Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: [Phys-l] A relativity/thermodynamics "dilemma"



Let me take another stab at this.

This is a nifty topic because it highlights some foundational
questions about the definition of temperature and the definition
of equilibrium.

I am reminded of a passage from Goodstein _States of Matter_ :
He opens the section on "Variational Principles in Thermodynamics"
by saying:
Fundamentally there is only one variational principle in
thermodynamics. According to the Second Law, an isolated
body in equilibrium has the maximum entropy that physical
circumstances will allow.

Let's review the classic derivation. We can express the variational
principle as an equation
δS = 0 [1]

which is shorthand for
∂S/∂x = 0 [2]

where x represents any transformation that can is permitted by
the laws of physics and by the constraints of the situation.

One famous application of this principle concerns a thermally
isolated system that has been divided (artificially or otherwise)
into two subsystems. The two subsystems can exchange energy.
This may or may not be the only interaction between the two
subsystems, but we don't care; equation [2] applies with x
representing energy whether or not it applies with x representing
something else.

So, we choose to focus on what happens when equation [2] describes
the transfer of x amount of energy from subsystem A to subsystem B,
so that
E_A(x) = E_A(0) - x [3]
E_B(x) = E_B(0) + x

Hence
(d/dx) E_A = -1 [4]
(d/dx) E_B = +1

We assume that the energy and entropy are extensive, which is
an assumption not a law of nature. Generally this is a good
approximation when the subsystems are large, so we can neglect
boundary terms. In symbols, we have
S = S_A + S_B [5]
E = E_A + E_B

We arranged equation [3] to uphold conservation of energy, so that
E = E_A + E_B
= constant (independent of x) [6]

We assume the entropy of each subsystem is known as a function of
energy.
S_A = f_A(E_A) [7]
S_B = f_B(E_B)

Putting it all together we have
∂S/∂x = ∂S_A/∂x + ∂S_B/∂x
= ∂S_A/∂E_A ∂E_A/∂x + ∂S_B/∂E_B ∂E_B/∂x
= (-1) ∂S_A/∂E_A + (+1) ∂S_B/∂E_B

but ∂S/∂x = 0 so
∂S_A/∂E_A = ∂S_B/∂E_B [9]

and if we define the inverse temperature to be ∂S/∂E then we have
demonstrated that in equilibrium, the two subsystems have the same
inverse temperature ... which usually means they have the same
temperature.

There is a nice figure that relates ∂S_A/∂E_A and ∂S_B/∂E_B to
the overall δS. See e.g. figure 4 in
Moore and Schroeder
"A Different Approach to Introducing Statistical Mechanics"
http://www.math.utah.edu/~palais/pcr/statmech.pdf

This is the end of the review.

Looking back, we see there was an unstated assumption of dubious
validity, namely the assumption is that there was a single frame
of reference where E_A and E_B are defined. In the presence of
significant differences in gravitational potential, this is tricky.
In particular, let us define "@" to mean "according to", so that
E_A@X = E_A according to observations in frame X (for any X)

The issue we need to tackle is that E_A@A may well be different
from E_A@B.

The rule is that energy is still conserved by the physical processes
/in any particular frame/ but if you switch frames the energy will
generally be different. This is the rule in general relativity.
It is also the rule in special relativity, and even in Galilean
relativity.

If we define temperature properly in any given frame, then things
like the melting point of a material at rest in that frame will
be determined by the temperature in that frame. Each observer
can define his own temperature in this way, quite independently
of what other observers (if any) are doing.

1 / T@A = ∂S_A / ∂E_A@A
1 / T@B = ∂S_B / ∂E_B@B

There is no requirement for these to be equal, because the interaction
does not transfer energy from E_A@A to E_B@B. It transfers a conserved
amount of energy from E_A@A to E_B@A, but that's not the same thing.

So, to derive the Tolman relation you need to know how E in one frame
is related to E in another frame ...

... and that's essentially all you need to know. In particular,
note that this discussion does not depend on whether the two
subsystems interact by exchanging photons or something else. You
don't need to know the law for gravitational redshift of photons.
The details of the interaction don't matter. The only thing that
matters is the law connecting E in one frame to E in another frame.

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

As a tangent, possibly helpful (but possibly just confusing),
note that the foregoing is predicated on the assumption that we
have two subsystems that can directly exchange energy. If we
have three subsystems and some tricky constraint such that no
two of them can directly exchange energy, then all bets are
off. We see this every day in the atmosphere of the earth,
where the temperature decreases with altitude. That's because
an air parcel at one altitude cannot very effectively exchange
energy directly with an air parcel at another altitude. The
most effective process involves adiabatically stirring the
air. An air parcel cools as it rises, doing work against the
earths gravitational field (which is the aforementioned third
subsystem). So by the time it gets to a new altitude, its
energy is different from what it started with.

You might get objections if you call this situation "thermal
equilibrium" ... but the point remains that
a) it is what we observe, and
b) it is entirely consistent with the laws of thermodynamics,
including the fundamental variational principle δS = 0,
since δS means the variation of S with respect to whatever
physics processes are actually occurring.

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

At a higher level, this is yet another reminder that it is
better to remember the argument that leads to this-or-that
conclusion than to just learn the conclusion by rote. In
this case, if you just remember that "equilibrium is when
the temperature is the same everywhere" then you have no
hope of understanding (let alone discovering) the Tolman
relation.

To say the same thing in different words, it is important to
have good judgment about which rules are most reliable and
which are less reliable. The variational principle δS = 0
is in the "profound and reliable" category. The rule about
temperature being the same everywhere is incomparably weaker.