Phases And Limits

Use $\mathbb{N}$ from The natural numbers $\mathbb{N}$ (von Neumann), which includes $0$, and define ${\mathbb{N}}_{>0}:=\mathbb{N}\setminus \{0\}$. A finite-state thermodynamic-limit contract specifies a sequence $(X_k,E_k,g_k,V_k,n_k{)}_{k\in {\mathbb{N}}_{>0}}$, where each $X_k$ is a finite nonempty state set, $E_k:X_k\to \mathbb{R}$ assigns finite energies in joules, $g_k:X_k\to (0,\mathrm{\infty })$ assigns positive dimensionless weights, $V_k>0$ is physical volume in ${\mathrm{m}}^3$, and, for one declared species, $n_k>0$ is its amount in moles. A fixed-density sequence has $V_k\to \mathrm{\infty }$ and $n_k\mathrm{/}{V}_k\to \rho \in (0,\mathrm{\infty })$ in $\mathrm{m}\mathrm{o}\mathrm{l}{\mathrm{m}}^{-3}$. Boundary conditions, interaction rules, and the choice of a volume or amount normalization are part of the contract and must be held fixed or explicitly tracked as $k$ varies.

For $T>0$, set $\beta :=(k_{B}T{)}^{-1}$ in ${\mathrm{J}}^{-1}$, with $k_B$ and units from SI quantities and unit conventions. The finite-state canonical partition sum is

as the finite case of Countable canonical model. It is a positive finite real number. Define $F_k(\beta ):=-{\beta }^{-1}\log Z_k(\beta )$ in joules and $f_k(\beta ):=F_k(\beta )\mathrm{/}{V}_k$ in $\mathrm{J}{\mathrm{m}}^{-3}$. The specified thermodynamic-limit value is $f(\beta ,\rho ):={\lim }_{k\to \mathrm{\infty }}{f}_k(\beta )$ if this limit exists. A limit per mole or per lattice site is a different normalization and must be stated separately.

This is a contract for what a limit claim must specify, not a theorem that the limit exists. It asserts no uniformity in $\beta$, derivative interchange, ensemble equivalence, convergence rate, or nonanalyticity. A finite model's regularity and any limiting singularity require their own arguments; neither is inferred from the word “thermodynamic limit.”

Fix a closed system $s$ of constitution $c$, with equilibrium domain $D_c$ from Equilibrium state domains, and fix $T_0>0$ and $p_0>0$. Define, for $x\in D_c$,

where $U_c,V_c$ are the dimensioned energy and volume maps of Equilibrium state domains and $S_c$ is the adopted entropy of Concave macroscopic fundamental entropy. Thus ${\mathrm{\Phi }}_{T_0,p_0}$ has units joules.

Let $P_s\in {\mathcal{P}}_{\mathrm{c}\mathrm{l}}(s)$ be an admitted closed system process record as defined in Thermodynamic process records and transfer conventions, with system endpoints $x_{P_s}^{-},x_{P_s}^{+}\in D_c$. Let $R$ be its ideal thermal reservoir and $B$ its ideal pressure store; both lie inside a declared isolated-compound outer boundary. Let $\widehat{P}\in {\mathcal{P}}_{\mathrm{c}\mathrm{l}}(s\bigsqcup R\bigsqcup B)$ be an admitted core closed process record for that isolated compound, whose supplied total mass is positive, whose endpoint center-of-mass velocities are defined in one fixed Galilean chart, and whose compound center-of-mass kinetic and separately tracked external-field potential energies have zero endpoint change. These are conditions on the whole compound, additional to the system-record conditions. By an extension, require that $\widehat{P}$ has the same time interval, system endpoints, system configuration path, and system-side protocol and transfer records as $P_s$, together with the reservoir and store records. Write its compound endpoint association as

where $Y_R,Y_B$ are the model-declared reservoir and store equilibrium state sets, represented as admitted constituent state domains (or subsets thereof) in the second-law endpoint split; $E_R,E_B$ and their entropies are the corresponding dimensioned state maps. The split is admitted as independent separated macroscopic systems by Concave macroscopic fundamental entropy. The set $\mathcal{F}$ is the exact feasible endpoint set from Isolated compound and feasible equilibrium set. The system protocol assigns heat $Q[P_s]$ and work $W_{\mathrm{o}\mathrm{n}}[P_s]$ to $P_s$; the auxiliary endpoint changes in $\widehat{P}$ satisfy the reservoir and store relations in [A1]–[A2] below. No matter crosses the compound boundary, no work channel other than pressure-store work acts on the system, and no other energy-coordinate changes are omitted.

