Thermodynamics and Statistical Physics WS 2018/19

Thermodynamics and Statistical Physics WS 2018/19 Roser Valentí Institute for Theoretical Physics Goethe University Frankfurt, Germany Manuscript of the ITP members Roser Valentí, Claudius Gros and, partly Peter Kopietz.

Contents 1 Introduction 9 1.1 Literature.................................... 9 1.2 What is thermodynamics? What is statistical physics?.......................... 10 1.3 Historical development............................. 11 1.3.1 Thermodynamics............................ 12 1.3.2 Ideal Gas................................ 12 1.3.3 Statistical physics............................ 13 1.3.4 Twentieth Century........................... 13 1.3.5 Further developments.......................... 14 1.4 Effective vs. fundamental theories....................... 14 2 Thermodynamics basics 17 2.1 Thermodynamic systems and variables.................... 17 2.1.1 Thermodynamic limit and state.................... 18 2.2 Temperature................................... 18 2.2.1 Thermal equilibrium and transformations............... 19 3 Work, heat and the first law of thermodynamics 21 3.1 Mechanical work................................ 21 3.2 Heat....................................... 22 3.3 Exact differentials................................ 23 3.4 First law of thermodynamics internal energy................ 24 3.4.1 Internal energy of an ideal gas..................... 25 3.5 Energy and heat capacity for various processes................ 26 3.5.1 Isochoric process............................ 26 3.5.2 Isobaric process............................. 26 3.5.3 Isothermal processes for the ideal gas................. 27 3.5.4 Free expansion of an ideal gas..................... 28 3.5.5 Adiabatic processes for the ideal gas................. 29 3.6 Enthalpy..................................... 29 3.7 Magnetic systems................................ 30 3

4 CONTENTS 4 Entropy and the second law of thermodynamics 33 4.1 Heat engines................................... 33 4.1.1 Carnot cycle............................... 34 4.2 Second law of thermodynamics........................ 34 4.2.1 Universality of the Carnot cycle.................... 35 4.3 Absolute temperature.............................. 37 4.3.1 Clausius s inequality.......................... 38 4.3.2 Entropy................................. 39 4.4 Entropy as a thermodynamic variable..................... 41 4.4.1 Entropy of the ideal gas........................ 41 4.4.2 Maxwell equations........................... 42 4.4.3 Energy equation............................. 42 4.5 Eulers s cyclic chain rule............................ 42 4.5.1 Entropy differentials.......................... 44 4.6 Third law of thermodynamics (Nernst law).................. 45 5 Thermodynamic potentials 47 5.1 Internal energy U................................ 47 5.1.1 Monoatomic ideal gas.......................... 48 5.1.2 Thermodynamic potential vs. equation of state............ 49 5.1.3 Classical mechanics vs. thermodynamics............... 49 5.2 Legendre transformation............................ 50 5.2.1 Legendre transformations in thermodynamics............ 50 5.2.2 The four basic thermodynamic potentials............... 51 5.2.3 Maxwell relations............................ 52 5.2.4 Entropy................................. 52 5.3 Homogeneity relations............................. 53 5.4 Grand canonical potential........................... 54 5.5 Equilibrium conditions............................. 55 5.5.1 Principle of maximal entropy..................... 55 5.5.2 Principle of minimal free energy.................... 57 5.5.3 Stationarity principles in information theory............. 58 5.6 Summary of thermodynamic potentials.................... 60 6 Phase transitions 63 6.1 Concept of phase................................ 63 6.2 First-order phase transition.......................... 64 6.2.1 Condition for phase coexistence.................... 64 6.2.2 Clausius-Clapeyron equation...................... 65 6.3 Ehrenfest classification of phase transitions.................. 66 6.4 Van der Waals equation of state........................ 69 6.4.1 Virial expansion............................. 71 6.4.2 Critical point.............................. 72 6.4.3 Maxwell construction.......................... 73

