Statistical Physics

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• Publié par: mtoledan
• Publié le: 30 septembre 2011
• Nombre de lectures: 36
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Statistical Physics Yuri Galperin and Jens Feder FYS 203 The latest version at www.uio.no/ yurigContents 1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Deterministic Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Atomistic Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3.1 Cooperative Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.5 What will we do? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Historical Background 7 2.1 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Steam Engines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Sadi Carnot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 The Maxwell’s Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3 Principles of statistical mechanics 21 3.1 Microscopic states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 Statistical treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.3 Equal weight, . . . microcanonical ensemble . . . . . . . . . . . . . . . . . . . . . 24 3.4 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.5 Number of states and density of states . . . . . . . . . . . . . . . . . . . . . . . 27 3.6 Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.7 Normal systems in statistical thermodynamics . . . . . . . . . . . . . . . . . . . 30 4 Thermodynamic quantities 31 4.1 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2 Adiabatic process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.3 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.4 Equations of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.5 Work and quantity of heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.6 Thermodynamics potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.6.1 The heat function (enthalpy) . . . . . . . . . . . . . . . . . . . . . . . . 36 4.6.2 Free energy and thermodynamic potential . . . . . . . . . . . . . . . . . 37 4.6.3 Dependence on the number of particles . . . . . . . . . . . . . . . . . . 38 iii CONTENTS 4.6.4 Derivatives of thermodynamic functions . . . . . . . . . . . . . . . . . . 39 4.6.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.7 Principles of thermodynamic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.7.1 Maximum work and Carnot cycle . . . . . . . . . . . . . . . . . . . . . 42 4.7.2 Thermodynamic inequalities. Thermodynamic stability . . . . . . . . . . 45 4.7.3 Nernst’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.7.4 On phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5 The Gibbs Distribution 49 15.1 Spin in a magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 5.2 The Gibbs distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.2.1 The Gibbs distribution for a variable number of particles . . . . . . . . . 53 6 Ideal gas 55 6.1 Classical gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 6.2 Ideal gases out of equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 6.3 Fermi and Bose gases of elementary particles . . . . . . . . . . . . . . . . . . . 65 6.4 Black body radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 6.5 Lattice Vibrations. Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 7 Statistical ensembles 81 7.1 Microcanonical ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 7.2 Canonical ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 7.3 Ensembles in quantum statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 87 8 Fluctuations 91 8.1 The Gaussian distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 8.2 Fluctuations of thermodynamic quantities . . . . . . . . . . . . . . . . . . . . . 93 8.3 Correlation of fluctuations in time . . . . . . . . . . . . . . . . . . . . . . . . . 95 8.4 Fluctuation dissipation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 101 8.4.1 Classical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 8.4.2 Quantum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 8.4.3 The generalized susceptibility . . . . . . . . . . . . . . . . . . . . . . . 104 8.4.4 Fluctuation dissipation theorem: Proof . . . . . . . . . . . . . . . . . . 107 9 Stochastic Processes 111 9.1 Random walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 9.1.1 Relation to diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 9.2 Random pulses and shot noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 9.3 Markov processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 9.4 Discrete Markov processes. Master equation . . . . . . . . . . . . . . . . . . . . 117 9.5 Continuous Markov Fokker Planck equation . . . . . . . . . . . . . . 119 9.6 Fluctuations in a vacuum tube generator . . . . . . . . . . . . . . . . . . . . . . 121CONTENTS iii 10 Non Ideal Gases 129 10.1 Deviation of gases from the ideal state . . . . . . . . . . . . . . . . . . . . . . . 129 11 Phase Equilibrium 135 11.1 Conditions for phase equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 135 12 Continuous Phase Transitions 139 12.1 Landau Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 12.2 Scaling laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 12.3 Landau Ginzburg theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 12.4 Ginzburg Landau of superconductivity (SC) . . . . . . . . . . . . . . . . 153 12.5 Mean field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 12.6 Spatially homogeneous ferromagnets . . . . . . . . . . . . . . . . . . . . . . . . 160 12.7 Renormalization framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 13 Transport phenomena 167 13.1 Classical transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 13.2 Ballistic transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 A Notations 175 B On the calculation of derivatives from the equation of state 179iv CONTENTSChapter 1 Introduction 1.1 Motivation Statistical physics is an unfinished and highly active part of physics. We do not have the the oretical framework in which to even attempt to describe highly irreversible processes such as fracture. Many types of nonlinear systems that lead to complicated pattern formation processes, the properties of granular media, earthquakes, friction and many other systems are beyond our present understanding and theoretical tools. In the study of these and other macroscopic systems we build on the established conceptual framework of thermodynamics and statistical physics. Thermodynamics, put into its formal and phenomenological form by Clausius, is a theory of impressive range of validity. Thermody namics describes all systems form classical gases and liquids, through quantum systems such as superconductors and nuclear matter, to black holes and elementary particles in the early universe 1in exactly the same form as they were originally formulated. Statistical physics, on the other hand gives a rational understanding of Thermodynamics in terms of microscopic particles and their interactions. Statistical physics allows not only the cal culation of the temperature dependence of thermodynamics quantities, such as the specific heats of solids for instance, but also of transport properties, the conduction of heat and electricity for example. Moreover, statistical physics in its modern form has given us a complete understand ing of second order phase transitions, and with Wilson’sRenormalization Group theory we may calculate the scaling exponents observed in experiments on phase transitions. However, the field of statistical physics is in a phase of rapid change. New ideas and concepts permit a fresh approach to old problems. With new concepts we look for features ignored in previous experiments and find exciting results. Key words are deterministic chaos, fractals, self organized criticality (SOC), turbulence and intermitency. These words represent large fields of study which have changed how we view Nature. Disordered systems, percolation theory and fractals find applications not only in physics and engineering but also in economics and other social sciences. What are the processes that describe the evolution of complex systems? Again we have an 1From the introduction of Ravndal’s lecture notes on Statistical Physics 1w 2 CHAPTER 1. INTRODUCTION active field with stochastic processes as a key word. So what has been going on? Why have the mysteries and excitement of statistical physics not been introduced to you before? The point is that statistical physics is unfinished, subtle, intellectually and mathematically demanding. The activity in statistical physics has philosophical implications that have not been discussed by professional philosophers—and have had little influence on the thinking of the general public. DETERMINISTIC behavior rules the day. Privately, on the public arena, in industry, man agement and in teaching we almost invariably assume that any given action leads predictably 2to the desired result—and we are surprised an