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Lecture 10—Ideas of Statistical Mechanics Chapter 4, Wednesday January 30 th

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Lecture 10—Ideas of Statistical Mechanics Chapter 4, Wednesday January 30th

- Finish Ch. 3 - Statistical distributions
- Statistical mechanics - ideas and definitions
- Quantum states, classical probability, ensembles, macrostates...
- Entropy
- Definition of a quantum state

Reading: All of chapter 4 (pages 67 - 88)

***Homework 3 due Fri. Feb. 1st****

Assigned problems, Ch. 3: 8, 10, 16, 18, 20

Homework 4 due next Thu. Feb. 7th

Assigned problems, Ch. 4: 2, 8, 10, 12, 14

Exam 1: Fri. Feb. 8th (in class), chapters 1-4

Statistical Mechanics (Chapter 4)

Statistical Mechanics

Thermal Properties

- What is the physical basis for the 2nd law?
- What is the microscopic basis for entropy?

Boltzmann hypothesis: the entropy of a system is related to the probability of its state; the basis of entropy is statistical.

Statistics + Mechanics

- Use classical probability to make predictions.
- Use statistical probability to test predictions.

Note: statistical probability has no basis if a system is out of equilibrium (repeat tests, get different results).

How on earth is this possible?

- How do we define simple events?
- How do we count them?
- How can we be sure they have equal probabilities?

REQUIRES AN IMMENSE LEAP OF FAITH

Statistical Mechanics – ideas and definitions

e.g. Determine: Position

Momentum

Energy

Spin

of every particle, all at once!!!!!

............

A quantum state, or microstate

- A unique configuration.
- To know that it is unique, we must specify it as completely as possible...

THIS IS ACTUALLY IMPOSSIBLE FOR ANY REAL SYSTEM

Statistical Mechanics – ideas and definitions

An ensemble

- A collection of separate systems prepared in precisely the same way.

A quantum state, or microstate

- A unique configuration.
- To know that it is unique, we must specify it as completely as possible...

Classical probability

- Cannot use statistical probability.
- Thus, we are forced to use classical probability.

Statistical Mechanics – ideas and definitions

The microcanonical ensemble:

Each system has same: # of particles

Total energy

Volume

Shape

Magnetic field

Electric field

and so on....

............

These variables (parameters) specify the ‘macrostate’ of the ensemble. A macrostate is specified by ‘an equation of state’. Many, many different microstates might correspond to the same macrostate.

Ensembles and quantum states (microstates)

Cell volume, DV

Many more states look like this, but no more probable than the last one

Volume V

There’s a major flaw in this calculation.

Can anyone see it?

It turns out that we get away with it.

Boltzmann hypothesis: the entropy of a system is related to the probability of its being in a state.

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