Re: [Phys-l] Temperture profile in a graviational field

[John Denker <jsd@av8n.com>]

On 01/26/2012 12:31 PM, Folkerts, Timothy J wrote:
> * The walls of the tube are meant to me perfectly insulating, so no heat
> can go in or out through the sides of the tube.  The walls are also
> assumed to have a negligible heat capacity and to be perfectly rigid.
> This sounds like both isentropic and thermally insulated to me.
> * the two pistons, being 1 km apart, are meant to be thermally isolated
> from each other.
> * Initially, assume the pistons on the ends are also perfectly insulated
> and have negligible heat capacity.  
> * Later, we can change this to make on or both of the ends a thermal
> reservoir with a constant temperature.
> * initial conditions for the gas are assumed to be isothermal.  The
> initial pressure distribution would depend on the acceleration of the
> spaceship (or the acceleration of gravity).
> * the time scale I am most interested in is the long-term results for
> the co-moving pistons (or for the equivalent cylinder ins a psace ship
> or cylinder in the earth's gravity. (But he other cases are also fun to
> consider.)

Thanks, those are important clarifications.

There is still one point that seems confusing and/or unphysical:
Since we are effectively setting all thermal conductivities to
zero, the timescale for the system to reach thermal equilibrium
is infinite.  So ... if you are interested in the "long-term
results", the question arises, long compared to what?  One of
the natural physical timescales is already infinite, so you can't
afford to wait longer than that, in hopes of obtaining long-term
results.

> * initial conditions for the gas are assumed to be isothermal.

OK, that probably means the initial condition is thermal equilibrium.
So we have equilibrium by fiat, by assumption ... not because of any
process of equilibration.  So far, so good.

If we also assume that the acceleration has been going on for so long
that it is part of the initial condition, then by Einstein's principle
of equivalence, that equivalent to a gravitational field, so the whole 
question reduces to the problem previously solved, i.e. a column of gas 
in a gravitational field.

OTOH if the acceleration ever changes, the transient will create
non-equilibrium effects that will (in this idealized system) never
die out.

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A different set of assumptions:

> Of course, the temperature gradient will cause heat flow, which will
> tend to equalize the temperatures.  (Without the partitions there might
> be convection, but now we can old have conduction). 

OK, change of plans, we are now including thermal conductivity.

>  The question is
> then "will the temperatures become the same throughout, or could some
> gradient remain because the systems is constantly being 'disturbed' by
> the accelerating ends?"

It will eventually become the same.  Principle of equivalence.