CONTENTS 5 7 Equilibrium statistical physics 75 7.1 Introduction................................... 75 7.2 Statistical mechanics fundamentals...................... 76 7.2.1 Classical mechanics........................... 77 7.2.2 Microstates............................... 77 7.2.3 Probability distribution functions................... 78 7.3 Liouville theorem................................ 79 7.4 Relaxing towards thermal equilibrium..................... 81 7.4.1 Ergodicity timescales.......................... 82 7.4.2 Time- vs. statistical averaging..................... 83 7.5 Postulate of equal a priori probabilities.................... 84 8 Microcanonical ensemble 85 8.1 Definition.................................... 85 8.2 Entropy..................................... 86 8.2.1 Additivity; Gibbs paradox....................... 88 8.2.2 Consistency with the definition of the temperature......... 89 8.2.3 Consistency with the second law of thermodynamics......... 90 8.2.4 How thick is the energy shell?..................... 90 8.3 Calculating with the microcanonical ensemble................ 91 8.3.1 Hyperspheres.............................. 92 8.4 The classical ideal gas............................. 93 8.4.1 Entropy................................. 94 8.5 Fluctuations and correlation functions.................... 96 8.5.1 Characteristic function and cumulants................ 97 8.5.2 Correlations between observables................... 97 8.6 Central limit theorem.............................. 98 8.6.1 Normal distribution........................... 98 8.6.2 Derivation of the central limit theorem................ 99 9 Canonical ensemble 103 9.1 System in contact with a heat reservoir.................... 103 9.1.1 Boltzmann factor............................ 105 9.2 Canonical partition function.......................... 107 9.3 Canonical vs. microcanonical ensemble.................... 108 9.4 Additivity of F(T,V,N)............................ 109 9.5 Ideal gas in the canonical ensemble...................... 110 9.6 Energy fluctuations............................... 112 9.7 Paramagnetism................................. 114 10 Grand canonical ensemble 117 10.1 Grand canonical partition function...................... 117 10.2 Grand canonical potential........................... 120 10.3 Fugacity..................................... 121 10.3.1 Particle distribution function..................... 121

6 CONTENTS 10.3.2 Particle number fluctuations...................... 122 10.3.3 Stability conditions........................... 122 10.4 The ideal gas in the grand canonical ensemble................ 125 10.4.1 Summary ensembles.......................... 127 11 Quantum states 129 11.1 Thermal wavelength.............................. 129 11.1.1 Fundamentals of quantum statistics.................. 130 11.2 One particle states............................... 131 11.3 The density matrix as the fundamental statistical operator......... 132 11.3.1 Pure states............................... 132 11.3.2 Mixed states............................... 133 11.3.3 Properties of ˆρ............................. 134 11.3.4 Correspondence principle........................ 135 11.4 Statistics of identical particles......................... 137 11.4.1 Slater determinant........................... 139 12 Quantum gases 141 12.1 Microcanonical ensemble............................ 141 12.1.1 Maximum entropy principle...................... 143 12.1.2 Third law of thermodynamics..................... 144 12.1.3 Two-level system and the concept of negative temperature..... 145 12.2 Canonical ensemble............................... 147 12.2.1 Quantum mechanical harmonic oscillators.............. 149 12.3 Grand canonical ensemble........................... 151 12.3.1 Entropy and the density matrix operator............... 152 12.4 Partition function of ideal quantum gases................... 153 12.5 Thermodynamic properties of ideal quantum gases.............. 154 12.5.1 Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac distribution.. 156 12.5.2 Particle number fluctuations...................... 157 13 Ideal Fermi gas 159 13.1 Equation of state................................ 159 13.1.1 Grand canonical potential....................... 160 13.2 Classical limit.................................. 163 13.3 Degenerated Fermi gas............................. 165 13.3.1 Ground state properties........................ 166 13.3.2 Fermi temperature........................... 166 13.4 Low temperature expansion.......................... 168 13.4.1 Density of states............................ 168 13.4.2 Sommerfeld expansion......................... 169 13.4.3 Internal energy at low temperatures.................. 170

CONTENTS 7 14 Ideal Bose gas 173 14.1 Equation of state................................ 173 14.2 Classical limit.................................. 177 14.3 Bose-Einstein condensation........................... 179 15 Landau-Ginzburg theory 181 15.1 The Landau model............................... 181 15.1.1 Solution in the absence of an external field.............. 182 15.1.2 Effect of an external field........................ 183 15.1.3 Specific heat............................... 184 15.2 Space-dependent order parameter....................... 185 16 Self-Organized Criticality 189 16.1 The sandpile model............................... 189 16.2 Probability Generating Function Formalism................. 191 16.3 Random branching theory........................... 193

8 CONTENTS

Chapter 1 Introduction 1.1 Literature 1) K. Huang, Introduction to Statistical Physics, CRC Press, 2001. 2) E. M. Lifschitz and L. P. Pitajewski, Statistical Physics, London, Landau Lifschitz Band 5. 3) S. K. Ma, Statistical Mechanics, World Scientific Publishing, 1985. 4) W. Nolting, Grundkurs Theoretische Physik, 2. Halfte Band 4: Thermodynamik, Band 5: Statistische Physik. 5) F. Reif, Fundamentals of Statistical and Thermal Physics, Mc Grow-Hill, Singapore, 1964. 6) R. Baierlein, Thermal Physics, Cambridge, 1999. 7) F. Mandl, Statistical Physics, Wiley, N. Y, 2 nd Edition 1988. 8) M. Toda, R. Kubo, N. Saito, Statistical Physics I, Equilibrium Statistical Mechanics, Springer, 2 nd Edition 1992. 9) P. Kopietz, Thermodynamik und Statistische Mechanik Script 2007. 10) Harrey Gould and Jan Tobochnik, Statistical and Thermal Physics with computer applications, Princeton University Press (2010). 9

