Adiabatic Thermodynamics

Total Page:16

File Type:pdf, Size:1020Kb

Adiabatic Thermodynamics 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.
Recommended publications
  • Critique of the Gibbs Volume Entropy and Its Implication
    Critique of the Gibbs volume entropy and its implication Jian-Sheng Wang∗ Department of Physics, National University of Singapore, Singapore 117551, Republic of Singapore (Dated: 8 July 2015) Dunkel and Hilbert, “Consistent thermostatistics forbids negative absolute temperatures,” Na- ture Physics, 10, 67 (2014), and Hilbert, H¨anggi, and Dunkel, “Thermodynamic laws in isolated systems,” Phys. Rev. E 90, 062116 (2014) have presented an unusual view of thermodynamics that sets aside several properties that have traditionally have been assumed to be true. Among other features, their results do not satisfy the postulates of thermodynamics originally proposed by Tisza and Callen. In their theory, differences in the temperatures of two objects cannot determine the direction of heat flow when they are brought into thermal contact. They deny that negative temperatures are possible. We disagree with most of their assertions, and we present arguments in favor of a more traditional interpretation of thermodynamics. We show explicitly that it is possible to deduce the thermodynamic entropy for a paramagnet along the lines of classical thermodynamics and that it agrees with Boltzmann entropy in equilibrium. A Carnot engine with efficiency larger than one is a matter of definition and is possible for inverted energy distributions, regardless of whether negative temperatures are used in the analysis. We elaborate on Penrose’s argument that an adiabatic and reversible process connecting systems with Hamiltonian H to −H is possible, thus negative temperatures logically must exist. We give a demonstration that the Boltzmann tempera- ture determines the direction of heat flow while Gibbs temperature does not, for sufficiently large systems.
    [Show full text]
  • Paul Ehrenfest: the Genesis of the Adiabatic Hypothesis, 1911–1914
    Arch. Hist. Exact Sci. 60 (2006) 209–267 Digital Object Identifier (DOI) 10.1007/s00407-005-0105-1 Paul Ehrenfest: The Genesis of the Adiabatic Hypothesis, 1911–1914 Luis Navarro and Enric Perez´ Communicated by R.H. Stuewer Contents 1. Introduction ...................................210 2. From black-body radiation to rotational specific heats (1913) .........212 2.1. EINSTEIN and STERN’s proposal in support of a zero-point energy . 212 2.2. EHRENFEST’s counter-proposal: Quantize the rotational motion of diatomic molecules ............................215 2.3. EHRENFEST versus EINSTEIN and STERN ..............217 2.4. Toward an adiabatic hypothesis ......................223 2.5. Some reactions ..............................228 3. From black-body radiation to periodic mechanical systems (1913) ......231 3.1. Application of a mechanical theorem of BOLTZMANN .........232 3.2. Adiabatic invariants appear on the quantum scene ............233 3.3. EINSTEIN’s objection and other considerations .............236 4. Inquiries on the validity of BOLTZMANN’s statistical mechanics (1914) . 237 4.1. Criticism of equiprobability ........................238 4.1.1. The “δG-condition” .......................239 4.1.2. Further details ..........................241 4.2. On the genesis of EHRENFEST (1914) as seen in his notebooks ....243 4.2.1. The significance of EHRENFEST (1911) ............243 4.2.2. FOKKER’s ensembles ......................246 4.2.3. Elaboration of EHRENFEST (1914) ...............250 5. EINSTEIN and EHRENFEST’s hypothesis ..................253 6. Epilogue: EHRENFEST’s adiabatic hypothesis (1916) ............259 References ......................................263 Abstract We analyze the evolution of EHRENFEST’s thought since he proved the necessity of quanta in 1911 until the formulation of his adiabatic hypothesis in 1914. We argue that his research contributed significantly to the solution of critical problems in quantum 210 L.
