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Stream: deprecated: thermodynamics

Topic: pressure equilibriation in thermostatics

Owen Lynch (Feb 01 2022 at 21:44):

If two thermostatic systems are constrained to have constant total volume, then what equilibriates is not $p$, but rather $p/T$. This does not compound with our experience of two systems that can do mechanical work on each other, but are thermally isolated; we expect that pressure should equilibriate.

The answer to this dilemna is that even if two thermostatic systems are thermally isolated, they still may exchange energy by doing work on each other. This work is not arbitrary, however, it is somehow in proportion to the volume transfer.

Owen Lynch (Feb 01 2022 at 21:45):

So maximizing entropy simply over the constraint that total volume is constant makes no sense, but on the other hand we still need a stronger constraint than simply maximizing over total energy.

Owen Lynch (Feb 01 2022 at 21:55):

The "curve" that we are maximizing over has different values of $(U_1,U_2)$.

Owen Lynch (Feb 01 2022 at 21:58):

In fact, here's a description of the curve that we are maximizing over. It is described by

$V_1 + V_2 = V$
$dU_1 = - p_1 d V_1$
$dU_2 = - p_2 d V_2$

This describes many curves; we must pick out one of these by picking "initial conditions" as it were; i.e. a point that the curve goes through.

Owen Lynch (Feb 01 2022 at 22:02):

Note that, for movement along this curve,

$dU_1 = T_1 dS_1 - p_1 dV_1$

is actually incorrect. The true equation for conservation of energy is actually

$dU_1 = dQ_1 - p_1 dV_1$

and it just so happens that in a reversible process $dQ_1 = T_1 dS_1$. But two systems coming into equilibrium are in fact increasing total entropy, so they are not undergoing a reversible process.

Owen Lynch (Feb 01 2022 at 22:08):

I think we can generalize this example, and it might even be worth putting into the paper. The general statement is that suppose that a manifold $X$ has a foliation by submanifolds, and let $Y$ be the set of leaves of the foliation. Then there is a function $X \to Y$ sending a point to the leaf that it is in, and we can push forward an entropy function on $X$ along this function.

Owen Lynch (Feb 01 2022 at 22:09):

However, this is not generally going to be convex!

Owen Lynch (Feb 01 2022 at 22:11):

I think perhaps to do this properly would involve some serious contact geometry, so perhaps it is best to hold off for now.

Owen Lynch (Feb 01 2022 at 22:13):

I think this is interesting though because it show that the intuition that "mixing increase entropy" doesn't necessarily imply convexity, because the path that the mixing takes might not be a straight line!

Owen Lynch (Feb 02 2022 at 00:59):

I think this is analogous to "holonomic vs. non-holonomic" constraints; the constraint that total energy is conserved is expressed by setting some function of the coordinates to a value, but the constraint that total volume is conserved cannot be expressed in this way so easily.

John Baez (Feb 02 2022 at 01:28):

If you have a single constraint, it's holonomic constraints if it's of the form $df = 0$, and nonholonomic if it's of the form $\omega = 0$ where $\omega$ is a 1-form that's not exact.

John Baez (Feb 02 2022 at 01:28):

So this is connected to some important themes in thermodynamics, like how thermodynamicists sometimes write $d$ when they don't really mean it.

Owen Lynch (Feb 02 2022 at 15:49):

John Baez said:

If you have a single constraint, it's holonomic constraints if it's of the form $df = 0$, and nonholonomic if it's of the form $\omega = 0$ where $\omega$ is a 1-form that's not exact.

Right, I think the $\omega$ that is not exact is $dU_1 + p_1 V_1$.

Owen Lynch (Feb 02 2022 at 15:51):

I think that our thermostatic framework currently works best with holonomic constraints; making it work with non-holonomic constraints would be good future work