Re: ENERGY WITH Q

["John S. Denker" <jsd@MONMOUTH.COM>]

At 02:05 PM 10/25/01 -0400, Carl E. Mungan wrote:
> I stop a block with a string. No change in the block's
> thermal energy occurs. Now I stop the block via sliding friction. The
> block's thermal energy changes. Why does friction change the thermal
> energy of an object but tension does not (in these particular
> situations)? Give me an algorithm for deciding how a particular force
> changes the thermal energy, in other words, for partitioning the
> changes in the two terms on the RHS of the First Law

This is a good question.  It seems like a simple question, but it does not 
have a simple answer.  The question is almost tantamount to asking "What is 
thermodynamics?"  It's a reasonable question, but don't expect a 
super-simple answer.

I think it is important to consider _three_ cases.  Put wheels on the 
block, and call it a cart.

  Case 1)  Stop the cart with a string.  No change in the cart's thermal 
energy occurs.

  Case 2)  Stop the cart using disc brakes (connected to the wheels in the 
usual way).  The initial energy of overall motion is dissipated by 
friction, becoming thermal energy in the brakes.

  Case 3)  Rather than disc brakes, the cart is equipped with fancy 
dynamic brakes.  Each wheel has a little dynamo that generates electrical 
energy which is stored in a capacitor aboard the cart.  This is nonthermal 
energy.

Note that at this level of detail, case (3) is indistinguishable from case 
(2).  All we know is that the cart stopped itself, no strings attached.  We 
need to make additional careful observations if we wish to distinguish 
thermal from nonthermal energy.

In general, the answer to all such questions involves entropy.

In this particular case, the answer will involve counting modes.

 -- In the capacitor, there is basically one mode.  There is only one 
variable, Q, the charge on the capacitor.  There will be 1/2 k T of thermal 
energy in this mode, but that is a microscopic number, totally negligible 
compared to the energy involved in stopping the cart.

 -- In the brake disk, there will be something like 10^23 modes, namely 
thermal phonon modes.  There will be 1/2 k T of energy in each of these 
modes.  Stopping the cart via friction raises the temperature T.  That is 
to say, it creates more thermal phonons in these modes.  Lots and lots of 
thermal phonons.  We can't keep track of these phonons, so there is lots of 
entropy.

OK, that's the theoretical background.  How do we turn that a recipe or 
algorithm for something we can observe, as Carl requested?

Answer #1:  Brownian motion.

All those thermal phonons in the disc brake will cause the brake disc to 
jiggle.  The wheel will jiggle.  The whole cart will jiggle.  It won't 
necessarily jiggle very much, but the jiggles will be observable if you 
have a sufficiently small cart.  The mean-square amplitude of the of 
jiggling will be proportional to the temperature of the brake, and to the 
number of modes therein.

The cart with the dynamic brakes will jiggle much less, because of the 
vastly fewer modes.

Albert Einstein "Über die von der molekularkinetischen Theorie der Wärme 
geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen",
Annalen der Physik und Chemie, IV. Folge, Band 17 549-560 (1905).
  http://www.wiley-vch.de/berlin/journals/adp/549_560.pdf   (400KB)

Answer #2:  See if you can get the energy back out.

-- The cart with the dynamic brakes can start moving again if it 
chooses.  It just takes energy out of the capacitor, and runs the dynamos 
backwards (i.e. as motors).  This can be made nearly 100% efficient.

-- In contrast, the cart with the frictional brakes can't do this.  You 
might be able to get a little bit of energy out of the hot brakes, but 
nowhere near 100%.

Note that it is part of the definition of entropy that you can't predict 
the jiggles.  If you could predict them, you could set up some sort of 
Maxwell demon to get energy out of them with high efficiency.

To summarize:  A key part of the algorithm is noticing where the energy 
went, and whether you can keep track of the details.  If you can't keep 
track of the details, it's probably thermal.

==============

Remark:  Answer #1 is intimately related to answer #2.  This is called the 
fluctuation/dissipation theorem.  Whatever causes dissipation (energy going 
away) also causes fluctuations (jiggles coming in).