    [Show full text]
  • Quantization Conditions, 1900–1927
    Quantization Conditions, 1900–1927 Anthony Duncan and Michel Janssen March 9, 2020 1 Overview We trace the evolution of quantization conditions from Max Planck’s intro- duction of a new fundamental constant (h) in his treatment of blackbody radiation in 1900 to Werner Heisenberg’s interpretation of the commutation relations of modern quantum mechanics in terms of his uncertainty principle in 1927. In the most general sense, quantum conditions are relations between classical theory and quantum theory that enable us to construct a quantum theory from a classical theory. We can distinguish two stages in the use of such conditions. In the first stage, the idea was to take classical mechanics and modify it with an additional quantum structure. This was done by cut- ting up classical phase space. This idea first arose in the period 1900–1910 in the context of new theories for black-body radiation and specific heats that involved the statistics of large collections of simple harmonic oscillators. In this context, the structure added to classical phase space was used to select equiprobable states in phase space. With the arrival of Bohr’s model of the atom in 1913, the main focus in the development of quantum theory shifted from the statistics of large numbers of oscillators or modes of the electro- magnetic field to the detailed structure of individual atoms and molecules and the spectra they produce. In that context, the additional structure of phase space was used to select a discrete subset of classically possible motions. In the second stage, after the transition to modern quantum quan- tum mechanics in 1925–1926, quantum theory was completely divorced from its classical substratum and quantum conditions became a means of con- trolling the symbolic translation of classical relations into relations between quantum-theoretical quantities, represented by matrices or operators and no longer referring to orbits in classical phase space.1 More specifically, the development we trace in this essay can be summa- rized as follows.
    [Show full text]
  • Adiabatic Theory for Slowly Varying Hamiltonian Systems with Applications to Beam Dynamics
    Alma Mater Studiorum · Università di Bologna Scuola di Scienze Dipartimento di Fisica e Astronomia Corso di Laurea Magistrale in Fisica Adiabatic theory for slowly varying Hamiltonian systems with applications to beam dynamics Relatore: Presentata da: prof. Armando Bazzani Federico Capoani Correlatore: dott. Massimo Giovannozzi Anno Accademico 2017/2018 Qual è ’l geomètra che tutto s’affige per misurar lo cerchio, e non ritrova, pensando, quel principio ond’elli indige, tal era io a quella vista nova: veder voleva come si convenne l’imago al cerchio e come vi s’indova. Pd., xxxiii, 133–138 iv Abstract In questo lavoro si presentano due modelli in cui la teoria dell’in- varianza adiabatica può essere applicata per ottenere peculiari ef- fetti nel campo della dinamica dei fasci, grazie all’attraversamento di separatrici nello spazio delle fasi causato dal passaggio attraverso determinate risonanze d’un sistema. In particolare, si esporrà un modello bidimensionale con cui è possibile trasferire emittanza tra due direzioni nel piano trasverso: si fornirà una spiegazione del mec- canismo per cui tale fenomeno ha luogo e si mostrerà come prevedere i valori finali di emittanza che un sistema raggiunge in tale configu- razione. Questi risultati sono confermati da simulazioni numeriche. Simulazioni numeriche e relativi studî parametrici sono anche i risul- tati che vengono presentati per un altro modello, stavolta unidi- mensionale: si mostrerà infatti come un eccitatore esterno oscillante dipolare la cui frequenza passi attraverso un multiplo del tune della macchina permetta di catturare le particelle d’un fascio in un certo numero d’isole stabili. vi Abstract Contents Contents viii Introduction 1 I The toolbox 5 1 Elements of Hamiltonian mechanics 7 1.1 Hamiltonian mechanics .
    [Show full text]
  • A Microscopic, Mechanical Derivation for the Adiabatic Gas Relation
    A microscopic, mechanical derivation for the adiabatic gas relation P. M. Bellan Applied Physics, California Institute of Technology, Pasadena CA 91125 Abstract It is shown that the ideal gas adiabatic relation, P V γ = constant, can be derived by considering the motion of a particle bouncing elastically between a stationary wall and a moving wall. arXiv:physics/0310015v1 [physics.ed-ph] 2 Oct 2003 1 I. INTRODUCTION The simplest form of the adiabatic gas relation is the observation that the temperature of a thermally insulated gas increases when it is compressed and decreases when it is expanded. According to the historical review of this subject by Kuhn,1 the first publication document- ing this behavior was by the Scottish physician William Cullen in the mid 17th century. Experimental observations were summarized by the relation PV γ = constant, where the exponent γ was determined to exceed unity. The deviation of γ from unity is what allowed Sadi Carnot to develop his eponymous cycle. (Apparently Carnot did not have the correct value of γ, but the use of an incorrect value did not affect his fundamental result that heat engine efficiency depends only on inlet and outlet temperatures.) Serious attempts to develop a theoretical explanation for the adiabatic gas relation were undertaken by Laplace, Poisson, and others in the early 19th century, but no single individ- ual has been identified as being the first to provide the correct theoretical explanation. Since the mid-19th century development of thermodynamics, the adiabatic gas relation has been established from first principles using thermodynamic arguments. The standard thermody- namic derivation is based on considering the temperature change of the gas in a cylinder at constant pressure or constant volume while taking into account specific heats at constant pressure and constant volume.2 The purpose of this paper is to show that the adiabatic gas relation PV γ = constant is a direct consequence of an important property of periodic mechanical motion, namely adiabatic invariance.
