lecture 10—ideas of statistical mechanics chapter 4, wednesday january 30 th finish ch. 3 -...

Lecture 10—Ideas of Statistical Lecture 10—Ideas of Statistical Mechanics Chapter 4, Mechanics Chapter 4, Wednesday Wednesday

January 30January 30thth•Finish Ch. 3 - Statistical distributions•Statistical mechanics - ideas and definitions

•Quantum states, classical probability, ensembles, macrostates...

•Entropy•Definition of a quantum state

Reading: Reading: All of chapter 4 (pages 67 - 88)All of chapter 4 (pages 67 - 88)***Homework 3 due Fri. Feb. 1st*******Homework 3 due Fri. Feb. 1st****Assigned problems, Assigned problems, Ch. 3Ch. 3: 8, 10, 16, 18, : 8, 10, 16, 18,

2020Homework 4 due next Thu. Feb. 7thHomework 4 due next Thu. Feb. 7thAssigned problems, Assigned problems, Ch. 4Ch. 4: 2, 8, 10, 12, : 2, 8, 10, 12,

1414Exam 1: Exam 1: Fri. Feb. 8th (in class), chapters 1-4Fri. Feb. 8th (in class), chapters 1-4

Statistical distributionsStatistical distributions

ni

xi

16

, wherei iiii

n xx N n

N Mean:

Statistical distributionsStatistical distributions

ni

xi

16

, wherei iiii

n xx N n

N Mean:

Statistical distributionsStatistical distributions

ni

xi

16

, where lim ii i ii N

nx p x pN

Mean:

N

Statistical distributionsStatistical distributions

ni

xi

16

2 2i ii

x p x x Standard deviation

2

2

1( ) exp22x x

p x

Statistical distributionsStatistical distributions

Gaussian distribution(Bell curve)

64

Statistical Mechanics (Chapter 4)Statistical Mechanics (Chapter 4)•What is the physical basis for the 2nd law?What is the physical basis for the 2nd law?•What is the microscopic basis for entropy?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 + MechanicsStatistics + Mechanics

Statistical MechanicsStatistical Mechanics

Thermal PropertiesThermal Properties

Statistical MechanicsStatistical Mechanics•Use classical probability to make predictions.Use classical probability to make predictions.•Use statistical probability to test 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 on earth is this possible?

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

REQUIRES AN IMMENSE LEAP OF FAITHREQUIRES AN IMMENSE LEAP OF FAITH

Statistical Mechanics – ideas and Statistical Mechanics – ideas and definitionsdefinitionsA quantum state, or microstateA quantum state, or microstate

•A unique configuration.A unique configuration.•To know that it is unique, we must specify it To know that it is unique, we must specify it

as completely as possible...as completely as possible...e.g. Determine:e.g. Determine: PositionPosition

MomentumMomentumEnergyEnergySpinSpin

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

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

THIS IS ACTUALLY IMPOSSIBLE FOR ANY REAL SYSTEMTHIS IS ACTUALLY IMPOSSIBLE FOR ANY REAL SYSTEM

Statistical Mechanics – ideas and Statistical Mechanics – ideas and definitionsdefinitionsA quantum state, or microstateA quantum state, or microstate

•A unique configuration.A unique configuration.•To know that it is unique, we must specify it To know that it is unique, we must specify it

as completely as possible...as completely as possible...Classical probabilityClassical probability

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

probability.probability.An ensembleAn ensemble•A collection of separate systems prepared in A collection of separate systems prepared in

precisely the same way.precisely the same way.

Statistical Mechanics – ideas and Statistical Mechanics – ideas and definitionsdefinitionsThe microcanonical ensemble:The microcanonical ensemble:

Each system has same:Each system has same: # of particles# of particlesTotal energyTotal energyVolumeVolumeShapeShapeMagnetic fieldMagnetic fieldElectric fieldElectric field

and so on....and so on....

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

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

64Statistical Mechanics – ideas and Statistical Mechanics – ideas and

definitionsdefinitions

An example:An example: Coin toss again!!

width

Ensembles and quantum states Ensembles and quantum states (microstates)(microstates)

Cell volume, Cell volume, VV

Volume Volume VV 10 particles, 36 cells10 particles, 36 cells

10

16

136

3 10

ip

Ensembles and quantum states Ensembles and quantum states (microstates)(microstates)

Cell volume, Cell volume, VV

Volume Volume VV 10 particles, 36 cells10 particles, 36 cells

10

16

136

3 10

ip

Ensembles and quantum states Ensembles and quantum states (microstates)(microstates)

Cell volume, Cell volume, VV

Volume Volume VV 10 particles, 36 cells10 particles, 36 cells

10

16

136

3 10

ip

Ensembles and quantum states Ensembles and quantum states (microstates)(microstates)

Cell volume, Cell volume, VV

Volume Volume VV 10 particles, 36 cells10 particles, 36 cells

10

16

136

3 10

ip

Ensembles and quantum states Ensembles and quantum states (microstates)(microstates)

Cell volume, Cell volume, VV

Volume Volume VV 10 particles, 36 cells10 particles, 36 cells

10

16

136

3 10

ip

Ensembles and quantum states Ensembles and quantum states (microstates)(microstates)

Cell volume, Cell volume, VV

Volume Volume VV 10 particles, 36 cells10 particles, 36 cells

10

16

136

3 10

ip

Ensembles and quantum states Ensembles and quantum states (microstates)(microstates)

Cell volume, Cell volume, VV

Volume Volume VV 10 particles, 36 cells10 particles, 36 cells

10

16

136

3 10

ip

Ensembles and quantum states Ensembles and quantum states (microstates)(microstates)

Cell volume, Cell volume, VV

Volume Volume VV 10 particles, 36 cells10 particles, 36 cells

10

16

136

3 10

ip

Ensembles and quantum states Ensembles and quantum states (microstates)(microstates)

Cell volume, Cell volume, VV

10

16

136

3 10

ip

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

Volume Volume VV

There’s a major flaw in this calculation.There’s a major flaw in this calculation.Can anyone see it?Can anyone see it?It turns out that we get away with it.It turns out that we get away with it.

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

1 np S f W WW

lnBS k W