Re: [Phys-l] Temp & Energy density

[John Denker <jsd@av8n.com>]

On 08/03/2009 07:17 AM, Paul Lulai wrote:
> Hello.
> I am curious about the conditions for which energy density = temp of a  
> gas.
> For this question I have a small cylinder if gas with only a few gas  
> particles.
> Using work, if I squeeze a piston down when the few gas particles are  
> at the bottom of the cylinder, then I have not given them any extra  
> energy through a work process. Their temperature should remain constant.
> Using energy density, this sneaky work  can't be ignored. Energy  
> density would show a temp difference regardless of how the piston was  
> compressed.
> This leads me to think the energy density solution includes some  
> assumptions that I do not know.

If you move the piston down when the particles already happen
to be at the bottom, there is no work.  That's because there 
is no gas pressure acting against the piston.  Remember, pressure
on the piston depends on gas particles hitting the piston.

There is no "sneaky work" and therefore no problem accounting
for the energy.

===========

On the other hand, if you continue the analysis, you find that
the gas afterward has the same energy, same temperature, same
number of particles, smaller volume, larger density, and (!) 
smaller entropy.

The original question asked about violation of the first law of
thermodynamics -- conservation of energy -- but there was no 
violation.  What we have instead is a blatant violation of the 
_second_ law of thermodynamics.

We have re-invented Maxwell's Demon.

This is discussed at length in Feynman.  The short answer is that
you can't do the experiment, not for a large number of particles
or even for a small number of particles, because you don't know
when to move the piston.  If you move the piston at a random time,
you are equally likely to encounter an abnormally-favorable 
configuration as an abnormally-unfavorable configuration ... and 
remember that entropy is a property of the ensemble, not a property 
of any microstate, so you are absolutely required to do the experiment
several times and take the average.  Meanwhile, if you want to move 
the piston at a non-random time, you need to look at the particles, 
and the act of looking transfers entropy in such a way that the 
second law is upheld.  It's not the piston but the _looking_ that
matters.

A javascript demo that illustrates (among other things) how looking
affects the entropy can be found at
  http://www.av8n.com/physics/entropy-sim.htm