    [Show full text]
  • 1 the Ehrenfest Adiabatic Hypothesis and the Old Quantum Theory, Before Bohr
    1 The Ehrenfest Adiabatic Hypothesis and the Old Quantum Theory, before Bohr Enric Perez´ Canals and Marta Jordi Taltavull HQ1 - S1 The Ehrenfest Adiabatic Hypothesis and the Old Quantum Theory, before Bohr Enric Pérez Canals Marta Jordi Taltavull Universitat de Barcelona Paul Ehrenfest at University of Leiden This talk deals with the role of the adiabatic hypothesis in the development of the old quantum theory. This hypothesis was formulated by Ehrenfest in a paper published in 1916,This but talk practically deals with all the role results of the that adiabatic appeared hy therepothesis had in been the publisheddevelopment by himof theduring old thequa previousntum theory. ten years. This What hypothesis Ehrenfest was did formulated in 1916, was by to Ehrenfest collect all in those a pap earlierer publishedresults on thein 1916, adiabatic but transformations practically all andthe theirresults relations that appeared to the quantum there theory,had been with the idea that they should become widely known. Far from that, the Ehrenfest's 1916 published by him during the previous ten years. What Ehrenfest did in 1916, was to paper had little impact during the next few years. It was only after Bohr in 1918 collect all those earlier results on the adiabatic transformations and their relations to the published an essential work about the quantum theory, where he used the adiabatic quantum theory, with the idea that they should become widely known. hypothesis, that its importance began increasing. To sumFar upfrom the that, role thatthe Ehrenfest’s the adiabatic 191 hypothesis6 paper had played little inimpact the development during the n ofext the few old years.quantum It was theory only up after to 1918, Bohr whatin 1918 follows published is centred an essent in theseial fourwork axes: about the quantum theory, where he used the adiabatic hypothesis, that its importance began increasing.
    [Show full text]
  • Paul Ehrenfest
    ELEGANT CONNECTIONS IN PHYSICS The Struggles of Paul Ehrenfest by Dwight E. Neuenschwander Professor of Physics at Southern Nazarene University in Bethany, OK During the founding years of quantum mechanics, the Niels Bohr Institute in Co- Paul Ehrenfest always insisted on hon- penhagen, Denmark, hosted an annual spring meeting where the hottest topics in esty in thought and action. In the preface physics were discussed. It became traditional to close the meeting with a skit that to Ehrenfest’s collected papers, H.B.G. Ca- parodied the state of physics at the time. The 1932 conference coincided with the simir wrote that Ehrenfest’s lectures were tenth anniversary of the institute’s founding and needed to end on a special note. brilliant in an unconventional way:[2] A few years earlier, Wolfgang Pauli had suggested the existence of a particle carrying zero mass and zero electric charge that could explain the missing energy and momentum He emphasized salient points rather in beta decay, a type of radioactive decay of the nucleus. He called this hypothetical par- than continuity of argumentation; the ticle the “neutron.” When in 1932 James Chadwick discovered the massive neutral par- essential formulae appeared on the ticle we now call the neutron, Enrico Fermi suggested that the name of Pauli’s particle be blackboard almost as aesthetical enti- changed to neutrino, the “little neutral one.” ties and not only as links in a chain of At the Bohr Institute’s 1932 meeting, Pauli’s neutrino was still a speculative teaser, deductions. One had very little incli- with many doubters.