Then the system endpoint difference obeys

If the final compound endpoint ${\rho }_{\widehat{P}}(+)$ is an attained global entropy maximum on $\mathcal{F}$, then $x_{P_s}^{+}$ attains a global minimum of ${\mathrm{\Phi }}_{T_0,p_0}$ on the system projection

This is a conditional model deduction; it does not assert that arbitrary fixed-$T,p$ reactions have this preparation or that the system reaches the minimum dynamically.

The system process record and its isolated-compound extension are distinct records. The former carries $Q$ and $W_{\mathrm{o}\mathrm{n}}$ in the first-law balance; the latter contains the reservoir and store endpoint changes used by the second law and the global entropy maximum. The argument does not cover an open matter reservoir, which would contribute chemical-potential terms, or an external work channel not included in [A2].

Fix a constitution $c$ with $C\ge 1$ named species and its equilibrium state domain $D_c\subseteq {\mathbb{R}}^{C+2}$ from Equilibrium state domains. An interior composition is represented by

with $x_C:=1-{\sum }_{i<C}{x}_i>0$; for $C=1$, ${\mathbb{R}}^0=\{0\}$ and $x_1=1$.

A phase branch $\alpha$ is model data consisting of an open chart $W_{\alpha }\subseteq (0,\mathrm{\infty })\times \mathbb{R}\times {\mathrm{\Delta }}_{C-1}^{\circ }\times (0,\mathrm{\infty })$ with coordinates $(T,p,x,m)$ and a $C^1$ diffeomorphism ${\sigma }_{\alpha }:W_{\alpha }\to U_{\alpha }$ onto an open subset $U_{\alpha }\subseteq {\mathbb{R}}^{C+2}$ contained in the equilibrium state domain $D_c$. Here $T$ is in kelvins, $p$ is in pascals, $m$ is total amount in moles, and $C^1$ means continuous first coordinate derivatives as in $C^k$ Euclidean maps and diffeomorphisms. The chart dimension is $C+2$, matching the ambient dimension of $D_c$. The map ${\sigma }_{\alpha }$ takes values in numerical state coordinates; its physical SI state map is explicitly ${\chi }_{\alpha }={\mathrm{\Phi }}_c\circ {\sigma }_{\alpha }$, where ${\mathrm{\Phi }}_c$ is the numerical-to-SI unit map of Entropic temperature, pressure, and molar chemical potentials. Thus the physical amounts at this point are $n_j=m{x}_j$, and the numerical amount coordinates in ${\sigma }_{\alpha }$ are $n_j\mathrm{/}(1\mathrm{m}\mathrm{o}\mathrm{l})$. Derivative-defined physical state functions are evaluated on ${\chi }_{\alpha }(T,p,x,m)$, or equivalently pulled back along ${\mathrm{\Phi }}_c$ to ${\sigma }_{\alpha }(T,p,x,m)$. At ${\sigma }_{\alpha }(T,p,x,m)$ in this precise pullback sense, the derivative-defined state functions of Entropic temperature, pressure, and molar chemical potentials give temperature $T$, pressure $p$, and mole fractions $x_j=n_j\mathrm{/}m$; here $n_{\mathrm{t}\mathrm{o}\mathrm{t}}:={\sum }_{i=1}^C{n}_i=m$ uses the finite-sum convention of Finite sums and finite products, by recursion. An intensive phase branch additionally has each derivative-defined chemical potential ${\mu }_j(T,p,x,m)$ independent of $m$ throughout $W_{\alpha }$; write its value as ${\mu }_j^{\alpha }(T,p,x)$. This scale invariance is an explicit constitutive model property, not a consequence of Additive separated systems and tagged bulk scaling. A valid local response chart and its inverse are as in Thermodynamic response functions on valid local charts and Local thermodynamic Legendre potentials. Existence of $W_{\alpha }$ or ${\sigma }_{\alpha }$ is a model premise, not a consequence of the equations of thermodynamics.

