Re: [Phys-L] ?conservation of _internal_ energy

[John Denker <jsd@av8n.com>]

On 01/13/2016 01:31 AM, David Bowman wrote:

> The way I understand the situation is what is typically/usually 
> called the 'internal energy' in thermodynamics/statistical physics
> is simply the total (plain old as JD says) energy but *as seen* in a
>  particular frame in which the system's center-of-mass is at rest
> and not rotating, or, more specifically, the  frame is one in which
> the system's total momentum is zero and the total angular momentum
> about its CM is zero.

As I understand it, the «internal energy» also excludes
things like the gravitational potential energy.  So we
need to adjust not only the special frame's velocity,
but also its height.

> Such a frame may be either inertial or non-inertial

The inertial case is trivial;  in that case the «inertial 
energy» is the plain old energy, plus a constant.

> in any inertial frame it possesses a number of extensive constants of
> motion stemming from a combination of Noether's Theorem and Poincare
> invariance.

But I'm interested in the non-inertial parcels, such as one
encounters in even the simplest fluid dynamics situations.
Example:  Bernoulli's principle.

> If the system is not fully isolated but is in weak interaction with 
> an environment of a number of fixed intensive potentials, [....]

But I'm interested in the non-weak non-fixed interactions,
such as one encounters in even the simplest fluid dynamics
situations.

> Another way to keep a system at rest for which the use of a container
> is not appropriate (e.g. an expanding gas cloud) is to rewrite the
> system's Hamiltonian in a coordinate system that treats the CM
> degrees of freedom as their own dynamical variables, and treat the
> rest of the degrees of freedom in terms of relative coordinates
> measured WRT the CM.

That's fine up to a point, but the validity of that approach
is contingent upon an eeeenormous number of hitherto-unstated
assumptions.

The books that glorify U tend to write things like
	ΔU = heat + work
and assert that this expresses conservation of energy.  I find
that very very odd, since in reality U is not conserved.

You can talk about Noether's Theorem and Poincaré invariance
all you want, but it won't convince me that U is conserved.
It's bad luck to prove things that can't possibly be true.

> It makes the problem analysis simpler.

It seems like an odd sort of simplicity.  It makes some
equations simpler to write down ... but harder to interpret.
It conceals the meaning of the equation.

Suggestion:  Calculate the heat capacity of a ten-mile-high
column of air in a standard gravitational field.  I predict
it will lessen your attachment to U as the allegedly central
focus of thermodynamics.  Hint:  As you heat the air, its 
center of mass goes up, changing the gravitational potential
energy, even though the container has not moved.

=========

Here's how I think about it:  Very often there are some degrees
of freedom that do not equilibrate with others on the timescale
of interest.  For example:
 a) I can measure the temperature of a flashlight battery even
  though the electrical degree of freedom is verrrrry far out
  of equilibrium, i.e. a bazillion kT higher than would be
  expected from equipartition.  This one mode is protected by
  conservation of charge in conjunction with well-engineered
  electrical insulators.
 b) I can measure the temperature of a car that is moving
  relative to the lab frame, even though the overall motion
  of the CM is verrrrry far out of equilibrium.  This one
  special mode is protected by conservation of momentum in
  conjunction with well-engineered bearings.

These special modes contribute to the energy in the usual
way, even though they do not equilibrate in the usual way.
It is necessary to identify them and give them special
treatment, including assigning them their own thermodynamic
variables.  OTOH AFAICT it is not necessary or even helpful
to define new thermodynamic potentials (such as U).

Some very basic questions are still on the table:
 *) We know that U is conserved in trivial situations, but do
  you claim it conserved in nontrivial situations, when the
  parcels are actually changing speed and changing elevation?
 *) Is dU = work + heat?   (assuming const # of particles)
 *) If so, is that consistent with the work/KE theorem?
  It seems to express a work/U theorem, which is not the same.
 *) Is it consistent with the usual definition of force?
  It seems that if (M g h) is missing from U, then (M g)
  is missing from the force, which seems like a problem.
 *) Is it consistent with the principle of virtual work?