Introduction Overview of Statistical & Thermal Physics Basic Definitions & Terminology Thermodynamics (“Thermo”): The study of the Macroscopic properties 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 makes NO direct 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 by Cp– Cv = R. But, thermo gives no prescription for calculating numerical values for Cp, Cv. Calculating these requires a microscopic model.

Kinetic Theory: A microscopic theory! Applies the Laws of Mechanics (Classical or Quantum) to a microscopic model of the individual molecules of a system. Allows the calculation of various Macroscopically measurable quantities on the basis of a Microscopic theory 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.

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).

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

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

Statistical Mechanics: Preliminary Remarks Where 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

Partition Function E = E(r1,r2,r3,…rN,p1,p2,p3,…pN) 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!

P(E) ≡ e(-E/kT)/Z CLASSICAL 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(E) ≡ e(-E/kT)/Z <A>≡ ∫∫∫(A)d3r1d3r2…d3rN d3p1d3p2…d3pNP(E) (Another 6N Dimensional Integral!) Don’t panic! We’ll derive this later!

Statistical Mechanics: 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

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

QUANTUM Statistical Mechanics: <A> ≡ ∑n <n|A|n>P(En) 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!

“P(E) & Z are the summit of both 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 the Partition Function Z (similar in classical & quantum cases). Quoting Richard P. Feynman: “P(E) & Z are the summit of both Classical & Quantum Statistical Mechanics.”

Statistical Mechanics (Classical or Quantum) P(E), Z Statistical Mechanics (Classical or Quantum) / \ / \ / \ / ________ \ The entire subject is either the “climb” UP to the summit (calculation of P(E), Z) or the slide DOWN (the use of P(E), Z to calculate measurable properties). On the way UP: We’ll rigorously define Thermal Equilibrium & Temperature. On the way DOWN, we’ll derive all of Thermodynamics beginning with microscopic theory. Calculation of Measurable Quantities   Equations of Motion