On a branch with a valid natural-variable chart, $G^{\alpha }$ denotes the Gibbs potential of Local thermodynamic Legendre potentials restricted to the branch. For a one-component branch with amount $m_{\alpha }>0$, define $g_{\alpha }:=G^{\alpha }\mathrm{/}{m}_{\alpha }$, $s_{\alpha }:=S_{\alpha }\mathrm{/}{m}_{\alpha }$, and $v_{\alpha }:=V_{\alpha }\mathrm{/}{m}_{\alpha }$, in $\mathrm{J}\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}$, $\mathrm{J}\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$, and ${\mathrm{m}}^3\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}$, respectively. These quotients need not be independent of amount, and $g_{\alpha }={\mu }^{\alpha }$ is not assumed by this definition. Such an equality requires a separate same-tag homogeneity hypothesis on the branch.

A phase mixture of a finite list of branches is a feasible separated compound whose positive phase amounts $m_{\alpha }$ have fractions ${\lambda }_{\alpha }:=m_{\alpha }\mathrm{/}{\sum }_{\beta }{m}_{\beta }>0$ with ${\sum }_{\alpha }{\lambda }_{\alpha }=1$. Its extensive coordinates and entropy are those supplied by Additive separated systems and tagged bulk scaling and Concave macroscopic fundamental entropy, on the exact feasible set of Isolated compound and feasible equilibrium set. The mixture is in coexistence when it is an attained global entropy-maximizing endpoint under the declared totals and permitted exchanges, as in Stable equilibrium requires an attained global entropy maximum. This definition does not assert that a maximum exists, that all branches can exchange every coordinate, or that metastable states are global equilibria.

The bulk scaling postulate may send a material tag $c$ to a different tag $c^{(\lambda )}$. It therefore does not supply same-tag degree-one homogeneity of $G^{\alpha }$ or $U^{\alpha }$ on a phase branch. Any result using $g_{\alpha }={\mu }^{\alpha }$ must state that additional hypothesis.

Let $P\ge 1$ named phase branches of the same finite species list occupy a separated compound at an attained global entropy-maximizing endpoint with the fixed totals and exact feasible set declared by Phase branches and coexistence domains and Isolated compound and feasible equilibrium set. Assume each phase lies in an open C2 single-phase chart with finite positive temperature. For every pair of phases, assume the following about the feasible set at this endpoint: each named exchange channel permits a sufficiently small nonzero amount of transfer in both directions, while all other state coordinates may be held fixed.

Then, for every pair $\alpha ,\beta$:

• if energy exchange is permitted, $T_{\alpha }=T_{\beta }$;
• if volume exchange is permitted, $p_{\alpha }\mathrm{/}{T}_{\alpha }=p_{\beta }\mathrm{/}{T}_{\beta }$, and if energy exchange is also permitted then $p_{\alpha }=p_{\beta }$;
• if species $j$ is transferable, ${\mu }_{\alpha ,j}\mathrm{/}{T}_{\alpha }={\mu }_{\beta ,j}\mathrm{/}{T}_{\beta }$, and if energy exchange is also permitted then ${\mu }_{\alpha ,j}={\mu }_{\beta ,j}$.

The pressure and chemical potentials are the derivative-defined state functions of Entropic temperature, pressure, and molar chemical potentials. The theorem makes no equality claim for a retained coordinate, a one-sided boundary exchange, or a channel absent from the declared feasible set. It is a model deduction, not evidence that these phases or exchange channels occur in nature.

