Entropic temperature, pressure, and molar chemical potentials

Fix a material constitution $c$, with $r$ named species and dimensionless equilibrium state domain $D_c\subseteq {\mathbb{R}}^{r+2}$ from Equilibrium state domains. For $x=(\widehat{U},\widehat{V},{\widehat{n}}_1,\dots ,{\widehat{n}}_r)\in D_c$, define the diagonal SI coordinate map

${\mathrm{\Phi }}_c(x):=(U_c(x),V_c(x),n_{c,1}(x),\dots ,n_{c,r}(x))=(\widehat{U}\mathrm{J},\widehat{V}{\mathrm{m}}^3,{\widehat{n}}_1\mathrm{m}\mathrm{o}\mathrm{l},\dots ,{\widehat{n}}_r\mathrm{m}\mathrm{o}\mathrm{l})\mathrm{.}$

In SI numerical coordinates this is a diagonal linear bijection with a diagonal linear inverse, so it maps ambient open sets to open sets and its inverse is defined on its image.

Choose an open subset $O\subseteq D_c$ in the ambient dimensionless Euclidean coordinates that is designated as a single-phase chart, and set $\mathrm{\Omega }:={\mathrm{\Phi }}_c(O)$, an open chart of physical SI coordinate values. Here $U$ is the internal-energy coordinate evaluated in the center-of-mass rest frame of Galilean inertial charts and the center-of-mass rest frame. If $S_c:D_c\to \mathbb{R}$ is the entropy adopted by Concave macroscopic fundamental entropy, define its physical-coordinate representative on $\mathrm{\Omega }$ by

$S_c^{\mathrm{S}\mathrm{I}}:=S_c\circ {\mathrm{\Phi }}_c^{-1}:\mathrm{\Omega }\to \mathbb{R}\mathrm{.}$

When its arguments are the physical coordinates $(U,V,n_1,\dots ,n_r)$, write this representative as $S_c(U,V,n)$ for brevity. On $\mathrm{\Omega }$, assume the finite representative is $C^2$ and has $S_{c,U}>0$, as adopted by Concave macroscopic fundamental entropy. Here $C^2$ means that all coordinate partial derivatives through order two exist and are continuous ($C^k$ Euclidean maps and diffeomorphisms). A coordinate partial such as $S_{c,U}$ is the ordinary derivative in the physical $U$ coordinate while every other physical coordinate is held fixed, as in Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$; in particular the conversion factor from $\widehat{U}$ is included.

Explicitly, with the partial derivatives on the right taken in the dimensionless coordinates on $O$ while holding every other hatted coordinate fixed, the physical partials are

$S_{c,U}^{\mathrm{S}\mathrm{I}}=(1\mathrm{J}{)}^{-1}{\mathrm{\partial }}_{\widehat{U}}{S}_c,S_{c,V}^{\mathrm{S}\mathrm{I}}=(1{\mathrm{m}}^3{)}^{-1}{\mathrm{\partial }}_{\widehat{V}}{S}_c,S_{c,n_j}^{\mathrm{S}\mathrm{I}}=(1\mathrm{m}\mathrm{o}\mathrm{l}{)}^{-1}{\mathrm{\partial }}_{{\widehat{n}}_j}{S}_c,$

using the coordinate-conversion convention of Concave macroscopic fundamental entropy.

Define the physical-coordinate representatives of the absolute thermodynamic temperature, entropic pressure, and molar chemical potential of species $j$ by

$T_c^{\mathrm{S}\mathrm{I}}:=\frac{1}{S_{c,U}^{\mathrm{S}\mathrm{I}}},p_c^{\mathrm{S}\mathrm{I}}:=T_c^{\mathrm{S}\mathrm{I}}{S}_{c,V}^{\mathrm{S}\mathrm{I}},{\mu }_{c,j}^{\mathrm{S}\mathrm{I}}:=-T_c^{\mathrm{S}\mathrm{I}}{S}_{c,n_j}^{\mathrm{S}\mathrm{I}}\mathrm{.}$

For a state-domain point $x\in O$, define the corresponding state functions by pullback, $T_c(x):=T_c^{\mathrm{S}\mathrm{I}}({\mathrm{\Phi }}_c(x))$, $p_c(x):=p_c^{\mathrm{S}\mathrm{I}}({\mathrm{\Phi }}_c(x))$, and ${\mu }_{c,j}(x):={\mu }_{c,j}^{\mathrm{S}\mathrm{I}}({\mathrm{\Phi }}_c(x))$. When the argument is a physical tuple $(U,V,n)$, write $T_c(U,V,n)$, $p_c(U,V,n)$, and ${\mu }_{c,j}(U,V,n)$ for the physical-coordinate representatives. Thus both notations refer to the same state functions through ${\mathrm{\Phi }}_c$.

The entropy unit is $\mathrm{J}{\mathrm{K}}^{-1}$, the energy $U$ is in joules, $V$ is in cubic metres, and $n_j$ is in moles (SI quantities and unit conventions). Thus $S_{c,U}^{\mathrm{S}\mathrm{I}}$ has units ${\mathrm{K}}^{-1}$, $T_c$ is finite and positive in kelvins, $S_{c,V}^{\mathrm{S}\mathrm{I}}$ has units $\mathrm{J}{\mathrm{K}}^{-1}{\mathrm{m}}^{-3}$ and $p_c$ has units pascals, and $S_{c,n_j}^{\mathrm{S}\mathrm{I}}$ has units $\mathrm{J}{\mathrm{K}}^{-1}\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}$ and ${\mu }_{c,j}$ has units $\mathrm{J}\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}$. If a microscopic model instead uses a dimensionless particle count $N_j=N_{\text{\hspace{0.17em}⁣}A}{n}_j$, the corresponding per-particle chemical potential is ${\mu }_{c,j}\mathrm{/}{N}_{\text{\hspace{0.17em}⁣}A}$ joules, with the exact SI Avogadro constant from SI quantities and unit conventions.

These are scalar state functions of the stated rest-frame equilibrium chart. The pressure in this definition is a derivative-defined conjugate; identifying it with force per area requires a separate adopted mechanical coupling. Neither an empirical thermometer calibration nor any equation of state is supplied by this definition.