Potentials And Responses

Given: The open domain $C$, the $C^1$ function $f$, and for each $x\in C$ the stated radial homogeneity on a nondegenerate interval containing $\lambda =1$.

The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$: If $f$ and $g$ are totally differentiable at the relevant points, then $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$.

Given: An open domain, a $C^2$ function $f$, and a $C^1$ inverse chart $g$ for $(u,z)\mapsto (D_{u}f(u,z),z)$.

The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$: Differentiating a composition of totally differentiable maps gives the composed derivative.

$C^k$ Euclidean maps and diffeomorphisms: A $C^1$ diffeomorphism is a bijection between open neighborhoods whose map and inverse are both $C^1$.

Chain sum product and composition rules for Banach derivatives: Differentiation of a bounded-bilinear product obeys the sum of the two product terms.

Given: A fixed constitution $c$, an open C2 single-phase chart in coordinates $(s,V,n_1,\dots ,n_r)$, the local energy representation, and the valid natural-coordinate inverse charts named in the Statement. Use SI units: $U,H,F,G,\mathrm{\Omega }$ in joules; $s$ in $\mathrm{J}{\mathrm{K}}^{-1}$; $T$ in kelvins; $V$ in ${\mathrm{m}}^3$; $p$ in pascals; $n_j$ in moles; and ${\mu }_j$ in $\mathrm{J}\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}$.

Local energy representation and thermodynamic conjugates: Its Statement gives $U_{c,s}=T_c$, $U_{c,V}=-p_c$, and $U_{c,n_j}={\mu }_{c,j}$ on a local C2 chart. The carrier is registered in batch 8; its current proof uses the C1 graph interface of the batch-4 draft thm-tdmath-local-implicit-rank. That supplier's Step 1 readiness record is authoring status, not independent proof acceptance, so this use remains conditional on its mathematical review.

Local thermodynamic Legendre potentials: $H=U+pV$, $F=U-Ts$, $G=F+pV$, and $\mathrm{\Omega }=F-{\sum }_j{\mu }_j{n}_j$ on the stated inverse charts.

Differential of a C2 partial Legendre transform on a local inverse chart: For a C2 scalar function and an assumed C1 local inverse of its conjugate-coordinate map, the partial Legendre transform has differential $-u\cdot dx+D_{z}f\cdot dz$.

Given: The local C2 energy representation on $C$, the open subdomain $W$, and the same-tag homogeneity hypothesis in the Statement, either assumed independently or supplied by the designated common-law branch as specified there. Coordinates are entropy $s$ in $\mathrm{J}{\mathrm{K}}^{-1}$, volume $V$ in ${\mathrm{m}}^3$, and whole-system amounts $n_j$ in $\mathrm{m}\mathrm{o}\mathrm{l}$.

Additional same-tag premise: for each $q\in W$ there is an open interval $\mathrm{\mid }\lambda -1\mathrm{\mid }<{\epsilon }_q$ on which ${\mathcal{U}}_c(\lambda q)=\lambda {\mathcal{U}}_c(q)$ whenever $\lambda q\in C$. The common-law branch in [A2] supplies this premise on the corresponding local inverse subdomain.

Common constitutive entropy law for a designated bulk family: For the supplied bulk branch, the inverse states $\mathrm{\Psi }(W)$ lie in its designated open single-phase cone, after the fixed SI unit conversion. Coordinate-preserving entropy transport and tagged scaling give $S_c(\lambda x)=\lambda S_c(x)$ there. The material law has fixed species and intensive parameters and common units and energy/entropy reference conventions; distinct physical system tags are retained. This premise is required only for the assertion that the framework supplies [A1], not for the general implication from independently assumed [A1].

Additive separated systems and tagged bulk scaling: The scaling map may send $c$ to a different material tag $c^{(\lambda )}$; the statement explicitly does not identify those tags.

Concave macroscopic fundamental entropy: The entropy scaling statement is $S_{c^{(\lambda )}}({\delta }_{\lambda }x)=\lambda S_c(x)$ for the tagged family; it does not state fixed-tag homogeneity of the energy representation.