Fix $C\ge 1$ species and $R\ge 1$ declared internal reaction extents. Let $\nu \in {\mathbb{R}}^{C\times R}$ be the dimensionless stoichiometric matrix, let $n^0\in [0,\mathrm{\infty }{)}^C$ have units mol, and let $\xi \in {\mathbb{R}}^R$ have units mol. The amount vector is $n(\xi )=n^0+\nu \xi$, and the nonnegative-amount feasible extent set is

Assume the exact local reaction-feasible extent set is ${\mathcal{F}}_{\nu }\cap U$; no additional local constraint removes any direction in its tangent cone. Fix $T_0>0$ and $p_0>0$ in a declared natural-variable chart. On an open neighborhood $U$ of a candidate endpoint ${\xi }_{\ast }$, let $\mathrm{\Phi }:U\to \mathbb{R}$ be a finite $C^1$ constrained Gibbs-potential extension, in joules, satisfying

where every ${\mu }_j$ is in $\mathrm{J}\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}$. At positive amounts, require $\mathrm{\Phi }(\xi )=U_c(x(\xi ))+p_0{V}_c(x(\xi ))-T_0{S}_c(x(\xi ))$ for the system state $x(\xi )$, with $U_c,V_c,S_c$ from Equilibrium state domains and Concave macroscopic fundamental entropy; its derivative is that of the local Gibbs chart supplied by Local thermodynamic Legendre potentials and Differentials of the thermodynamic potentials.

If all amounts at ${\xi }_{\ast }$ are positive, assume ${\xi }_{\ast }$ is the system endpoint of a closed system process record and its admitted isolated-compound extension exactly as in Fixed-temperature and fixed-pressure potential minimum with internal reactions, is a global entropy-maximizing endpoint on that extension's exact feasible set, and the record's system projection contains every state $x(\xi )$ for $\xi \in {\mathcal{F}}_{\nu }\cap U$. If at least one amount is zero, it is outside Equilibrium state domains; in that boundary case, assume instead that the model supplies a boundary-inclusive state domain with finite $C^1$ maps ${\bar{U}}_c,{\bar{V}}_c,{\bar{S}}_c:U\to \mathbb{R}$ in joules, ${\mathrm{m}}^3$, and $\mathrm{J}{\mathrm{K}}^{-1}$, respectively, that $\mathrm{\Phi }(\xi )={\bar{U}}_c(\xi )+p_0{\bar{V}}_c(\xi )-T_0{\bar{S}}_c(\xi )$, and that its exact isolated-compound model has fixed total energy, an ideal reservoir with $S_R(E_R)=S_{R,0}+E_R\mathrm{/}{T}_0$, a constant-entropy pressure store satisfying $E_B-p_0{\bar{V}}_c(\xi )=C_B$, and a feasible-set projection equal to ${\mathcal{F}}_{\nu }\cap U$. Assume ${\xi }_{\ast }$ is an attained global entropy maximum on that boundary-inclusive set. This additional boundary model premise is not supplied by the positive-mole state chart or by Fixed-temperature and fixed-pressure potential minimum with internal reactions.

Under the relevant interior or boundary assumptions above, ${\xi }_{\ast }$ locally minimizes $\mathrm{\Phi }$ on ${\mathcal{F}}_{\nu }\cap U$. Let $J_{\ast }:=\{j:n_j^0+(\nu {\xi }_{\ast }{)}_j=0\}$ be the active species set. Then the first-order necessary condition is

where

If ${\xi }_{\ast }$ is interior to ${\mathcal{F}}_{\nu }$, this reduces to ${\nu }^{\mathsf{T}}\mu ({\xi }_{\ast })=0$. If, in addition, $\mathrm{\Phi }$ is convex and differentiable on a convex neighborhood containing the relevant feasible set, the same tangent inequality is sufficient for global minimization there.

For the interior route, the potential minimum follows from the admitted process record and its reservoir/store model. A boundary point with a zero species amount is not in the positive-mole chart. The boundary route therefore needs the explicitly extended entropy and endpoint model in [A2]; a finite C1 extension of $\mathrm{\Phi }$ alone would not establish physical stable equilibrium.

Let $\alpha ,\beta$ be two one-component phase branches from Phase branches and coexistence domains on a common open chart $O\subset (0,\mathrm{\infty })\times \mathbb{R}$ with coordinates $(T,p)$. Assume each branch has a local $C^2$ energy representation ${\mathcal{U}}_i(s,V,n)$, $i\in \{\alpha ,\beta \}$, and that within each fixed branch tag it satisfies the same-tag homogeneity

for all $\lambda$ in a neighborhood of $1$ for which the scaled state remains in the chart. This is a phase-branch hypothesis: the general tagged law of Additive separated systems and tagged bulk scaling does not imply it. It may be stipulated independently on each local branch, or supplied by the common-law bulk subclass of Euler and Gibbs-Duhem identities under same-tag homogeneity when each phase's local inverse states lie in its designated entropy cone with fixed species and intensive parameters and common SI units and energy/entropy reference conventions. That supplied subclass uses local inverse uniqueness and does not require a global energy inverse.

