BELIEVE   ME   NOT!    - -     A   SKEPTIC's   GUIDE  

. . . Physics^15.1

I have "borrowed" the notation, general approach, basic derivations and most of the quotations shown here from the excellent textbook of the same name by Kittel & Kroemer, who therefore deserve all the credit (and none of the blame) for the abbreviated version displayed before you.

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

. . . way,^15.2

In the present case, we have a choice of whether to treat the pennies as "indistinguishable" or not. No two pennies are really indistinguishable, of course; even without our painted-on numbers, each one has unique scratches on its surface and was crystallized from the molten state in a unique microscopic pattern. We could tell one from another; we just don't care, for circumstantial reasons. In QUANTUM MECHANICS, however, you will encounter the concept of elementary particles [e.g. electrons] which are so uncomplicated that they truly are indistinguishable [i.e. perfectly identical]; moreover, STATISTICAL MECHANICS provides a means of actually testing to see whether they are really absolutely indistinguishable or just very similar!

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

. . .  tails.^15.3

It might be that we get to keep all the pennies that come up heads, but for every penny that comes up tails we have to chip in another penny of our own. In that case our profit would be   n - (N-n) = 2n - N  cents.

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

. . . filled].^15.4

If you were the parking lot owner and were charging $1 per space [cheap!], your profit would be $n. I keep coming back to monetary examples - I guess cash is the social analogue of energy in this context.

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

. . .  particles,^15.5

The term "particle" is [in this usage] meant to be as vague as possible, just like "system:" the particles are "really simple things that are all very much alike" and the system is "a bunch of particles taken together."

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

. . . plotted^15.6

Actually what is plotted in Fig. 15.1 is the probability function

$${\cal P}(n) \; \equiv \; {1 \over 2^N} \cdot \Omega(n,N) \; = \; {1 \over 2^N} \cdot {N! \over n! \; (N-n)! }$$

vs.   ${n \over N}$,  as explained in the caption. Otherwise it would be difficult to put more than one plot on the same graph, as the peak value of   $\Omega(n,N)$  gets very large very fast as  N  increases!

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

. . . variance^15.7

Recall your Physics Lab training on MEASUREMENT!

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

. . . huge.^15.8

A good estimate of the size of  N!  for large  N  is given by Stirling's approximation:

$$N! \approx \sqrt{2 \pi N} \cdot N^N \cdot e^{-N}$$

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

. . . paradigm^15.9

Count on it!

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

. . . "particles"^15.10

Remember, a "particle" is meant to be an abstract concept in this context!

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

. . . other.^15.11

If I flip my coin once and you flip your coin twice, there are  2¹ = 2  ways my flip can go [h, t] and  2² = 4  ways your 2 flips can go [HH, HT, TH, TT]; the total number of ways the combination of your flips and mine can go [hHH, hHT, hTH, hTT, tHH, tHT, tTH, tTT] is   $2 \times 4 = 8$. And so on.

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

. . . states^15.12

Nothing precludes finding the system in states with other values of  U₁, of course. In fact we must do so sometimes! Just less often.

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

. . . follows:^15.13

Perhaps the converse is actually true: human "wants" are actually manifestations of random processes whose variety is greater in the direction of perceived desire. I find this speculation disturbing.

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

. . . assumptions.^15.14

Or, at least, none that are readily apparent . . . .

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

. . . configuration^15.15

Note the distinction between the words configuration and state. The latter implies we specify everything about the system - all the positions and velocities of all its particles, etc. - whereas the former refers only to some gross overall macroscopic specification like the total energy or how it is split up between two subsystems. A state is completely specified while a configuration is only partly specified.

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

. . . entropy.^15.16

This is the same as maximizing the probability, but from now on I want to use the terminology "maximizing the entropy."

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

. . . mathematics.^15.17

We have already done this once, but it bears repeating! To avoid complete redundancy, this time we will reverse the order of hot and cold.

