gas law. I.4. The internal energy and the first law Page 21 Internal Energy. Entropy. Carnot cycle. The free energy. I.5. The kinetic potential, and a warning Page 26 The energy concept. The danger of defining a theory by a set of equations. I.6. A Lagrangian in case of dissipation Page 28 I.7. Hydrodynamics of potential flow Page 29 I.8. The ‘Lagrangian’ formulation of hydrodynamics Page 33 The limitations of potential hydrodynamics. ii II. ACTION PRINCIPLE FOR THERMODYNAMICS II.1. The internal energy and the free energy Page 38 Functional relations. Legendre transformations II.2. The ideal gas Page 39 The free energy of the ideal gas II.3. The two time scales of thermodynamics Page 41 Adiabatic process. Dissipative process II.4. Towards an Action Principle for Thermodynamics Page 44 Interpretation of the action principle. The free energy and the Gibbs function II.5. Lowest energy and maximal entropy Page 48 Gibbs’ theorem III. LOCALIZATION Page 50 III.1. Hamiltonian density. Adiabatic equations Page 51 Adiabatic condition. Hydrostatic condition. The specific entropy density. Thermodynamic pressure and the pressure on the walls. III.2. The kinetic energy Page 55 Poisson bracket. III.3. The Lagrangian and adiabatic dynamics Page 57 On-shell Lagrangian. Legendre transformations III.4. The second law Page 60 The increasing entropy III.5. Work and pressure Page 62 Pressure on the walls. Adiabatic systems III.6. Important lesson Page 68 Entropic rubber. Some questions III.7. Local thermodynamical potentials Page 71 The Clapeyron equation. Open and closed systems. III.8. Conservation laws and the stress tensor Page 72 Noether’s theorems. The equalisation of pressure III.9. Minimal energy and maximal entropy Page 78 The classical Lagrangian paradigm. Maximal entropy at fixed energy. The two laws of thermodynamics III.10. The third law and the normalization of entropy Page 82 III.11. Galilei invariance Page 82 III.12. On the space of variations Page 86 Taking stock. Assigned entropy. iii IV. APPLICATIONS TO SIMPLE SYSTEMS Page 89 IV.1. The propagation of sound in an ideal gas Page 90 Harmonic perturbations. Adiabatic propagation. Radial waves IV.2. The van der Waals fluid Page 94 Excluded volume. Critical point. Data for Helium IV.3. Saturation Page 98 Interpretation of the velocity potential IV.4. Dew point, bubble point and Maxwell’s rule Page 101 The adiabatic relation. Calculating the dew point of Argon. Normalization of f. Common tangent IV.5. The critical point Page 109 Coexistence IV.6. Sound propagation near the critical point Page 111 IV.7. Refinements of the internal energy Page 113 The heat capacity. The speed of sound IV.8. The cubic equation of state Page 115 IV.9. Stability Page 117 Tables V. A THEORY OF FLUID MIXTURES Page 122 V.1. Variables and potentials Page 125 Molar fractions and ‘free energy of mixing’ V.2. Entropy and equations of motion Page 129 Adiabatic systems. Entropy path. Partial pressures V.3. Mixture of 2 ideal gases Page 134 Dalton’s law V.4. The Gibbs-Dalton hypothesis for ideal gases ... Page 137 Strong Gibbs-Dalton hypothesis V.5. .. and what lies behind it Page 140 Condensation V.6. Gibbs-Dalton hypothesis for some real gases Page 144 V.7. The simplest mixtures. Boltzmann theory Page 147 V.8. Chemical reactions. Dissociation Page 148 Gibbs’ phase rule. Solving the equations. Minimal energy V.9. Other chemical reactions Page 154 Constraints. Reaction constant. General case V.10. Taking stock Page 157 Project 1. Project 2. Project 3. iv VI. IMMISCIBLE FLUIDS Page 159 VI.1. Ideal gases with mutual interaction Page 160 Data. Conditions for equilibrium. The method of the common tangent. Symmetry. Direct search. Summary. VI.2. Mixtures of two van der Waals gases Page 169 The spinodal. Mixtures. Calculation of the line of coexistence VI.3. Sample applications - first example Page 174 Binary van der Waals mixtures. Points that are critical to the experimenter. VI.4. Mixture of van der Waals fluids. Second example Page 180 The square root interaction density. Nitrogen and Argon VI.5. Helium and Xenon Page 187 Project 4. VII. SPEED OF SOUND IN MIXTURES Page 190 VII.1. Propagation of sound in a Gibbs-Dalton mixture Page 191 Equations of motion. Propagation matrix. Types of mixtures. Summary. Type 1, similar gases. Type 2, disparate gases VII.2. A simple model of interactions Page 198 The square root interaction. Variants VII.3. Two van der Waals gases. Sound speed Page 202 Using the Gibbs - Dalton hypothesis for vand der Waals fluids VIII. SUPERFLUID HELIUM VIII.1. Experiments and interpretation Page 205 Fountain experiment. Entropy. Equilibrium configuration. Second sound. Pressure VIII.2. First analytic model. Page 