Timeline for The "geometry" of thermodynamics
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 14, 2023 at 23:55 | comment | added | EE18 | My multivariable calculus is very weak/not practiced. I need to brush up! Thanks @TobiasFünke | |
| May 14, 2023 at 22:00 | comment | added | Tobias Fünke | @EE18 In case you don't know already: The implicit function theorem. | |
| May 14, 2023 at 20:52 | comment | added | EE18 | Ah, I think your first comment gets at the crux of my question, with my "geometric" phrase coming from my lack of math background here. The existence of said implicit function wasn't stated in the main body of the text (Postulate II is couched in terms of the explicit function), so I wasn't sure whether there was some mathematical theorem guaranteeing that we could cast things in terms of an implicit function or something like that. | |
| May 14, 2023 at 20:27 | comment | added | hyportnex | Callen in Appendix A5 (I think that is what you are referring to) proves a very useful and frequently employed identity A.28 among the various partial derivatives of the smooth implicit function $\psi(x,y,z) =const$. This is just straight multivariable calculus without any direct geometric meaning, differential or otherwise. In comparison, based on the principle of adiabatic inaccessibility, Caratheodory's "proof" that the thermostatic space can be foliated is more or less equivalent to the existence of "entropy" and "absolute temperature". | |
| May 14, 2023 at 20:19 | comment | added | hyportnex | You can write a functional relationship connecting the parameters $S, U, X_1,X_2,..X_N$ as an implicit function $\psi(S, U, X_1, X_2,..X_N)=const$ or as an explicit function $S=\phi(U, X_1,X_2,..X_N$. While they are not exactly equivalent the existence of the explicit function follows from some mild smoothness requirements that will fail at phase boundaries, and some such, but there we impose a kind of compatibility relationship (Clausius-Clapeyron equation) between the two of functions that are joined at the boundary. | |
| May 14, 2023 at 19:56 | comment | added | EE18 | Can you comment then on what underlies the partial derivatives in my question? In the end, that's the crux of my question, but I had thought (after following the derivation in Callen's Appendix A) that one had to begin at $\psi(S,U,N,X_k) = const$ to arrive at these relationships between derivatives "on this surface"? | |
| May 14, 2023 at 19:42 | history | answered | hyportnex | CC BY-SA 4.0 |