Local energy representation and thermodynamic conjugates: On its local inverse chart, ${\mathcal{U}}_{c,s}=T_c$, ${\mathcal{U}}_{c,V}=-p_c$, and ${\mathcal{U}}_{c,n_j}={\mu }_{c,j}$. It supplies a unique local C1 inverse from the draft local-rank graph interface and derives C2 regularity from the C2 entropy and its positive energy derivative; draft readiness does not constitute independent proof acceptance.

Differentials of the thermodynamic potentials: $d{\mathcal{U}}_c=Tds-pdV+{\sum }_j{\mu }_{j}d{n}_j$ on the same chart, conditional on [F3].

Euler identity for a differentiable degree-one homogeneous function: A C1 function with degree-one homogeneity along each local ray satisfies $f(x)=Df(x)[x]$.

Chain sum product and composition rules for Banach derivatives: Differentiating finite scalar products obeys the product rule.

Given: The local C2 potential charts and their exact differentials. Temperature is positive; derivatives are taken only inside the relevant open natural-coordinate chart.

Differentials of the thermodynamic potentials: The coefficient lists are $(T,-p,\mu )$ for $U$, $(T,V,\mu )$ for $H$, $(-s,-p,\mu )$ for $F$, $(-s,V,\mu )$ for $G$, and $(-s,-p,-n)$ for $\mathrm{\Omega }$.

Local thermodynamic Legendre potentials: The listed natural variables are local inverse coordinates only on the stated valid transform charts.

Clairaut--Schwarz theorem for continuous second partial derivatives: For a C2 scalar function on an open Euclidean set, every pair of mixed second coordinate partials is equal.

Given: A fixed composition, an explicit feasible endpoint set for a closed system plus its named reservoirs, equilibrium endpoints, fixed excluded kinetic and external-field energy changes, and the following ideal reservoir model. Energies are in joules, entropy in $\mathrm{J}{\mathrm{K}}^{-1}$, $T_0$ in kelvins, $V$ in ${\mathrm{m}}^3$, and $p_0$ in pascals.

Ideal thermal reservoir: its energy $E_R$ ranges in an open interval and its entropy is $S_R(E_R)=S_{R,0}+E_R\mathrm{/}{T}_0$ there. Heat $Q$ positive into the system changes its energy by $\mathrm{\Delta }{E}_R=-Q$, so $\mathrm{\Delta }{S}_R=-Q\mathrm{/}{T}_0$.

Ideal pressure store: at fixed external pressure $p_0$, its stored energy changes by $p_0\mathrm{\Delta }V$ when the system volume changes by $\mathrm{\Delta }V$; the only system work is $W_{\mathrm{o}\mathrm{n}}=-p_0\mathrm{\Delta }V$. The store's entropy is constant. At fixed volume this store is absent. No other work coordinate or matter transfer is admitted.

The system and bath, with the pressure store when present, form a complete isolated compound with fixed total energy. In the fixed-volume case $\mathrm{\Delta }U+\mathrm{\Delta }{E}_R=0$; in the fixed-pressure case $\mathrm{\Delta }U+\mathrm{\Delta }{E}_R+p_0\mathrm{\Delta }V=0$. These system energy balances are adopted ideal-environment constraints. Each operation includes an admitted core closed record of the complete compound as in Thermodynamic process records and transfer conventions, with positive fixed total mass, defined endpoint center-of-mass velocities in one fixed Galilean chart, and zero endpoint changes of compound center-of-mass kinetic and separately tracked external-field energies. The model supplies the full constituent and auxiliary endpoint sets with their energy and entropy maps, and an endpoint association ${\rho }_P:\{-,+\}\to \mathcal{F}$ into one exact feasible set, representing both aggregate equilibrium endpoints and their system, bath and store coordinates. Every named auxiliary is retained; the endpoint split is admitted by the additive rule required by Macroscopic second law for isolated compounds. Any additional retained auxiliary has constant total energy and entropy contributions throughout $\mathcal{F}$. On every tuple of $\mathcal{F}$, total energy is one fixed constant and total entropy is the sum of the system, bath and constant store or other entropy-neutral contributions. In the fixed-pressure case require $E_p-p_{0}V=C_p$ on every tuple of that same $\mathcal{F}$ for one fixed constant $C_p$; at fixed volume $V$ is fixed and the pressure store is absent. Thus the constraints used for global minimization hold throughout the feasible set, including disconnected components. No core-record assumption is imposed on the subsystem alone: its displayed energy balances are already adopted above.