    [Show full text]
  • Einstein's Physics
    OUP-SECOND UNCORRECTED PROOF, November 29, 2012 Einstein’s Physics Atoms, Quanta, and Relativity—Derived, Explained, and Appraised TA-PEI CHENG University of Missouri–St. Louis Portland State University 3 OUP-SECOND UNCORRECTED PROOF, November 29, 2012 Preface Einstein explained in equations Albert Einstein’s achievement in physics is proverbial. Many regard him as the greatest physicist since Newton. What did he do in physics that’s so import- ant? While there have been many books about Einstein, most of these explain his achievement only in qualitative terms. This is rather unsatisfactory as the language of physics is mathematics. One needs to know the equations in order to understand Einstein’s physics: the precise nature of his contribu- tion, its context, and its influence. The most important scientific biography of Einstein has been the one by Abraham Pais: Subtle is the Lord ... The Science and the Life of Albert Einstein: The physics is discussed in depth; however, it is still a narrative account and the equations are not worked out in detail. Thus this biography assumes in effect a high level of physics back- ground that is perhaps beyond what many readers, even working physicists, possess. Our purpose is to provide an introduction to Einstein’s physics at a level accessible to an undergraduate physics student. All physics equa- tions are worked out from the beginning. Although the book is written with primarily a physics readership in mind, enough pedagogical support material is provided that anyone with a solid background in an introductory physics course (say, an engineer) can, with some effort, understand a good part of this presentation.
    [Show full text]
  • Plasma-Part1
    Fields of active research 1- Plasma theory Plasma equilibria and stability Plasma interactions with waves and beams Guiding center Adiabatic invariant Debye sheath Coulomb collision 2- Dusty plasmas 3- Plasma sources 4- Plasma diagnostics Thomson scattering Langmuir probe Spectroscopy Interferometry Ionospheric heating Incoherent scatter radar Fields of active research 5 -Plasmas in nature The Earth's ionosphere Space plasmas, e.g. Earth's plasmasphere (an inner portion of the magnetosphere dense with plasma) Astrophysical plasma Industrial plasmas Plasma chemistry Plasma processing Plasma spray Plasma display Fields of active research 6- Plasma applications Fusion power Magnetic fusion energy (MFE) — tokamak, stellarator, reversed field pinch, magnetic mirror, dense plasma focus Inertial fusion energy (IFE) (also Inertial confinement fusion — ICF) Plasma-based weaponry Ion implantation Plasma ashing Food processing (nonthermal plasma, aka "cold plasma") Plasma arc waste disposal, convert waste into reusable material with plasma. Plasma acceleration Plasma equilibrium and stability 1- Plasma theory Plasma interactions with waves and beams Guiding center Adiabatic invariant Plasma stability Debye sheath Coulomb collision An important field of plasma physics is the stability of the plasma. It usually only makes sense to analyze the stability of a plasma once it has been established that the plasma is in equilibrium. "Equilibrium" asks whether there are net forces that will accelerate any part of the plasma. If there are not, then "stability"
    [Show full text]
  • Ehrenfest's Adiabatic Hypothesis in Bohr's Quantum Theory Enric
    Ehrenfest’s adiabatic hypothesis in Bohr’s quantum theory Enric Pérez * and Blai Pié Valls ** Abstract: It’s widely known that Paul Ehrenfest formulated and applied his adiabatic hypothesis in the early 1910s. Niels Bohr, in his first attempt to construct a quantum theory in 1916, used it for fundamental purposes in a paper which eventually did not reach the press. He decided not to publish it after having received the new results by Sommerfeld in Munich. Two years later, Bohr published “On the quantum theory of line- spectra.” There, the adiabatic hypothesis played an important role, although it appeared with another name: the principle of mechanical transformability . In the subsequent variations of his theory, Bohr never suppressed this principle completely. We discuss the role of Ehrenfest’s principle in the works of Bohr, paying special attention to its relation to the correspondence principle. We will also consider how Ehrenfest faced Bohr’s uses of his more celebrated contribution to quantum theory, as well as his own participation in the spreading of Bohr’s ideas. Key words: Adiabatic Principle, Correspondence Principle, Paul Ehrenfest, Niels Bohr. 1. Introduction Ninety years ago an issue of Die Naturwissenschaften was released to commemorate the tenth anniversary of Bohr’s atom (Figure 1). Ehrenfest contributed a paper on the adiabatic principle, and the contributions by Kramers, and by Bohr himself (in fact, his Nobel lecture) also mentioned Ehrenfest’s hypothesis. In a similar vein, it is not uncommon to find references to the adiabatic principle when considering Bohr’s quantum theory from a historiographical point of view, as one of its two pillars, next to the correspondence principle.1 Yet, the disproportion in the interest provoked and the treatment given to the two principles is significant.
