mon. 1/20 2.5 hw3,4,5 wed. 1/22 hw7: fri. 1/23 (c 10.3.1)...

Phys 344 Lecture 5 Jan. 16th

2009 1

Fri. 1/16 2.4, B.2,3 More Probabilities HW5: 13, 16, 18, 21; B.8,11

Mon. 1/20

Wed. 1/22

Fri. 1/23

2.5 Ideal Gas

(C 10.3.1) 2.6 Entropy & 2nd Law

(C 10.3.1) 2.6 Entropy & 2nd Law (more)

HW6: 26

HW7: 29, 32, 38

HW3,4,5

10_wells_oscillator.py & helix.py (note: helix must be lowercase)

BallSpring.mov

Statmech.exe

2. The Second Law

Motivation / transition

o Combinatorics

2.1 Two-State Systems

Microstate = state of the system in terms of microscopic details.

Macrostate = state of the system in terms of macroscopic variables

Multiplicity = : How many Microstates are consistent with a given Macrostate.

Fundamental assumption of statistical mechanics: In an isolated system in internal

thermal equilibrium, all accessible microstates are equally probable.

Probability:

N Fair Coins

o n

N

nNn

NnN

!!

!,

2.1.1 The Two-State Paramagnet

o !!

!,

NNN

N

N

NNN

2.2 The Einstein Model of a Solid

o Demo. BallSpring.mov

o Demo. 10_wells_oscillator.py

o !!1

!1,

qN

qNqN

2.3 Thermal equilibrium of two blocks

To address thermodynamic equilibrium, we need a way of describing two,

interacting objects. We’ll take two Einstein Solids. We’ll begin simple, with

each “solid” simply being an atom, i.e. 3 oscillators.

Two single-atom blocks

o We’re going to consider these two sharing a total of 4 quanta of energy,

so, at any given instant, one of the atoms may have all 4, 3, 2, 1, or none

of the quanta. So we’re going to need the…

Solid A

UA

qA

NA

Solid B

UB

qB

NB

Didn’t get through 2.3 last time, so doing it now

At the End, reassess what homework will be due Monday.


2009 2

o Multiplicities for one atom-solid with 4,3,2,1,0 quanta.

First, picking up where we left off last time, let’s consider the

multiplicities for such a single atom (3-oscillators) given 4, 3, 2,1,

or 0 quanta of energy.

!1!

!1,1

Nq

NqqN quantaosc .

Multiplicity for N=3, q = 4 15)4,13(

Demo:10_wells_oscillator.py . See some of the different ways that 4 quanta can

be distributed among the 3 oscillators of an atom.

Multiplicity for N = 3, q = 3 10)3,13(




o Note: This last result makes perfect sense – there is

only one way for no energy to be added to the

system. In our equation we encounter 0! At first

blush, you may guess that that is 0. But what works

with our intuition for our system is that 0!=1.

o Multiplicity table for 2 solids of 3 oscillators and 4 quanta

Ok. Now say we take two such “solids” and place them in

“thermal” contact, i.e., allow that 4 quanta of energy can flow

between the two.

Macrostate: how much energy each ‘solid’ has, q1 and q2.

Microstate: how that energy is distributed among the individual

oscillators.

Let’s tabulate the possibilities and the corresponding multiplicities.

Demo: PowerPoint Visuals of Multiplicitie

Constraints: Qtot = 4 NA = 3, NB = 3

Possibilities & Multiplicities

qA qB A(NA-1,qA) B(NB-1,qB) A&B = A× B

0 4 1 15 15

1 3 3 10 30

2 2 6 6 36

3 1 10 3 30

4 0 15 1 15

Total possible microstates: 126

o Multiplicities multiply

Ex. If I have 10 shirts and 10 pants, then there would be 100

possible outfits (all be it, some are really atrocious).