Macroscopic second law for isolated compounds: on the admitted isolated core compound record with its full endpoint association, admitted additive split and zero excluded-energy changes specified in [A3], $\mathrm{\Delta }{S}_{\mathrm{t}\mathrm{o}\mathrm{t}}\ge 0$.

Stable equilibrium requires an attained global entropy maximum: Stable equilibrium is an attained global maximum of the sum of constituent entropies on the explicitly declared feasible set; uniqueness is not assumed.

Local thermodynamic Legendre potentials: The local Helmholtz and Gibbs potentials are $F=U-TS$ and $G=U-TS+pV$ on their valid natural-variable charts.

SI quantities and unit conventions: $U$, $Q$, and work transfers are in joules, $T$ in kelvins, $S$ in $\mathrm{J}{\mathrm{K}}^{-1}$, $p$ in pascals, and $V$ in ${\mathrm{m}}^3$, with $\mathrm{P}\mathrm{a}{\mathrm{m}}^3=\mathrm{J}$.

Given: A system $\sigma$ of fixed constitution $c$, an admitted record $P\in {\mathcal{P}}_{\mathrm{c}\mathrm{l}}(\sigma )$ with equilibrium endpoints $x_i,x_f$, $T_0>0$, $p_0>0$, and the ideal reservoir/work-store model below. The core-record conditions and fixed-chart bookkeeping are those of Thermodynamic process records and transfer conventions. Write $Q=Q[P]$ and $W_{\mathrm{o}\mathrm{n}}=W_{\mathrm{o}\mathrm{n}}[P]$. Use $W_{\mathrm{o}\mathrm{u}\mathrm{t}}>0$ for energy delivered by the system, in joules.

Ideal thermal reservoir of constitution $c_R$, with fixed volume and composition: its accessed equilibrium states have entropy $S_R(E_R)=S_{R,0}+E_R\mathrm{/}{T}_0$ on the accessed energy interval, where $E_R$ is its internal energy in joules and entropy is in $\mathrm{J}{\mathrm{K}}^{-1}$. Heat $Q$ into the system changes reservoir energy by $\mathrm{\Delta }{E}_R=-Q$.

Ideal pressure and work stores: the only work on the system is $W_{\mathrm{o}\mathrm{n}}=-p_0\mathrm{\Delta }V-W_{\mathrm{o}\mathrm{u}\mathrm{t}}$, where the pressure store has energy change $p_0\mathrm{\Delta }V$ and the work store receives $W_{\mathrm{o}\mathrm{u}\mathrm{t}}$. Both stores have constant entropy; no other work coordinate is admitted.