10 CHAPTER 1. INTRODUCTION 1.2 What is thermodynamics? What is statistical physics? Thermodynamics is the phenomenological theory of the macroscopic properties of systems with a large number of constituents (particles, degrees of freedom). This theory describes the many-body system in terms of a few experimentally measurable parameters like Volume V, Pressure P, Temperature T, Density ρ. Phenomenological theory means that it does not provide the microscopic origin of a phenomenon. A large number of constituents means that the fluctuation effects of the constituents are negligible at the scale at which we are studying the system. A macroscopic object (1 cm 3 ) has usually N N Avogadro = 1 mole 6.022 10 23 particles. In such a system, the fluctuation effects of a macroscopic observable O are of the order: O O 1 N 10 12, which is a negligible effect if we study the system at the meter scale but will gain in importance if we consider shorter scales as would be required, for instance, for present electronic devices, which have sizes of 1nm=10 9 m. Statistical physics provides a description of the macroscopic properties of a many-body system in terms of the microscopic structure of the system. Statistical physics considers the basic equations of Classical Mechanics(classical statistical physics) or Quantum Mechanics (quantum statistical physics) and describes the macroscopic quantities in terms of the microscopic behavior. Statistical physics is more general than thermodynamics and the laws of thermodynamics can be derived from statistical physics. Classical statistical physics. As a reminder, the basic equations in classical mechanics for a system of N particles are given in the Hamilton formalism as q i = H p i, ṗ i = H q i, i = 1,2,...,3N, (1.1)

1.3. HISTORICAL DEVELOPMENT 11 with H = H(q 1...q 3N,p 1...p 3N,t) being the Hamilton function and q i and p i the generalized position and momentum coordinates. Our objective in this course will be to relate equations (1.1) to such concepts as temperature, pressure, internal energy, i.e., to establish relations microscopic equations macroscopic equations. Quantum statistical physics. In quantum mechanics, the basic equation for a system with N particles is the Schrödinger equation: i h Ψ t ( r 1σ 1,..., r N σ N, t) = [ N i=1 ] h 2 2m 2 i +V( r 1,..., r N, t) Ψ( r 1 σ 1,..., r N σ N, t), where Ψ( r 1 σ 1, r 2 σ 2,..., r N σ N, t) is the wave-function of the N-particle system. The individual particles are characterized here by their real-space positions r j and their spins σ j. We have in quantum mechanics two types of particles: Bosons: particles with integer spin. Their wave-function is symmetric with respect to interchange of two particles: Ψ( r 1 σ 1,..., r i σ i,..., r j σ j,..., r N σ N ) = Ψ( r 1 σ 1,..., r j σ j,..., r i σ i,..., r N σ N ); Fermions: particles with half-integer spin. Their wave-function is antisymmetric with respect to interchange of two particles: Ψ( r 1 σ 1,..., r i σ i,..., r j σ j,..., r N σ N ) = Ψ( r 1 σ 1,..., r j σ j,..., r i σ i,..., r N σ N ). As in the classical statistical physics, our aim in quantum statistical physics is to obtain, out of the microscopic equations, a description of the macroscopic properties, which can be related to experiment. 1.3 Historical development Thermodynamics was born out of a few unresolved important questions like: what is heat? what is temperature? why do some reactions happen and others do not? is it possible to convert heat into useful energy?

12 CHAPTER 1. INTRODUCTION 1.3.1 Thermodynamics 1600 s - Concept of heat: already in the seventeenth century some scientists thought that heat is associated with the motion of microscopic constituents of matter but the idea was too revolutionary at that time. 1700 s - there was a general belief that heat is a separate fluid-like substance. 1840 s - experiments of James Joule invalidating the theory of a separate fluid. 1850 s - it became accepted that heat is a form of energy. 1824- Sadi Carnot(French engineer and physicist) showed that the work produced by a steam engine is proportional to the heat transferred from the boiler to the condenser and that, in general, work could only be gained from heat by a transfer from a warmer to a colder body (Carnot s law). 1842 - Julius Robert von Mayer (German physicist) formulated the general law of conservation of energy and defined heat as a form of energy. 1843 - James Joule (British physicist) formulated the Joule s laws and determined the mechanical equivalent of heat. The heat Q generated by a current I flowing through a conductor is Q = I 2 Rt, where R is the resistance and t time. IR is the voltage drop. The internal energy of an ideal gas depends only on its temperature. 1850 - Rudolf Clausius (German physicist) formulated the second law of thermodynamics and introduced the entropy concept (1865) (in the same year Maxwell postulated the Maxwell equations in electrodynamics). The second law of thermodynamics reads: Heat cannot be transformed from a colder to a warmer body. 1.3.2 Ideal Gas In the nineteenth century, there were also a few advances concerning the behavior of the gas phase: 1807- Joseph Gay-Lussac(French chemist and physicist) investigated the changes in volume of a gas at different temperatures and showed that at constant pressure V T = V 0 T 0, (1.2) where T is absolute temperature in K (remember T = 273.15 K + t (in Celsius)). At constant temperature VP = V 0 P 0.