So, if there are 6 ways of arranging 2 quanta in solid A and 6 ways

of arranging 2 quanta in solid B, then there are a total of 6×6=36

ways of doing both.

o Multiplicity Plot


2009 3

Noteworthy: The whole system’s multiplicity is maximized when

the multiplicities of the two solid’s are balanced, not individually

maximized.

o Multiplicities and Probabilities

If the microstates that we have defined are indeed each equally

probable, then the probability of a macrostate is proportional to the

number of microstates compatible with it.

So, the macrostate with the greatest multiplicity is the most

probable and the macrostate with the smallest multiplicity is the

least probable.

Ex. 36 out of 126 possible microstates correspond to an equal split

of energy among equal “solids,” or 36/126 = 29% of the time or a

probability of 0.29 that when I look at the system I find the energy

evenly split.

o Peak width. Though there clearly is a peak, and it’s where we would

intuitively imagine it to be, it isn’t very sharp – only 29% of the time is the

energy evenly split. 15% of the time, atom 1 has all the energy. This is a

fairly broad peak.

o Equilibrium & Probability. Now, recall, we said that two objects in

thermal contact exchange energy until they achieve thermal equilibrium.

Looking at this system, energy can be swapped back and forth and the

system will generally tend toward the most probable state, though with

only a 29% chance, it is far form inevitable that the system will be there.

We can identify the most probable state with that of equilibrium.

0

10

20

30

40

0

4

1

3

2

2

3

1

4

0

=qA

=qB

A

B

A&B


2009 4

You will be asked in the homework to use Excel to tabulate the possibilities for

the same system with 6 quanta of energy. While it could be done by hand, later

we’ll be considering much larger systems, and those, you don’t want to do by

hand. Here’s a start.

At a given instant our total system looks like this

Set up for Problems 9 and 10: Two 3-oscillator solids share 6 quanta. Model with

Excell.

What math is executed in cell B4?

)!1(!

)!1(1,

AA

AA

A

AA

AANq

Nq

q

NqqN

where NA is 3 (found in cell B2) and qA is 0 (found in cell A4).

What must be the code for cell B4?

=COMBIN($B$2-1+A4,A4)

What math is executed in cell C4?

Calculates the number of ways of distributing 0 energy units in body A AND 6 in body

B. It should be the product of the number of ways of doing each individually:

BBAA qNqN ,,

What must be the code for cell C4?

=B4*D4

A B C D E F 1 Two Einstein Solids 2 NA= 3 NB= 3 q_total= 6 3 qA A qB B Total 4 0 1 6 28 28 5 1 3 5 21 63 6 2 6 4 15 90 7 3 10 3 10 100 8 4 15 2 6 90 9 5 21 1 3 63 10 6 28 0 1 28

total possible = 462=6

166

qA to

tal


2009 5

o Average / Distribution for large sets

Increasing the numbers of members and quanta by two orders of magnitude

has a significant effect on the sharpness of the peak.

V. Lab 5.2 Two Einstein Solids. In StatMech.exe, start with Na = 1 (one atom, thus three

oscillators), Nb=1, and q=6. See the same table, same plot.

Increase particles & offset (Na>Nb) and quantum # to Na 300, Nb 200,q 100. See a

much stronger peak

o Irreversibility

Scenario

Imagine the following, in a system of 300 members of A, 200

members of B, and 100 energy quanta, the most probable state (60

quanta in A) is 1033

times more probable than is the least probable

state (no energy in solid A). Say then that you started with the system

in this most probable state, then checked up on it periodically hoping

to find no more than 10 quanta in A – almost all the energy

spontaneously shifted to B. Collectively, these states have a

probability around 10-20

, or you’d have to check about 1020

times, or

100 times a second for the lifetime of the universe, to stand a good

chance of finding it in such a state. Then again, if you started with the

system in such a state – say B got heated, then touched A, it wouldn’t

be long before the energy redistributed itself, and though only 7% of

the time you’d find it split 60 – 40, the vast majority of the time you’d

find it near this.

Conclusion

It is extremely likely that a system will progress from any initial state

to the vicinity of its most probable state, but it is prohibitively

improbable that it will progress away – the approach of equilibrium is

‘irreversible.’

