Representation And Equilibrium

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.

For a simple-compressible system whose only admitted mechanical equilibrium coordinate is volume, model the piston as a planar boundary of fixed area $A>0$ in ${\mathrm{m}}^2$ that translates only along its outward unit normal without changing shape. Let $\mathrm{\ell }$ in metres be its outward displacement coordinate, so the enclosed volume is $V_{\ast }+A\mathrm{\ell }$ for a fixed reference value $V_{\ast }$. Let $\mathrm{\Omega }$ be an open positive-temperature single-phase chart of equilibrium states, with equilibrium states as in Equilibrium state domains.

A quasistatic piston leg in this postulate is a map $\theta \mapsto (U(\theta ),V(\theta ),n,\mathrm{\ell }(\theta ))$ on a compact interval $J=[\alpha ,\beta ]\subseteq \mathbb{R}$, with $\alpha <\beta$, of a dimensionless parameter, fixed composition $n$, $V(\theta )=V_{\ast }+A\mathrm{\ell }(\theta )$, and the system in an equilibrium state at every $\theta \in J$. Each coordinate function is continuous on $J$, differentiable on $(\alpha ,\beta )$, and has a derivative extending continuously to $J$; equivalently the leg is $C^1$ in its Euclidean coordinates. A piecewise-$C^1$ quasistatic piston path is a finite concatenation of these legs with matching endpoint state and piston position. The path is frictionless when the tangential contact force vanishes on every leg. It is in mechanical equilibrium when the net normal force on the piston is zero at every path point; thus the external load balances the system's outward normal force. Write $F_{\mathrm{s}\mathrm{y}\mathrm{s}}(\theta )$ for the signed outward normal force component, measured in newtons, exerted by the system on the piston, and define the oriented mechanical pressure by $p_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}(\theta ):=F_{\mathrm{s}\mathrm{y}\mathrm{s}}(\theta )\mathrm{/}A$, in pascals.

For every such path, adopt the identification

$p_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}=p_c,$

where $p_c$ is the derivative-defined pressure of Entropic temperature, pressure, and molar chemical potentials, evaluated at $(U(\theta ),V(\theta ),n)$. With $W_{\mathrm{o}\mathrm{n}}$ positive for work done on the system, the system's outward work on the piston is $F_{\mathrm{s}\mathrm{y}\mathrm{s}}d\mathrm{\ell }$, so the coupling assigns the work increment along the path

$\delta W_{\mathrm{o}\mathrm{n}}=-p_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{h}}Ad\mathrm{\ell }=-p_{c}dV\mathrm{.}$

Here $\delta W_{\mathrm{o}\mathrm{n}}$ denotes an increment along the stated path; no path-independence is asserted. This is an adopted, regime-limited physical assumption, not a deduction from the entropy derivative or from the first-law balance First-law energy balance for closed process records. It makes no claim for friction, paths outside the quasistatic regime, changing piston area, shear, surface forces, fields, composition transfer, or other generalized coordinates.

Let $c$ be a material constitution with $r$ named species, and let $O\subseteq D_c$ be an open dimensionless single-phase subset from Equilibrium state domains. Its physical SI chart is $\mathrm{\Omega }={\mathrm{\Phi }}_c(O)$ under the diagonal unit map ${\mathrm{\Phi }}_c$ defined in Entropic temperature, pressure, and molar chemical potentials. Let $(U_0,V_0,n_{0,1},\dots ,n_{0,r})\in \mathrm{\Omega }$, with $U$ in joules, $V>0$ in cubic metres, and each $n_j>0$ in moles (SI quantities and unit conventions). On $\mathrm{\Omega }$, write $S_c(U,V,n)$ for the physical-coordinate representative $S_c\circ {\mathrm{\Phi }}_c^{-1}$; its physical-coordinate partials include the unit factors as specified in Concave macroscopic fundamental entropy. Assume this representative is $C^2$ and $S_{c,U}(U_0,V_0,n_0)>0$. Then there are open neighborhoods of $(S_c(U_0,V_0,n_0),V_0,n_0)$ and $(U_0,V_0,n_0)$ on which the equation

