Lecture Notes, Statistical Mechanics (Theory F)

1 Lecture notes , Statistical Mechanics (Theory F)J org SchmalianApril 14, 20142 Institute for Theory of Condensed Matter (TKM) Karlsruhe Institute ofTechnologySummer Semester, 2014 Contents1 Introduction72 Equilibrium and the laws of thermodynamics .. Thermodynamic potentials .. of a Legendre transformation .. Gibbs Duhem relation .. 183 Summary of probability theory214 Equilibrium Statistical The maximum entropy principle .. The canonical ensemble .. within an external field (paramagnetism) harmonic oscillator.

2 Heat capacity and Maxwell relations .. The micro-canonical ensemble .. harmonic oscillator .. 365 Ideal Classical ideal gases .. non-relativistic classical ideal gas .. classical ideal gas .. ultra-relativistic classical ideal gas .. theorem .. quantum gases .. number representation .. canonical ensemble .. function of ideal quantum gases .. limit .. of the ideal fermi gas .. ideal Bose gas .. in equilibrium .. model for hadrons and the quark-gluon plasma.

3 Fermi gas .. 646 Interacting systems and phase The classical real gas .. Classification of Phase Transitions .. Gibbs phase rule and first order transitions .. The Ising model .. solution of the one dimensional model .. field approximation .. Landau theory of phase transitions .. Mechanics motivation of the Landautheory: .. criterion .. Scaling laws .. Renormalization group .. theory .. and slow variables .. behavior of the correlation function.

4 -expansion of the 4-theory .. interactions .. 977 Density matrix and fluctuation dissipation Density matrix of subsystems .. Linear response and fluctuation dissipation theorem .. 1038 Brownian motion and stochastic Langevin equation .. Random electrical circuits .. 1079 Boltzmann transport Transport coefficients .. Boltzmann equation for weakly interacting fermions .. integral for scattering on impurities .. time approximation .. the transition rates .. relaxation time.

5 Equilibrium, Chapman-Enskog Expansion .. 120 PrefaceThese Lecture notes summarize the main content of the course Statistical Me-chanics (Theory F), taught at the Karlsruhe Institute of Technology duringthe summer semester 2012 and 2014. They are based on the graduate courseStatistical Mechanics taught at Iowa State University between 2003 and 1 IntroductionMany-particle systems are characterized by a huge number of degrees of free-dom. However, in essentially all cases a complete knowledge of all quantum orclassical states is neither possible nor useful and necessary.

6 For example, it ishard to determine the initial coordinates and velocities of 1023Ar-atoms in ahigh-temperature gas state, needed to integrate Newton s equations. In addi-tion, it is known from the investigation of classical chaos that in classical systemswith many degrees of freedom the slightest change ( lack of knowledge) inthe initial conditions usually causes dramatic changes in the long time behaviorof the individual particles are concerned. On the other hand the macroscopicproperties of container of gas or a bucket of water are fairly generic and don tseem to depend on how the individual particles have been initialized.

7 This in-teresting observation clearly suggests that there are principles at work, ensuringthat only a few variables are needed to characterize the macroscopic propertiesof a macroscopic system. It is obviously worthwhile trying to identify theseprinciples instead of going through the effort to identify all particle momentaand tools and insights of Statistical Mechanics enable us to determine themacroscopic properties of many particle systems with known microscopic Hamil-tonian, albeit in many cases only approximately. This bridge between the micro-scopic and macroscopic world is based on the concept of a lack of knowledge in the precise characterization of the system and therefore has a probabilisticaspect.

8 This is indeed a lack of knowledge which, different from the probabilisticaspects of quantum Mechanics , could be fixed if one we were only able to fullycharacterize and solve the many particle problem. For finite but large systemsthis is an extraordinary tough problem. It becomes truly impossible in the limitof infinitely many particles. It is this limit of large systems where statisticalmechanics is extremely powerful. One way to see that the lack of knowledge problem is indeed more fundamental than solely laziness of the physicist is thatessentially every physical system is embedded in an environment.

9 Only completeknowledge of systemandenvironment allows for a complete the observable part of our universe seems to behave this way, denying us78 CHAPTER 1. INTRODUCTION full knowledge of any given system as a matter of 2 ThermodynamicsEven though this course is about Statistical Mechanics , it is useful to summarizesome of the key aspects of thermodynamics. Clearly these comments cannotreplace a course on thermodynamics itself. Thermodynamics and statisticalmechanics have a relationship which is quite special. It is well known that clas-sical Mechanics covers a set of problems which are a subset of the ones coveredby quantum Mechanics .

10 Even more clearly is non-relativistic Mechanics a partof relativistic Mechanics . Such a statement cannot be made if one tries to re-late thermodynamics and Statistical Mechanics . Thermodynamics makes verygeneral statements about equilibrium states. The observation that a system inthermodynamic equilibrium does not depend on its preparation in the past forexample is being beautifully formalized in terms of exact and inexact differ-entials. However, it also covers the energy balance and efficiency of processeswhich can be reversible or irreversible.