V. Lab 4.1 Conduction(equilib.exe)– just play around & see how the probabilities play

out. Note, that on a short time scale, not all distributions are equally probable, owing to

the time it takes for energy to be conducted from one local to the next. Also note that the

energy flows into the macrostate with the greatest number of microstates: evenly

distributed.

Heating is a probabilistic phenomenon. Energy flows from hotter to

cooler until equilibrium is reached because equilibrium is the most

probable state, not because any specific mechanism drives or requires

it.

This observation is of fundamental importance in thermodynamics. It is the

physics content of what’s known as the Second Law of Thermodynamics.

This law actually gets phrased many different ways, depending on the

application, but the underlying content is always the same.

Second Law of Thermodynamics: the spontaneous flow of energy stops when a system is

very near its most likely macrostate, that is, the one with the greatest multiplicity. i.e. Heat

flow maximizes multiplicity.


2009 6

Temperature: Recall that the book said Temperature quantified the tendency of bodies to

spontaneously give up energy. Temperature, energy change, and multiplicity change are

inextricably enter twined.

Summing up

Thermodynamic systems are characterized by macroscopic variables, i.e., we can

determine their macrostates; however the fundamental physics is down on the

microscopic level and determines the microstates. If we assume that all

microstates are equally probable, then the probability of a macrostate depends

simply on the number of microstates which it encompasses – its multiplicity, and

the total number of possible microstates consistent with whatever constraints we

have on our system.

For example, last time, we considered two Einstein solids in thermal contact. We

constrained the system by saying there’s only so much energy to go around, qtotal

units. We then counted the multiplicities of each macrostate (solid A has all the

energy, solid B has all the energy,…) and thus determined the probability of each

state.

Irriversibility…

If you’ve ever encountered the 2nd

Law of Thermodynamics before, it was

probably in terms of temperature or entropy. So we’ve got a little work ahead of

us relating multiplicities to these properties. The first step, which we’ll focus on

today, is extending our multiplicities to easily handle systems of realistic sizes

(huge numbers of particles).

Agenda suggestions 1. So, what was important from today’s reading? What do you get asked to

do in the homework?

If they don’t know, take a moment to let them look over the

homework questions.

2. Part of what we’ll do today is see quantitatively where irreversibility

comes form – why systems evolve in one direction and not another, and

when that behavior should and shouldn’t be expected. We already have a

qualitative sense, but this is physics, so we have to back it up

quantitatively.

StatMech.exe Through this intro, have them look at small, medium, and large

systems.

2.4 Large Systems

Very Small. To construct our statistical tools, we first considered a very small

system: three coins, with two possible states each. This was few enough that we


2009 7

could count micro and macrostates by hand and thus directly determine the

multiplicities.

Small. We developed the tools so that we could calculate, rather than count, the

multiplicities. This allowed us to get quickly through, say 4 oscillators with

infinite states but only 3 energy units to share.

o Tell the program that you want each Einstein solid to have 3 members and

there to be 6 units of energy. Note the breadth of the peak.

Medium. In the homework, you asked a computer to calculate the multiplicities

for a system of 200 or so oscillators. This could be extended to 500, 1000,

10,000, 100,000, 1,000,000 oscillators.

o Tell the program that you want each Einstein solid to have 30 members

and there to be 60 units of energy. Note the breadth of the peak.

o Tell the program that you want each Einstein solid to have 300 members

and there to be 600 units of energy. Note the breadth of the peak.

large. But what about 1010

, 1023

, you know, the actual number of particles in a

typical thermodynamic system? It won’t do to ask a computer to handle this

many particles. Today, we’ll evolve our tools to handle them. The main thrust

will be approximating our factorials (well defined, but too discrete for doing

much math with) in terms of more analytical functions – ones that can be

integrated rather than summed.

o Pay off. You can fairly imagine, as we consider more and more members

in our systems, the multiplicities, and thus probabilities, get more and

more sharply defined. When we consider a very large number of

members, we get such a sharp peak that the few states it indicates, while

not inevitable, are terribly probable. For example, we can say, with no

fear of contradiction, that two identical solids, brought into thermal

contact, will come to share equal amounts of energy – that state (and it’s

near neighbors) is overwhelmingly more probable than any other.

