## Thermodynamics and Geometry I

Thermodynamics seems to me a bewildering array of equations and relationships between an equally bewildering array of quantities. It also seems intimately connected with the geometry of surfaces though I can’t seem to find what I want on it. The treatments seem to leap into abstraction and offer little for a novice thermo student. I’d like to think that thermo could be better organized for a student by appealing to what they already know about surfaces from their calculus sequence. Also it might be an opportunity to improve that aspect of their mathematics.

So we start with some equation of state P(V,T) which is a 2D surface embedded in $\mathbb{R}^3$. If we take, as is convention, P to be be the vertical axis then we can imagine the surface have a set of induced coordinates (V,T) from the plane of constant P=0 “below” it. For this first post I’m going to ignore surfaces which account for phase transitions as they’ll have lines of non-differentiability in them. We’re also looking at the octent where P>0,V>0, T>0.

There would be several other equally obvious ways to establish coordinates on the surface as well via P and T or with P and V (yes I am making a simplifying assumption here also..nevermind).

Given a set of coordinates, say V,T, we might define a scalar field $U:M\to\mathbb{R}$, $U=U(V,T)$ which represents the internal energy of the system which has M as its space of equilibria. On this surface at each point there will be a tangent space spanned by a coordinate basis of $\partial/\partial V$ and $\partial/\partial T$. Given a process (a path) $\gamma:\mathbb{R}\to M$ with parameter t we can compose this with U and look at the change in energy along the process $dU(\gamma(t))/dt$ We can expand this,

$\frac{dU}{dt}=\frac{\partial U}{\partial V}\frac{dV}{dt}+ \frac{\partial U}{\partial T}\frac{dT}{dt}$

But this looks confusing mathematically although it appeals to our idea of a process (with a changing V and T). When you see these sorts of things on surfaces its almost always the case that there’s been an identification between coordinates the the components of a curve $(\gamma^1(t), \gamma^2(t))\equiv (V(t),T(t))$ putting that in,

$\frac{dU}{dt}=\frac{\partial U}{\partial V}\frac{d\gamma^1}{dt}+ \frac{\partial U}{\partial T}\frac{d\gamma^2}{dt}$

But this is still not quite right. $\gamma$ takes on values in a manifold which may not have a metric and so it doesn’t make sense to talk about values of $\gamma$ being close or far apart. So for full disclosure we would have to invoke the coordinate charts $x:M\to\mathbb{R}^2$ and view the ordinary derivatives above as $d\gamma^i /dt \equiv d (x^i\circ\gamma)/dt$ since $x^i\circ\gamma: \mathbb{R}\to\mathbb{R}$. But we won’t clutter our above expression with that. But we can rewrite the above as,

$\nabla U \cdot \frac{d\gamma}{dt}$

I realize this isn’t rigorous for a number of reasons but I want to appeal a bit to your gut (we could do $dU(\dot{\gamma})$ instead if you like). But the idea is that we find the change of U along the process $\gamma$

We also can use the differential operator to obtain a 1-form from u. $dU: T_pM\to\mathbb{R}$ We can write out dU in terms of its components with respect to the (V,T) coordinates,

$dU=\frac{\partial U}{\partial V}dV+\frac{\partial U}{\partial T}dT$

There also apparently exist two other important 1-forms in $T_p^*M$ which are $Q,W$ for heat flow and work. Here we avoid the traditional d-bar notation. These are not exact and so, for example, we write the first law as $dU=Q-W$

There’s an initial sketch. Now for some questions:

(1) There are a number of other scalar fields and corresponding 1-form fields on the equilibrium surface: enthalpy, Gibbs free energy, Helmholtz free energy and entropy (sort of…it also plays the role of a ‘coordinate’?)
computationally we go from one ‘flavor’ of energy to another with the Legendre transformation…what does this mean geometrically?

(2) What are the physical consequences of a change of coordiantes on M, say from (V,T) to (P,T) so that now $U=U(P,T)$?

(3) Where does entropy fit into all of these things, is it a coordinate on the surface (another change of coordinates?) or is it a function on the surface?

(4) What can we say, geometrically, about surfaces which may have lines or points of discontinuity?

(5) What other questions can we ask…(cyclic relation?). What other thermodynamic relationships can be expressed on this surface? Would it help to just focus on the ideal gas surface first?