216 Thermodynamic variables. Structure of the Lagrangian. The pressure. Entropy. Sound speeds. Equilibrium. Heat capacity. Equilibrium. Heat capacity. VIII.3. Dynamical equations of the model. Page 223 The adiabatic index. Fountain. The adiabatic index Rotational flow VIII.4. For the future Page 233 Tables.241 241 Project 5. v IX. ATMOSPHERES IX.1. Introducing gravitation Page 235 The van der Waals atmosphere IX.2. Mixed, isentropic atmospheres Page 238 IX.3. Nitrogen and Oxygen Page 239 Gibbs-Dalton and gravity IX.4. Air and Argon Page 242 The unifore lapse rate. van der Waals atmospheres IX.5. Air and H2O. Introduction Page 244 van der Waals parameters for water. Standard method IX.6. Phase separation in mixture of Air and Water Page 247 Calculation. Interaction Project 6. Appendix to Chapter IX Page 251 Emden’s theorem. The effect of entropic forces. Isentropic atmospheres. What physics is involved? Ask Maxwell. Interaction X. CONSERVATIVE HYDRODYNAMICS Page 258 X.1. The variables of hydrodynamics Page 259 Importance of velocity potential. Gravitation. Gauge theories. Why the scalar velocity potential is essential. X.2. Constraint and Euler-Lagrange equations Page 262 X.3. Cylindrical Couette flow Page 263 Laminar Couette flow. A classical mistake and what it tells us. X.4. Conservative hydrodynamics a kinetic potential Page 270 Viscosity X.5. Stability of laminar Couette flow Page 271 Boundary conditions. X.6. Conclusions Page 281 Main result. Tension. Entropic forces. Future experiments. Action Methodology. Other applications. A theory of incompressible fluids? X.8. The energy - momentum tensor Page 285 Irrotational case. Angular momentum. General case Appendix to Chapter 10. Plane Couette flow Page 290 Interpretation vi XI. ROTATING PLANETS Page 293 XI.1. Introduction Page 294 The equatorial bulge. Conservative Hydrodynamics XI.2. Compressible fluid rotating in gravitational field Page 298 Equatorial bulge. Regularity at the poles. Overview of results XI.3. Earth and Mars Page 304 Quality of fits, Density profiles. Rings or not. Evolution XI.4. Neptune and Uranus. Venus and Pluto Page 308 XI.5. Saturn and Jupiter. The Sun. Haumea. Galaxies Page 309 Saturn’s hexagon XI.6. Summary and conclusions Page 312 XII. SPECIAL RELATIVITY Page 314 XII.1. The relativistic, antisymmetric tensor field Page 315 Gauge theory. Field equations and constraints. Photon mass. The covariant gauge. Conservative hydrodynamics XII.2. Non Relativistic limit and Galilei transformations Page 322 Constraints XII.3. Dark Matter Page 325 XIII. GENERAL RELATIVITY XIII.1. Astrophysics and hydrodynamics Page 326 XIII.2. Introduction to General Relativity, the metric Page 327 XIII.3. The dynamical metric Page 328 Connections XIII.4. The Bianchi identity Page 330 Implications of the Bianchi identity. Particles as sources. Gravity waves XIII.5. Sources for G.R. Tolman’s formula Page 333 XIII.6. Conservative sources for Relativity Page 334 REFERENCES Page 336 INDEX Page 345 vii It is a pleasure to thank my friends Joseph Rudnick, Gary Williams, Tore Haug-Warberg, Alexander Zheltukhin, Frank Hansen, Hidenori Son- oda, Elliott Lieb and Joel Lebovitz for illuminating conversations. I thank the Bogoliubov Institute in Dubna and my much regretted friend Volodya Kadeshevskij for hospitality. Special thanks are due to Elliott Lieb for hos- pitality at Princeton during a prolonged visit. Credits 1 Foreword Thermodynamics is usually subdivided into a theory dealing with equilibrium and into one concerned with irreversible processes”. The quotation (Tisza 1961) is testimony to the meaning that was and still is ascribed to the word “thermodynamics”. Why is the extensive lit- erature on adiabatic, non equilibrium thermodynamics excluded from con- sideration? Why is the study of dissipative thermodynamics referred to as “non-equilibrium thermodynamics”? Laplace explained the speed of prop- agation of sound in air by postulating that the process is adiabatic; it is neither equilibrium nor dissipative. What about 200 years of atmospheric science? If it is regarded as an application of mere hydrodynamics then it is high time for an upgrade, for it is not possible to understand atmospheres without the concept of temperature. 2 This book is a my attempt to understand thermodynamics, in the for- mulation as a Eulerian field theory of temperature, pressure, densities and entropies, and completely integrated with hydrodynamics. Another hope was, and remains, to make classical thermodynamics more easily accessi- ble to physicists with other specialities, people like myself who neglected to study the subject in school. To this end the difficulties are emphasized rather than glossed over.