2.4.1 Very Large Numbers

Very Large Numbers

o If you have a system of a large number of particles, the multiplicities

become very large, as in 231010 ; and that’s huge.

Logs and Very Large Numbers

o A simple device to make such numbers a tad more manageable is to take

their log. We’ll be doing this a bit, so it’s worth remembering and

confirming the basic properties of the natural log. So off we go into Log-

Math-Land for a little while.


2009 8

Example: 2.12 The natural logarithm function, ln is defined so that elnx

=

x for any positive number x.

a. Sketch a graph of the natural logarithm function.

A few particular values can be found by asking “e raised to the what gives

1?”: ln (1) = 0, “e raised to the what gives e?”: ln (e ) = 1, “e raised to the

what gives 0?”: ln (0) = - . Plotting these out then gives

b. Prove the identities ln(ab) = ln(a) + ln(b) and ln(ab) = b ln(a).

Appealing to the defining relation, abe ab)ln , but )ln(aea and beb ln , so baba eeeab lnlnlnln

Similarly, bababa eeaeb lnlnln

c. Prove that x

xdx

d 1ln .

Again, we’ll appeal to the defining relation xe xln

xex

dx

d

exdx

d

xdx

de

dx

d

x

x

x

11ln

1ln

ln

ln

ln

using )()( )( xfxf e

dx

xdfe

dx

d

Where the last step again appeals to the definition.

d. Derive the useful approximation xx)1ln( for |x| << 1

This approximation will of course only be good for small x. Let’s look at

a plot of this function for small x

1

1

ln x

2 3 0 x

1

1

2 3 0 x

F(x) = x

Actually go

over this

one

Do this

one

Just

remind

them of

this

identity

Just

remind

them of

this

identity

F(x) = ln(1+x)


2009 9

In the spirit of a Taylor series, ...)0()(0

xdx

dffxf

x

at x = 0, the function = 0, approximating it with a straight line in that

vicinity, with intercept 0 and the same slope as our function:

11

1)1ln(

00 xx

dx

d, so F(x) = ln(x+1) is approximated by F(x) = x:

xx )1ln( for small x.

2.4.2 Stirling’s Approximation

The reason we get very large numbers is that in calculating multiplicities, we take

the factorials of large numbers.

For Large N, Ne

NN

N

2! or, just a little less precise, but fine for large N,

N

e

NN!

Why Bother?

o Calculators will balk at N! for large enough N.

o More importantly, the relationship on the right hand side, though messier,

is continuous instead of discrete – you can take its derivative, integrate it,

or do whatever math you desire to it.

How Good?

o For N = 1 , there’s 7.7% error (and that’s a decidedly small number)

o For N = 10 there’s 0.81% error (note: the difference between the numbers

is merely Medium sized, while the numbers themselves are rather Large,

so % error is small)

o For N = 100 there’s only 0.083% error

o By the time we’re actually using large numbers for N, the error is a

negligible percent.

Why True?

o Roughly: B.3 Stirling’s Approximation Derivation 1.

o

xxxnnnn

nnnnn

nnnnn

nx

x

nx

x 11

)ln()ln()1ln(...)2ln()1ln(ln!ln

1...)3()2(1ln!ln

1...)3()2(1!

The last step is a free-be because x = 1 (the size of each step in x)

o 1ln)11ln1(lnln)ln()ln(lim!ln1

110

nnnnnnxxxdxxxxnn

nx

x

nx

xx

The first approximation is because we’re pretending that a

demonstrably discrete sum is a continuous one, i.e., x is much

smaller than x. If n is large, then that’s true for most of the range

of the integral.

o Error

PowerPoint

, or can

they do?


2009 10

Graphically, the difference is between integrating the area under

this smooth curve and adding the area in these discrete boxes.

