Noteson STATISTICALMECHANICS - Chennai Mathematical …

1 MurthyNotes onSTATISTICAL MECHANICSA ugust 28, 2017 DRAFTIt was certainly not by design that the particles fell into orderThey did not work out what they were going to do,but because many of them by many chancesstruck one another in the course of infinite timeand encountered every possible form and movement,that they found at last the disposition they have,and that is how the universe was Lucretius Carus (94 BC - 55BC)de Rerum NaturaEverything existing in the universe is the fruit of chance and (370 BC)The moving finger writes; and, having writ,moves on : nor all your piety nor witshall lure it back to cancel half a linenor all your tears wash out a word of itOmar Khayyam(1048 - 1131)Whatever happened,happened for is happening,is happening for will happen,will happen for Gita Ludwig Boltzmann, who spent much of his life studying sta-tistical mechanics, died in 1906, by his own hand. Paul Ehrenfest,carrying on the work, died similarly in 1933.

2 Now it is our turn ..to study statistical mechanics. Perhaps it will be wise to approachthe subject rather cautiously. David Goodstein,States Matter, Dover (1975) (opening lines)All models are wrong, some are E P BoxDRAFTDRAFTC ontentsQuotes .. i1. Micro-Macro Synthesis.. Aim of Statistical Mechanics .. Micro - Macro Connections .. Boltzmann Entropy .. Boltzmann-Gibbs-Shannon Entropy .. Heat .. Work .. Helmholtz Free Energy .. Energy Fluctuations and Heat Capacity .. Micro World : Determinism andTime-Reversal Invariance .. Macro World : Thermodynamics .. Books .. Extra Reading : Books .. Extra Reading : Papers .. 132. Maxwell s Mischief.. Experiment and Outcomes.. Sample space and events .. Probabilities .. Rules of probability .. Random variable .. Maxwell s mischief : Ensemble .. Calculation of probabilities from an ensemble.

3 Construction of ensemble from probabilities .. Counting of the elements in events of the sample space :Coin tossing .. Gibbs ensemble .. Why should a Gibbs ensemble be large ? .. 223. Binomial, Poisson, and Gaussian.. Binomial Distribution .. Moment Generating Function .. Binomial Poisson .. Poisson Distribution .. Binomial Poisson `a la Feller .. Characteristic Function .. Cumulant Generating Function .. The Central Limit Theorem .. Poisson Gaussian .. Gaussian .. 384. Isolated System: Micro canonical Ensemble.. Preliminaries.. Configurational Entropy .. Ideal Gas Law : Derivation .. Boltzmann Entropy Clausius Entropy .. Some Issues on Extensitivity of Entropy .. Boltzmann Counting .. Micro canonical Ensemble .. Heaviside and his Function .. Dirac and his Function .. Area of a Circle .. Volume of anN-Dimensional Sphere.

4 Classical Counting of Micro states .. Counting of the Volume .. Density of States .. A Sphere Lives on its Outer Shell : Power Law canbe Intriguing.. Entropy of an Isolated System .. Properties of an Ideal Gas .. Temperature .. Equipartition Theorem .. Pressure.. Ideal Gas Law .. Chemical Potential .. Quantum Counting of Micro states .. Energy Eigenvalues : Integer Number of Half WaveLengths inL.. Chemical Potential : Toy Model .. 605. Closed System : Canonical Ensemble.. What is a Closed System ? .. Toy Model `a la H B Callen .. Canonical Partition Function .. Derivation `a la Balescu.. Helmholtz Free Energy .. Energy Fluctuations and Heat Capacity .. Canonical Partition Function : Ideal Gas .. Method of Most Probable Distribution .. Lagrange and his Method.. Generalisation toNVariables .. Derivation of Boltzmann Weight.

5 Mechanical and Thermal Properties.. Entropy of a Closed System.. Free Energy to Entropy .. Microscopic View : Heat and Work .. Work in Statistical Mechanics :W=XipidEi.. Heat in Statistical Mechanics :q=XiEidpi.. Adiabatic Process - a Microscopic View .. (T,V,N) for an Ideal Gas .. 836. Open System : Grand Canonical Ensemble.. What is an Open System ? .. Micro-Macro Synthesis :QandG.. Statistics of Number of Particles .. Euler and his Theorem .. : Connection to Thermodynamics .. Gibbs-Duhem Relation .. Average number of particles in an open system,hNi. ProbabilityP(N), that there areNparticles in anopen system .. Number Fluctuations .. Number Fluctuations and Isothermal Compressibility Alternate Derivation of the Relation : 2N/hNi2=kBT kT/V.. Energy Fluctuations .. 997. Quantum Statistics.. Occupation Number Representation.. Open System andQ(T,V, ).