$S_c(U,V,n)=s$

has a unique solution $U={\mathcal{U}}_c(s,V,n)$ of class $C^2$. On this local chart,

${\mathcal{U}}_{c,s}=T_c,{\mathcal{U}}_{c,V}=-p_c,{\mathcal{U}}_{c,n_j}={\mu }_{c,j},$

with the intensive functions defined by Entropic temperature, pressure, and molar chemical potentials. This is a local representation only: it asserts neither global injectivity or domain coverage nor differentiability at a nonsmooth phase boundary.

The mathematical supplier Local graph form at a nonsingular derivative minor supplies the local $C^1$ graph interface. The argument above derives $C^2$ regularity from the entropy's $C^2$ hypothesis; it does not attribute a $C^2$ conclusion to that supplier.

Consider two independent separated subsystems $A,B$ of fixed material constitutions, with states in domains $D_{c_A},D_{c_B}$ as in Equilibrium state domains, and additive total energy, volume, and aligned mole amounts as in Additive separated systems and tagged bulk scaling using SI coordinates (SI quantities and unit conventions). Let $S_{\mathrm{t}\mathrm{o}\mathrm{t}}=S_A+S_B$ be their adopted total entropy, and let $\mathcal{F}$ be the exact feasible endpoint set fixed by the totals, retained partitions, and permitted exchanges in Isolated compound and feasible equilibrium set. Suppose $x_{\ast }=(x_A^{\ast },x_B^{\ast })\in \mathcal{F}$ is a stable equilibrium attaining the global entropy maximum adopted by Stable equilibrium requires an attained global entropy maximum. Assume both subsystems lie in open $C^2$ single-phase charts with finite positive temperatures, and assume every exchange direction named below is locally feasible for both signs at $x_{\ast }$. Then:

• If energy exchange is permitted, $T_A=T_B$.
• If volume exchange is permitted, then $p_A\mathrm{/}{T}_A=p_B\mathrm{/}{T}_B$; if energy exchange is also permitted, this gives $p_A=p_B$.
• If species $j$ can be exchanged, then ${\mu }_{A,j}\mathrm{/}{T}_A={\mu }_{B,j}\mathrm{/}{T}_B$; if energy exchange is also permitted, this gives ${\mu }_{A,j}={\mu }_{B,j}$.

The pressure equality concerns the derivative-defined pressures of Entropic temperature, pressure, and molar chemical potentials. It is equality of mechanical piston pressures only when the coupling of Mechanical pressure-volume work coupling applies. A retained volume or species constraint supplies no corresponding equality.

Let $C$ be an open convex region of dimensionless numerical coordinates for a compound of independent separated systems, and let $S_{\mathrm{t}\mathrm{o}\mathrm{t}}:C\to \mathbb{R}$ be the adopted $C^2$ concave total entropy on a smooth single-phase chart. Let the feasible set be the affine slice $F=C\cap \{x:Ax=b\}$ for a fixed real linear map $A$ and vector $b$. For the following conclusions, assume the mathematical input that for differentiable concave $f$ on an open convex $C$, an interior point of $C\cap \{x:Ax=b\}$ whose derivative vanishes on $\ker A$ is a global feasible maximizer; strict concavity gives at most one maximizer; for $C^2$ concave $f$, the Hessian quadratic form is negative semidefinite. Under this input, if $x_{\ast }\in F$ is interior relative to this slice and $D{S}_{\mathrm{t}\mathrm{o}\mathrm{t}}(x_{\ast })$ vanishes on $\ker A$, then $x_{\ast }$ is a global entropy maximum on $F$. If $S_{\mathrm{t}\mathrm{o}\mathrm{t}}$ is strictly concave on $F$, that maximum is unique. The Hessian conclusion does not by itself give strict stability.

