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Lecture 10—Ideas of Statistical Mechanics Chapter 4, Wednesday January 30 th. Finish Ch. 3 - Statistical distributions Statistical mechanics - ideas and definitions Quantum states, classical probability, ensembles, macrostates... Entropy Definition of a quantum state.

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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 distributions

ni

xi

16

Mean:

Statistical distributions

ni

xi

16

Mean:

Statistical distributions

ni

xi

16

Mean:

Statistical distributions

ni

xi

16

Standard

deviation

Statistical distributions

64

Gaussian distribution

(Bell curve)

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

Statistical 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.

Statistical Mechanics – ideas and definitions

64

An example:

Coin toss again!!

width

Ensembles and quantum states (microstates)

10 particles, 36 cells

Volume V

Cell volume, DV

Ensembles and quantum states (microstates)

10 particles, 36 cells

Volume V

Cell volume, DV

Ensembles and quantum states (microstates)

10 particles, 36 cells

Volume V

Cell volume, DV

Ensembles and quantum states (microstates)

10 particles, 36 cells

Volume V

Cell volume, DV

Ensembles and quantum states (microstates)

10 particles, 36 cells

Volume V

Cell volume, DV

Ensembles and quantum states (microstates)

10 particles, 36 cells

Volume V

Cell volume, DV

Ensembles and quantum states (microstates)

10 particles, 36 cells

Volume V

Cell volume, DV

Ensembles and quantum states (microstates)

Cell volume, DV

10 particles, 36 cells

Volume V

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.

Entropy

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