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

. . . was!^15.18

Well, to be fair, people had a pretty good working knowledge of the properties of temperature; they just didn't have a definition of temperature in terms of nuts-and-bolts mechanics, like Eq. (10).

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

. . . degrees^15.19

These silly units were invented by an instrument maker called Fahrenheit [1686-1736] who was selling thermometers to meteorologists. He picked body temperature [a handy reference, constant to the precision of his measurements] for one "fiducial" point and for the other he picked the freezing point of saturated salt water, presumably from the North Sea. Why not pure water? Well, he didn't like negative temperatures [neither do we, but he didn't go far enough!] so he picked a temperature that was, for a meteorologist, as cold as was worth measuring. [Below that, presumably, it was just "damn cold!"] Then he (sensibly) divided up the interval between these two fiducials into 96=64+32 equal "degrees" [can you see why this is a pragmatic choice for the number of divisions?] and voilá!  he had the Fahrenheit temperature scale, on which pure water freezes at 32$^\circ$F and boils at 212$^\circ$F. A good system to forget, if you can.

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

. . . "Kelvin"^15.20

Named after Thomson, Lord Kelvin [1852], a pioneer of thermodynamics.

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

. . . field.^15.21

The rate of change of this energy with the angle between the field and the compass needle is in fact the torque which tries to align the compass in the Earth's magnetic field, an effect of considerable practical value.

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

. . . laboratory.^15.22

[by reversing the direction of the magnetic field before the spins have a chance to react]

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

. . . system^15.23

A "small system" can even be a "particle," since both terms are intentionally vague and abstract enough to mean anything we want!

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

. . . "atoms,"^15.24

I will cover the history of "Atomism" in a bit more detail later on!

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

. . . suggest,^15.25

If you want the details, here they are: Suppose that p_i is the CANONICAL MOMENTUM characterizing the $i^{\rm th}$ degree of freedom of a system and that $\varepsilon(p_i) = b p_i^2$ is the energy associated with a given value of p_i. Assume further that p_i can have a continuous distribution of values from $-\infty$ to $+\infty$. Then the probability of p_i having a given value is proportional to $\exp(-bp_i^2/\tau)$ and therefore the average energy associated with that degree of freedom is given by

$$\langle \varepsilon(p_i) \rangle = { \int_{-\infty}^{+\infty . . . . . . \over \int_{-\infty}^{+\infty} e^{- bp_i^2 / \tau} dp_i }$$

These definite integrals have "well known" solutions:

$$\int_{-\infty}^{+\infty} x^2 e^{- ax^2} dx = {1\over2} \sqr . . . . . . int_{-\infty}^{+\infty} e^{- ax^2} dx = \sqrt{\pi \over a} ,$$

where in this case $a = b/\tau$ and x = p_i, giving

$$\langle \varepsilon(p_i) \rangle = {\tau \over 2} . \qquad {\cal{QED}}$$

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

. . . argument^15.26

We can, of course, make the explanation more elaborate, thus satisfying both the demands of rigourous logic and the Puritan conviction that nothing of real value can be obtained without hard work. I will leave this as an exercise for other instructors.

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

. . . pressure^15.27

Unfortunately, we use the same notation (p) for both momentum and pressure. Worse yet, the notation for number density (number of atoms per unit volume) is  n. Sorry, I didn't set up the conventions.

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

. . . sum.^15.28

We may say that the average kinetic energy "stored in the  x  degree of freedom" of an atom is   ${1 \over 2}m \langle v_x^2 \rangle$.

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

. . . write^15.29

This is equivalent to saying that the average energy stored in the  x  degree of freedom of one atom [or, for that matter, in any other degree of freedom] is   ${1 \over 2} \, \tau$  -- which is just what we originally claimed in the EQUIPARTITION THEOREM. We could have just jumped to this result, but I thought it might be illuminating to show an explicit argument for the equality of the mean energies stored in several different degrees of freedom.

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