Now for another approximation. Again, in the case that n is fairly

large, the 1 is negligible, so we’ll neglect it.

o n

nnnnnnn

e

neneen

nnnnnnn

nlnln!

ln1ln!ln

o More Precisely

You can choose better limits on the integral and get a better

approximation (homework B.10)

Derivation 2 in Appendix B.3. This gives you the factor of n2

(to fully appreciate it, you may need to look at Appendix B.1 also)

lnx

x


2009 11

2.4.3 Multiplicity of a Large Einstein Solid

Now let’s put this approximation to use in a very large Einstein solid.

o The exact formula: !!1

!1,

qN

qNqN

o First approximation

N >> 1 so !!

!,

qN

qNqN

o Apply Stirling’s

qN

qN

ee

e

qN

qN

qN

qN

qN

qN

qN

qN

e

q

e

N

e

qN

qN

qNqN

qN

qN

11

1

!!

!,

“High-Temperature” Limit: N << q (enough energy available that the average

particle has much more than just one unit – well above ground state)

o The book arrived at

N

ThighN

eqqN ., q>>N

o You do something very similar on Homework 2.19.

“Low-Temperature: Limit: N >> q (not nearly enough energy to go around, the

vast majority of particles are down in the ground state, without an additional

quantum of energy.)

o Picking up where the general case left off”

Nq

Nq

NNqq

NqNq

Nq

Nq

eNeq

eNqqN ,

Since q<<N, get a ratio q/N to prepare for another

approximation allowed by this term being small.

o q

Nq

q

Nq

Nq

Nq

q

N

qN

Nq

N

qN

qN

11

,

Here’s how I did it in class:

Take the log of both sides so we can invoke an

approximation there.

qqN

qNqNqqN ln1ln)(ln,ln

Invoke xx )1ln( for small x, in this case, N

q.

qqN

qNqNqqN ln)(ln,ln

qqqN

qNqqN lnln,ln

2

Based on

their notes,

ask them to

guide me.


2009 12

Final approximation,N

q 2

is much smaller than all the

other terms, so we can drop it.

q

q

q

q

q

q

q

eNqN

q

eNe

q

NqN

qqq

NqN

qqqNqqN

,

lnlnln,ln

lnln,ln

lnln,ln

Here’s the shorter alternative that Santino proposed

NqNqq

q

Nq

q

Nq

Nq

Nq

N

q

q

N

N

q

q

NqN

q

N

qN

Nq

N

qN

qN

11,

11

,

Last step follows from N>>q

Then Santino recalled that

N

N

q

N

qe 1lim . Our

situation of N>>q>>1 approximates that limit, so in our

case,

N

q

N

qe 1

o Note: this can be shown as equivalent to the

approximation made by the other approach:

1lnln

1lnln

1

N

qNA

N

qA

N

qA

N

N

Invoke N

q

N

q1ln for q/N >> 1

qeA

qN

qNAln

So

N

q

N

qe 1


2009 13

2.4.4 Multiplicity of a Large Einstein Solid

“High-Temperature” Limit: N << q (enough energy available that the average

particle has much more than just one unit – well above ground state)

o The book arrived at

N

ThighN

eqqN ., q>>N

“Low-Temperature” Limit: N>>q (most oscillators are in the ground state)

o

q

Tlowq

eNqN .,

This Time

Put main topics and headings on board

What you’ll be expected to do on homework:

Pr. 22 Alternative approach for estimating multiplicity peak width

Pr. 26 Find Multiplicity for 2-D Ideal Gas

Thermodynamic Limit: When the science of thermodynamics is absolutely

valid. That science treats systems as if energies, pressures, volumes,

temperatures, particle counts, etc. don’t randomly fluctuate. It would really

screw things up if half the air particles in the room gave all their kinetic

energy to the other half. We’d seen for Einstein solids of just one or two

atoms, it wasn’t at all unlikely that this kind of thing would happen. As we

added more and more atoms to the solids, that kind of thing became less and

less probable. For really large solids, that kind of thing is negligibly

improbable. In the thermodynamic limit, none of those shenanigans happen,

the most probable state is the only state.