Suppose a state $(T_0,p_0)\in O$ is an attained interior coexistence equilibrium with bilateral energy, volume, and species exchange between the two branches, and both branches admit the valid local Gibbs charts of Local thermodynamic Legendre potentials. Write $g_i=G_i\mathrm{/}{n}_i$, $s_i=S_i\mathrm{/}{n}_i$, and $v_i=V_i\mathrm{/}{n}_i$ in the units specified by SI quantities and unit conventions. If

then, on some open interval about $T_0$, the equal-Gibbs-per-mole coexistence condition $g_{\alpha }(T,p(T))=g_{\beta }(T,p(T))$ defines a unique local $C^1$ curve and

If, in addition, conversion from $\alpha$ to $\beta$ is admitted as a reversible process in contact with an ideal thermal reservoir at the same fixed $T$, and the full finite isolated compound (converted phases, reservoir, work stores and all auxiliaries) has equilibrium endpoints with additive component entropy, with every component other than the converted phases and that reservoir having zero entropy change, and the admitted forward and reverse complete-compound records have nonnegative total endpoint entropy changes for that same additive split with their full endpoint tuples interchanged, then its molar latent heat absorbed by the system is ${\mathrm{\ell }}_{\alpha \to \beta }=T(s_{\beta }-s_{\alpha })$. These are conditional model deductions. The slope conclusion is not asserted when $\mathrm{\Delta }v=0$ or where the branch charts are not $C^1$.

The statement's same-tag homogeneity is deliberately local to each branch. The Euler supplier preserves the general implication from independently stipulated local homogeneity and also supplies it on the separately adopted common-law bulk branch. The general tagged postulate still allows $c^{(\lambda )}\mathrm{\ne }c$; neither constitutive route is a proof of exact extensivity for every finite body, surface-sensitive or long-range system, or gravitating system. Valid Gibbs charts, admitted bilateral exchanges, positive amounts, the nonzero molar-volume jump, and the reversible ideal-reservoir qualification remain essential.

Let $d\ge 1$ and let $K\subseteq {\mathbb{R}}^d$ be a nonempty compact set of dimensionless generator coordinates. Each $x\in K$ labels one model-specified phase-generator state; the model assigns it entropy per mole $s(x)\in \mathrm{J}\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$, and $s:K\to \mathbb{R}$ is continuous, using the numerical entropy value in that fixed unit. Define $\mathrm{conv}K$ to be the set of all finite convex combinations of points of $K$, and fix $\bar{x}\in \mathrm{conv}K$. Use the Borel sigma algebra relative to $K$. A Borel probability mixture is a countably additive probability measure $P$ on that sigma algebra; its barycenter and entropy per mole are the coordinatewise integrals ${\int }_{K}xdP$ and ${\int }_{K}sdP$. For $x\in K$, define the unit point mass ${\delta }_x$ by ${\delta }_x(A)=1$ if $x\in A$ and ${\delta }_x(A)=0$ otherwise, for Borel $A\subseteq K$. Assume every Borel probability mixture with barycenter $\bar{x}$ is feasible and has entropy per mole ${\int }_{K}sdP$. Then there are $m\le d+1$, points $x_1,\dots ,x_m\in K$, and weights ${\theta }_i\ge 0$ with ${\sum }_{i=1}^m{\theta }_i=1$ such that $P_{\ast }:={\sum }_{i=1}^m{\theta }_i{\delta }_{x_i}$ is a feasible Borel probability mixture and ${\int }_{K}sd{P}_{\ast }\ge {\int }_{K}sdP$ for every feasible mixture $P$.

Here integrals use the measure convention of Measure space and integral. The entropy map is part of the model; this theorem derives neither its microscopic meaning nor compactness of a physical phase domain.