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Introduction Overview of Statistical & Thermal Physics

Introduction Overview of Statistical & Thermal Physics. Basic Definitions & Terminology

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Introduction Overview of Statistical & Thermal Physics

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  1. IntroductionOverview of Statistical & Thermal Physics Basic Definitions & Terminology • Thermodynamics (“Thermo”):The study of the Macroscopicproperties of systems based on a few laws & hypotheses. (The Laws of Thermodynamics!).Thermo derives relations between the macroscopic properties(& parameters)of a system(heat capacity, temperature, volume, pressure, etc.).Thermo makesNOdirect reference to the microscopic structure of matter. For example, from thermo, we’ll derive later that,for an ideal gas,the heat capacities are related byCp– Cv = R.But, thermo gives no prescription for calculating numerical values for Cp, Cv. Calculating these requires a microscopic model.

  2. Kinetic Theory:A microscopic theory! Applies theLaws of Mechanics(Classical or Quantum)to a microscopic model of the individual molecules of a system. • Allows the calculation of variousMacroscopicallymeasurable quantities on the basis of aMicroscopictheory applied to a model of the system. • For example, it might be able to calculate the specific heat Cv using Newton’s 2nd Law along with the known force laws between the particles. • Uses the equations of motion for individual particles.

  3. Statistical Mechanics (or Statistical Thermodynamics) Ignores a detailed consideration of molecules as individuals. A Microscopic, statistical approach to the calculation of Macroscopic quantities. • Applies the methods of Probability & Statistics to Macroscopic systems with HUGE numbers of particles. • For systems with known energy (Classical or Quantum) it gives BOTH 1. Relations between Macroscopic quantities (like Thermo) AND 2.NUMERICAL VALUES of them (like Kinetic Theory).

  4. This course covers all three!: Thermodynamics Kinetic Theory Statistical Mechanics Statistical Mechanics: Reproduces ALLof Thermodynamics & ALL of Kinetic Theory. More general than either!

  5. Statistical Mechanics (the most general theory) ___________|__________ || | | | | ThermodynamicsKinetic Theory (a general, macroscopic theory)(a microscopic theory, most easily applicable to gases)

  6. Preliminary RemarksWhere we are going, a general survey. Don’t worry about details yet! The Key Principle of CLASSICAL Statistical Mechanics: • Consider a system containing N particles with 3d positions r1,r2,r3,…rN, & momenta p1,p2,p3,…pN. The system is in Thermal Equilibrium at absolute temperature T. We’ll show that the probability of the system having energy E is: P(E) ≡ e(-E/kT)/Z Z ≡ “Partition Function”, T ≡ Absolute Temperature k ≡ Boltzmann’s Constant

  7. Partition Function Z ≡ ∫∫∫d3r1d3r2…d3rN d3p1d3p2…d3pN e(-E/kT) (A 6N Dimensional Integral!) E = E(r1,r2,r3,…rN,p1,p2,p3,…pN) Don’t panic! We’ll derive this later!

  8. CLASSICALStatistical Mechanics: • Let A ≡any measurable, macroscopic quantity. The thermodynamic average of A ≡<A>. This is what is measured. Use probability theory to calculate <A>. P(E) ≡ e(-E/kT)/Z <A>≡ ∫∫∫(A)d3r1d3r2…d3rN d3p1d3p2…d3pNP(E) (Another 6N Dimensional Integral!) Don’t panic! We’ll derive this later!

  9. The Key Principle of QUANTUM Statistical Mechanics: • Consider a system which can be in any one of N quantum states. The system is in Thermal Equilibrium at absolute temperature T. We’ll show that the probability of the system being in state n with energy En is: P(En) ≡ exp(-En/kT)/Z Z ≡ “Partition Function”, T ≡ Absolute Temperature k ≡ Boltzmann’s Constant

  10. Partition Function Z ≡ ∑nexp(-En/kT) Don’t panic! We’ll derive this later!

  11. QUANTUM Statistical Mechanics: • Let A ≡any measurable, macroscopic quantity. The thermodynamic average of A ≡<A>. This is what is measured. Use probability theory to calculate <A>. P(En) ≡ exp(-En/kT)/Z <A> ≡ ∑n<n|A|n>P(En) <n|A|n> ≡ Quantum Mechanical expectation value of A in quantum state n. Don’t panic! We’ll derive this later!

  12. Question:What’s the point of showing this now? • Classical & Quantum Statistical Mechanics both revolve around the calculation of P(E) or P(En). • To calculate the probability distribution, we need to calculate thePartition FunctionZ(similar in classical & quantum cases). Quoting Richard P. Feynman: “P(E)&Zare the summit of both Classical&Quantum Statistical Mechanics.”

  13. Calculation of Measurable Quantities  P(E), Z Statistical Mechanics(Classical or Quantum) /\ /\ / \ /________\ The entire subject is either the “climb”UPto the summit(calculation of P(E), Z)or the slideDOWN(the use of P(E), Z to calculate measurable properties).On the wayUP: We’ll rigorously defineThermal Equilibrium&Temperature. On the wayDOWN, we’ll derive all ofThermodynamicsbeginning with microscopic theory.  Equations of Motion

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