1.3. HISTORICAL DEVELOPMENT 13 1811 - Amadeo Avogadro (Italian physicist and chemist) postulated the Avogadro s law: The same volume of different gases under same conditions (T, P) contains the same number of particles. This postulate together with (1.2) can be summarized in the equation of state for an ideal gas, which we will analyze in detail in the coming chapters: PV = Nk B T, (1.3) where N is number of particles and k B =1.381 10 23 Joule/K is the Boltzmann constant. 1.3.3 Statistical physics 1860 - James Maxwell (British physicist) laid the foundations of the modern electrodynamic theory of light. He was also involved in the development of the kinetic gas theory. 1870 s - Ludwig Boltzmann (Austrian physicist). His contributions were fundamental. He was the first scientist to provide a statistical interpretation of the 2 nd law of thermodynamics: S = k B lnγ, namely that the entropy is proportional to the logarithm of the number of available states given by Γ. The fact that he used the concept of atoms in motion in order to explain the concept of heat provoked very strong criticisms at that time(ernst Mach, Wilhelm Otswald). 1902 - Josiah Willard Gibbs (US American physicist and mathematician) formulated the Gibbs phase rule which allows to calculate thermodynamical properties out of microscopic quantities. 1.3.4 Twentieth Century The twentieth century is the century of quantum mechanics and quantum statistical mechanics. 1900 - Max Planck (German physicist) postulated that the energy exchange of the radiation with the walls of a black body does not take place continuously but in discrete units of hω, ε n = n hω quantization of the radiation energy, so that the energy density of radiation of a black body is given by u(ω,t) = k BTω 2 π 2 c 3 hω/k B T e hω/k BT 1.

14 CHAPTER 1. INTRODUCTION For his derivation, Planck used concepts of electrodynamics, quantum mechanics (quantization of energy) and statistical physics, following Boltzmann. 1925 - Satyendranath Bose (Indian physicist) and Albert Einstein (German physicist) derived the statistics (Bose-Einstein statistics) for photons and, in general, for a system of bosons. predicted the phenomenon of Bose-Einstein condensation which was first corroborated experimentally 70 years later by W. Ketterle, E. Cornell and C. E. Wieman (Nobel Prize 2001). 1926 - Enrico Fermi (Italian physicist) and Paul Dirac (British physicist) derived the quantum statistics for fermions (Fermi-Dirac statistics). 1940: Wolfgang Pauli(Austrian physicist) stated the spin-statistics theorem which associates the value of the spin of a particle to the nature of the particle. Bosons : integer spin Fermions : half-integer spin 1.3.5 Further developments Some selected examples for the wide range of further developments. 1 - treatment of phase transitions, both macroscopically and microscopically(landau, Ginzburg, Bardeen, Cooper, Schrieffer,...). Important phase transitions are of type disordered phase ordered phase, which include gas-liquid-solid, paramagnetic-ferromagnetic (antiferromagnetic), liquid-superfluid (He 4, He 3 ) transitions, metal-superconductor. 2 - treatment of interacting many-particle systems, especially if the interactions are strong. Areas of application are solid state theory: correlated systems (strong Coulomb repulsion), high-energy physics: quantum chromodynamics (QCD). 3 - systems in equilibrium non-equilibrium systems. 1.4 Effective vs. fundamental theories We have defined thermodynamics as a phenomenological theory which was developed to describe phenomena at the scale of our naked-eye observations (length-scales 1cm,

1.4. EFFECTIVE VS. FUNDAMENTAL THEORIES 15 time-scales 1s). At these scales the knowledge of the atomic constituents is irrelevant. Thermodynamics is an effective theory and, in general, effective theories are useful for the understanding of observations at a given length- and time-scale. Statistical physics is, on the other hand, a fundamental theory since it intends to describe macroscopic phenomena out of the laws governing the microscopic constituents (atoms, electrons) of the systems.

16 CHAPTER 1. INTRODUCTION