6 Fermi-Dirac Statistics .. Bose-Einstein Statistics .. Classical Distinguishable Particles .. Maxwell-Boltzmann Statistics .. (T,V,N) QMB(T,V, ) .. (T,V, ) QMB(T,V,N) .. Thermodynamics of an open system .. Average number of particles,hNi.. Maxwell-Boltzmann Statistics .. Bose-Einstein Statistics .. Fermi-Dirac Statistics .. Study of a System with fixedNEmploying GrandCanonical Formalism .. Fermi-Dirac, Maxwell-Boltzmann and Bose-Einstein Statis-tics are the same at High Temperature and/or Low Densities Easy Method : 3 0 .. Easier Method : 0 .. Easiest Methodb (n1,n2, ) = 1 .. Mean Occupation Number .. Ideal Fermions .. Ideal Bosons .. Classical Indistinguishable Ideal Particles .. Mean Ocupation : Some Remarks.. Fermi-Dirac Statistics .. Bose-Einstein Statistics .. Maxwell-Boltzmann Statistics .. At HighTand/or Low all Statistics give the samehnki.

7 Occupation Number : Distribution and Variance .. Fermi-Dirac Statistics and Binomial Distribution .. Bose-Einstein Statistics and Geometric Distribution . Maxwell-Boltzmann Statistics and Poisson Distribution1248. Bose-Einstein Condensation.. Introduction .. Summation to Integration .. Graphical Inversion and Fugacity .. Treatment of the Singular Behaviour .. Bose-Einstein Condensation Temperature .. Grand Potential for bosons .. Energy of Bosonic System .. > TBEC.. TBEC.. Specific Heat Capacity of bosons .. > TBEC.. Third Relation :1 d dT= 32Tg3/2( )g1/2( ).. < TBEC.. Mechanism of Bose-Einstein Condensation ..1409. Elements of Phase Transition..147 DRAFTC ontents110. Statistical Mechanics of Harmonic .. Classical Harmonic Oscillators .. Helmholtz Free Energy .. Thermodynamic Properties of the Oscillator System . Quantum Harmonic Oscillator .. Specific Heat of a Crystalline Solid.

8 Einstein Theory of Specific Heat of Crystals .. Debye Theory of Specific Heat .. Riemann Zeta Function .. Bernoulli Numbers .. 167 Index..169 DRAFTDRAFT1. Micro-Macro Aim of Statistical MechanicsStatistical mechanics provides a theoretical bridge that takes you from themicro world1, to the macro world2. The chief architects of the bridge wereLudwig Eduard Boltzmann (1844 - 1906), James Clerk Maxwell(1831-1879),Josiah Willard Gibbs(1839-1903) and albert Einstein(1879-1953).Statistical Mechanics makes an attempt to derive the macroscopicprop-erties of an object from the properties of its microscopic constituents and theinteractions amongst them. It tries to provide a theoretical basisfor the em-pirical thermodynamics - in particular the emergence of the time-asymmetricSecond law of thermodynamics from the time-symmetric microscopiclaws ofclassical and quantum mechanics. When do we call something, a macroscopic object, and something a mi-croscopic constituent ?

9 The answer depends crucially on the object under study and the propertiesunder investigation. For example, if we are interested in properties like density, pressure, a bottle of water, then the molecules of water are the microscopicconstituents; the bottle of water is the macroscopic object. in a different context, an atom constitutes a macroscopic object;theelectrons, protons and neutrons form its microscopic constituents. A polymer is a macroscopic object; the monomers are its microscopicconstituents. A society is a macroscopic object; men, women, children and perhapsmonkeys are its microscopic mechanics asserts that if you know the properties of the micro-scopic constituents and how they interact with each other, then you can makepredictions about its macroscopic Newton, Schr odinger, Maxwell, and a whole lot of other scientists who devel-oped the subjects of classical mechanics, quantum mechanics, and electromag-netism2of Carnot, Clausius, Kelvin, Helmholtz and several others responsible for thebirth and growth of thermmodynamicsDRAFT41.

10 Micro-Macro Micro - Macro Boltzmann EntropyThe first and the most important link, between the micro(scopic) and themacro(scopic) worlds is given by,S=kBlnb (E,V,N).( )It was proposed by for entropy and belongs to themacro world described by is the number of micro statesof a macroscopic the Boltzmann constant5that establishescorrespondence of the statistical entropy of Boltzmann to the thermodynamicentropy of Boltzmann-Gibbs-Shannon EntropyMore generally we have the Boltzmann-Gibbs-Shannon entropy given by,S= kBXipilnpi.( )The micro states are labelled byiand the sum is taken over all the microstates of the macroscopic system; The probability is denoted interesting issue : Normally we resort to the use of probability only whenwe have inadequate data about the system or incomplete knowledge of the phe-nomenon under study. Thus, such an enterprise is sort of tiedto our track of the positions and momenta of some 1030moleculesviaNewton sequations is undoubtedly an impossible task.