    [Show full text]
  • Ta-Pei Cheng Talk Based on …
    Ta-Pei Cheng talk based on … Oxford Univ Press (2013) Einstein’s Physics Atoms, Quanta, and Relativity --- Derived, Explained, and Appraised Albert Einstein 1. Atomic Nature of Matter 1879 – 1955 2. Quantum Theory The book 3. Special Relativity explains his physics 4. General Relativity in equations 5. Walking in Einstein’s Steps Today’s talk 1. The quantum postulate: Planck vs Einstein provides w/o math details 2. Einstein & quantum mechanics some highlight 3. Relativity: Lorentz vs Einstein historical context 4. The geometric theory of gravitation & influence 5. Modern gauge theory and unification 3 Einstein, like Planck, arrived at the quantum hypothesis thru the study of BBR Blackbody Radiation cavity radiation = rad in thermal equilibrium ∞ ∞ 2nd law ν u(uT(T) =) =∫ ρρ(T(T,ν,ν)d)νdν KirchhoffTo find the (1860) u(T) and ρ ( T, ) functions:densities u ( T ) and ρ ( T ,ν) 0∫0 Stefan (1878) Boltzmann (1884) law: u(T ) = aT 4 = universal functions rep intrinsic property of radiation Thermodynamics + Maxwell EM (radiation pressure = u/3 ) electromagnetic radiation = a collection of oscillators u = E 2, B 2 ~ oscillator energy kx 2 oscillating energy ~ frequency : Their ratio of is an adiabatic invariant 3 What is the f function? ρ (T ,ν ) =ν f (ν / T ) Wien’s displacement law (1893) Wien’s distribution (1896) : ρ ( T , ν ) = αν 3 e − βν / T fitted data well…. until IR αν 3 ρ(T,ν ) = eβν /T −1 4 αν 3 ρ(T,ν ) = Blackbody Radiation Planck (1900): e βν / T − 1 3 2 U = (c 8/ πν )ρ and dS = dU /T → entropy S ε = nh ν n = 2,1,0 ,..
    [Show full text]
  • General Plasma Physics I Notes AST 551
    General Plasma Physics I Notes AST 551 Nick McGreivy Princeton University Fall 2017 1 Contents 0 Introduction5 1 Basics7 1.1 Finals words before the onslaught of equations . .8 1.1.1 Logical framework of Plasma Physics . .9 1.2 Plasma Oscillations . 10 1.3 Debye Shielding . 13 1.4 Collisions in Plasmas . 17 1.5 Plasma Length and Time Scales . 21 2 Single Particle Motion 24 2.1 Guiding Center Drifts . 24 2.1.1 E Cross B Drift . 25 2.1.2 Grad-B drift . 29 2.1.3 Curvature Drift . 31 2.1.4 Polarization Drift . 35 2.1.5 Magnetization Drift and Magnetization Current . 37 2.1.6 Drift Currents . 38 2.2 Adiabatic Invariants . 40 2.2.1 First Adiabatic Invariant µ ................. 44 2.2.2 Second Adiabatic Invariant J ................ 46 2.3 Mirror Machine . 46 2.4 Isorotation Theorem . 48 2.4.1 Magnetic Surfaces . 49 2.4.2 Proof of Iso-rotation Theorem . 50 3 Kinetic Theory 54 3.1 Klimantovich Equation . 57 3.2 Vlasov Equation . 60 3.2.1 Some facts about f . 62 3.2.2 Properties of Collisionless Vlasov-Maxwell Equations . 63 3.2.3 Entropy of a distribution function . 64 3.3 Collisions in the Vlasov Description . 66 3.3.1 Heuristic Estimate of Collision Operator . 66 3.3.2 Strongly and Weakly Coupled Plasmas . 67 3.3.3 Properties of Collision Operator . 68 3.3.4 Examples of Collision Operators . 69 3.4 Lorentz Collision Operator . 70 3.4.1 Lorentz Conductivity . 72 4 Fluid and MHD Equations 74 4.1 Deriving Fluid Equations .
    [Show full text]