Of course, that is just a limit, an ideal never achieved. To know whether or

not we can get away with approximating a real system as being in this limit,

whether or not the system can be successfully modeled with the tools of

thermodynamics, it’s therefore important to assess the error of the

approximation – how wide is the peak that we’re approximating as having no

width?

2.4.5 Sharpness of the Multiplicity Function

According to the Fundamental assumption of statistical mechanics: In an

isolated system in thermal equilibrium, all accessible microstates are equally

probable.

q/2

max

Width


2009 14

So, we have been finding how many of these equally probable microstates are

associated with individual macrostates – thus how probable those macrostates are.

Now that we know how to express the probability of an individual macrostate, we

can plot out the distribution of multiplicities // probabilities.

The reason we bother is that we’ve got something at stake, essentially all of

thermodynamics, on there being a very sharply peaked distribution – there being

one narrow window of macrostates that is far more popular than all the rest,

allowing us to, to first order, pretend that the system’s always in that single most

popular state – always has pressure P, energy E, particle density N/V,… Even

though all the air on the left side of the room is free to freeze while all that is free

to heat up, we can say instead that the energy essentially must be evenly

distributed, the temperature must be uniform.

Thermodynamic Limit.

o With the two Einstein solids, you saw how the probability distribution got

progressively tighter with each larger system. We will now quantify that

relationship between N and the width of the peak. In the limit that N is

absolutely huge, the peak becomes, for all intents a single spike of no

width – there is no likelihood of the system perceptibly varying from the

most likely state. This is the Thermodynamic limit.

Derivation

See PowerPoint

o Consider two interacting Einstein solids with very many members, each

with N. In the hot limit, more than enough energy to go around, q >> N

Multiplicity:

N

BA

NN

B

N

A

BABA qqN

e

N

eq

N

eqqqqq )(),(

2

Or in terms of q = qA + qB,

N

AA

N

qqqN

e2

What’s most probable qA?

By maximizing (take the derivative and set equal to 0), or

by symmetry arguments, it’s clear that this is maximized at

22

qq

qq BA , i.e., with ½ the energy in each solid.

How high?

Plugging this back into the multiplicity relation says that

the peaked multiplicity is

NNq

N

e22

max2

.

How Wide?

A typical measure of the width of a peak is its “full width at

½ max.”

We’ll call the full width q, then the value at which the

curve drops to ½ max is 22

qqqA . Setting the

multiplicity equal to ½ that of the peak and substituting this

in for qA, we can solve for q – the width of the peak.

Ask them to do much of

this. Perhaps tell them to

get to 1-… and then use

ln(1+x) ~ x.


2009 15

Nq

q

Nq

q

q

q

q

q

qqq

qqqqq

N

eqqqq

qq

N

e

qqqN

e

N

N

N

NN

NNN

N

AA

N

2ln2

)2ln(1

2

2

1ln

21ln

2

1

21

22

1

22

22

1

2222

22

1

2222

2

2

2

222

2

22

max21

2

1

1

1

o Classic result N

1 The bigger the system, the narrower the peak.

You’ll get this result for any system with random fluctuations around a

mean, for example a set of data from a single experiment that has been

repeated and repeated.

o Example numbers

Say we have 100 particles per solid, then the fractional width is

q

q0.166: 17%

Say we have 10,000, then it’s 0.0166: 1.7%

Say we have 1023

(one mole), then it’s only 5.3×10-12

: 5.3×10-10

%

That’s getting near our uncertainty in the values of some

fundamental constants!

o Conclusion

Specifically: if you have as much as two moles of Einstein solids

interacting, you won’t see the energy anyway but evenly

distributed.

Generally: For a large enough system, there exists a macrostate which is so much more

probable than any other perceptibly different state, that we can ignore fluctuations. When

this is the case, we can blissfully apply the laws of thermodynamics with never a care that

our values may fluctuate: This is the thermodynamic limit.

mon. 1/20 2.5 hw3,4,5 wed. 1/22 hw7: fri. 1/23 (c 10.3.1)...

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