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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 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 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

slide5

Statistical distributions

ni

xi

16

Standard

deviation

slide6

Statistical distributions

64

Gaussian distribution

(Bell curve)

slide7

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

slide8

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

slide9

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

slide10

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.
slide11

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.

slide12

Statistical Mechanics – ideas and definitions

64

An example:

Coin toss again!!

width

slide13

Ensembles and quantum states (microstates)

10 particles, 36 cells

Volume V

Cell volume, DV

slide14

Ensembles and quantum states (microstates)

10 particles, 36 cells

Volume V

Cell volume, DV

slide15

Ensembles and quantum states (microstates)

10 particles, 36 cells

Volume V

Cell volume, DV

slide16

Ensembles and quantum states (microstates)

10 particles, 36 cells

Volume V

Cell volume, DV

slide17

Ensembles and quantum states (microstates)

10 particles, 36 cells

Volume V

Cell volume, DV

slide18

Ensembles and quantum states (microstates)

10 particles, 36 cells

Volume V

Cell volume, DV

slide19

Ensembles and quantum states (microstates)

10 particles, 36 cells

Volume V

Cell volume, DV

slide20

Ensembles and quantum states (microstates)

Cell volume, DV

10 particles, 36 cells

Volume V

slide21

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.

slide22

Entropy

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

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