The system, bath, and both stores form a compound $C$ with an admitted record $\widehat{P}\in {\mathcal{P}}_{\mathrm{c}\mathrm{l}}(C)$ representing the same operation as $P$. Its outer boundary is isolated throughout, its constitution is fixed, and its preparations at both endpoints are equilibrium macrostates. For both records the center-of-mass kinetic and separately tracked external-field energy changes vanish. Supply store endpoint sets $Y_B,Y_W$, energy maps $E_B:Y_B\to \mathbb{R}\mathrm{J}$ and $E_W:Y_W\to \mathbb{R}\mathrm{J}$, and constant entropy maps $S_B^{\mathrm{a}\mathrm{u}\mathrm{x}}:Y_B\to \mathbb{R}$ and $S_W^{\mathrm{a}\mathrm{u}\mathrm{x}}:Y_W\to \mathbb{R}$, with values in $\mathrm{J}{\mathrm{K}}^{-1}$. The process model supplies ${\rho }_{\widehat{P}}:\{-,+\}\to \mathcal{F}\subseteq D_c\times D_{c_R}\times Y_B\times Y_W$ with ${\rho }_{\widehat{P}}(-)=(x_i,x_R^{-},y_B^{-},y_W^{-})$ and ${\rho }_{\widehat{P}}(+)=(x_f,x_R^{+},y_B^{+},y_W^{+})$. The store-energy increments are exactly those in [A2]. Here $\mathcal{F}$ is one declared feasible constituent endpoint set for the fixed totals, retained constraints and permitted exchanges of [F2]; one fixed total-energy value on $\mathcal{F}$ is $U_c(x)+U_{c_R}(x_R)+E_B(y_B)+E_W(y_W)$, including both stores from [A2]. The system and bath form an independent separated split admitted by the entropy-additivity rule required in [F2]. Every named auxiliary and work device is included in $C$; any device outside this two-constituent entropy split has explicitly constant endpoint entropy. Thus $S_{\mathrm{t}\mathrm{o}\mathrm{t}}(x,x_R,y_B,y_W)=S_c(x)+S_{c_R}(x_R)+S_B^{\mathrm{a}\mathrm{u}\mathrm{x}}+S_W^{\mathrm{a}\mathrm{u}\mathrm{x}}$ on $\mathcal{F}$ gives $\mathrm{\Delta }{S}_{\mathrm{t}\mathrm{o}\mathrm{t}}=\mathrm{\Delta }S+\mathrm{\Delta }{S}_R$.

First-law energy balance for closed process records: The adopted closed-record balance is $\mathrm{\Delta }U=Q+W_{\mathrm{o}\mathrm{n}}$, with positive $Q$ into and positive $W_{\mathrm{o}\mathrm{n}}$ on the system.

Macroscopic second law for isolated compounds: For an admitted core closed record of a compound isolated throughout, with fixed constitution and equilibrium endpoint tuples associated to the same declared feasible set, the entropy change of an admitted independent separated split is nonnegative. All named devices are included, omitted entropy contributions have declared zero change, and the kinetic and separately tracked external-field endpoint energy changes vanish.

Thermodynamic process records and transfer conventions: $Q$ and $W_{\mathrm{o}\mathrm{n}}$ are process-record transfers, not state functions; an adiabatic record has $Q=0$.

SI quantities and unit conventions: $U$, $Q$, and $W_{\mathrm{o}\mathrm{n}}$ are in joules, $V$ is in cubic metres, and $\mathrm{P}\mathrm{a}{\mathrm{m}}^3=\mathrm{J}$.

Local thermodynamic Legendre potentials: On a valid local $(T,p,n)$ inverse chart, the Gibbs potential is $G=U-Ts+pV$, with $s$ the entropy coordinate.

Given: A fixed-composition open C2 chart with $T,V>0$, the valid local pressure response inverse, and finite derivatives in the Statement.

Thermodynamic response functions on valid local charts: $C_V=(\mathrm{\partial }U\mathrm{/}\mathrm{\partial }T{)}_{V,n}$, $C_p=(\mathrm{\partial }H\mathrm{/}\mathrm{\partial }T{)}_{p,n}$, ${\kappa }_T=-V^{-1}(\mathrm{\partial }V\mathrm{/}\mathrm{\partial }p{)}_{T,n}$, and ${\alpha }_p=V^{-1}(\mathrm{\partial }V\mathrm{/}\mathrm{\partial }T{)}_{p,n}$; the definitions assert no response identities.

Differentials of the thermodynamic potentials: $dF=-sdT-pdV+{\sum }_j{\mu }_{j}d{n}_j$ and $dH=Tds+Vdp+{\sum }_j{\mu }_{j}d{n}_j$ on valid charts.

Maxwell relations on valid thermodynamic potential charts: $s_V{\mathrm{\mid }}_{T,n}=p_T{\mathrm{\mid }}_{V,n}$ on the Helmholtz chart.

The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$: Derivatives of compositions obey the total-derivative chain rule.

Chain sum product and composition rules for Banach derivatives: The derivative of a product of differentiable scalar maps is the sum of the two product terms.