The mathematical input is supplied by Concave supporting bound, affine maxima, and Hessian sign on the stated region and admissible affine segments.

There is a further conditional scaling conclusion. On a designated open single-phase bulk cone $C_c\subseteq D_c$ with the common constitutive law adopted by Common constitutive entropy law for a designated bulk family, the $C^2$ entropy Hessian has zero radial direction: $D^2{S}_c(x)[h,x]=0$ for every $x\in C_c$ and every vector $h$. This branch adopts coordinate-preserving entropy transport between the distinct scaled-system tag and one fixed material-law chart, with common species, units and reference conventions. The null-direction conclusion is not asserted for the general tagged family from Concave macroscopic fundamental entropy and Additive separated systems and tagged bulk scaling alone.

The actual entropy postulate uses the distinct tag $c^{(\lambda )}$ and does not identify it with $c$. The additional common-constitutive-law branch [A3] explicitly supplies the transport needed for the M4 radial conclusion; the general tagged postulates alone do not supply it. The exact concavity prerequisite [L1] supplies the affine-maximum and Hessian-sign clauses on the admitted chart. A finite-dimensional countermodel is recorded on the companion page.

Fix a constitution $c$ and an open dimensionless subset $O\subseteq D_c$ in the single-phase region of Equilibrium state domains. Its physical chart is $\mathrm{\Omega }={\mathrm{\Phi }}_c(O)$, where ${\mathrm{\Phi }}_c$ is the SI coordinate map defined in Entropic temperature, pressure, and molar chemical potentials. For fixed physical amounts $n=(n_1,\dots ,n_r)$ with nonempty slice ${\mathrm{\Omega }}_n:=\{(U,V):(U,V,n)\in \mathrm{\Omega }\}$, this slice is open in ${\mathbb{R}}^2$. Use $T,p$ from Entropic temperature, pressure, and molar chemical potentials as functions of the physical coordinates on ${\mathrm{\Omega }}_n$. For a response derivative, restrict to an open neighborhood in ${\mathrm{\Omega }}_n$ where the required natural-coordinate map is a $C^1$ diffeomorphism onto an open image, so its inverse expresses the physical $(U,V)$ as $C^1$ functions of those natural coordinates ($C^k$ Euclidean maps and diffeomorphisms). Specifically, $C_V$ uses $(U,V)\mapsto (T,V)$ at fixed $n$, while $C_p$, ${\kappa }_T$, and ${\alpha }_p$ use $(U,V)\mapsto (T,p)$ at fixed $n$. The existence of each such local chart is a hypothesis for its response derivative, not a consequence of this definition. Each derivative uses the indicated natural variables and holds all other listed coordinates fixed.

The constant-volume heat capacity and constant-pressure heat capacity of the whole system are, where the derivatives exist and are finite,

$C_V:={\left(\frac{\mathrm{\partial }U}{\mathrm{\partial }T}\right)}_{V,n},H:=U+pV,C_p:={\left(\frac{\mathrm{\partial }H}{\mathrm{\partial }T}\right)}_{p,n}\mathrm{.}$

Here $H$ is the enthalpy in joules; both heat capacities are in $\mathrm{J}{\mathrm{K}}^{-1}$, not per mole. The isothermal compressibility and isobaric thermal expansion coefficient are

${\kappa }_T:=-\frac{1}{V}{\left(\frac{\mathrm{\partial }V}{\mathrm{\partial }p}\right)}_{T,n},{\alpha }_p:=\frac{1}{V}{\left(\frac{\mathrm{\partial }V}{\mathrm{\partial }T}\right)}_{p,n}\mathrm{.}$

Since $V>0$ and the coordinates use SI units (SI quantities and unit conventions), ${\kappa }_T$ has units $\mathrm{P}{\mathrm{a}}^{-1}$ and ${\alpha }_p$ has units ${\mathrm{K}}^{-1}$. This definition asserts no sign for any response, no identity among them, and no finite value at a critical or otherwise singular chart.