Entropy Foundation

Let $P=(I_P,x_P^{-},x_P^{+},z_P,{\kappa }_P,r_P)$ be an admitted closed process record as defined in Thermodynamic process records and transfer conventions, with equilibrium endpoints $x_P^{-},x_P^{+}\in D_c$. The record is reversible relative to its declared boundary and protocol when the model also admits a reverse record $P^{\mathrm{r}\mathrm{e}\mathrm{v}}$ for the same system and material constitution, starting at $x_P^{+}$ and ending at $x_P^{-}$, such that every auxiliary and device state specified by the protocol is restored to its initial state and every signed boundary transfer is reversed. In particular,

$Q[P^{\mathrm{r}\mathrm{e}\mathrm{v}}]=-Q[P],W_{\mathrm{o}\mathrm{n}}[P^{\mathrm{r}\mathrm{e}\mathrm{v}}]=-W_{\mathrm{o}\mathrm{n}}[P]\mathrm{.}$

If the protocol records additional transfer channels, their signed values must likewise be negatives of their forward values. A record that is slow is not reversible merely because it is slow; it must satisfy the reverse-record condition above. Reversibility is therefore relative to what the model includes inside the boundary and which reverse controls it admits.

For each material constitution $c$, let $D_c$ be the equilibrium state domain from Equilibrium state domains. A thermal-contact family is a model-declared set $\mathcal{T}$ of tagged pairs $a=(c_A,x_A)$ with $x_A\in D_{c_A}$. Its tag and coordinate projections are $\mathrm{tag}(a)=c_A$ and $\mathrm{coord}(a)=x_A$; equal numerical coordinate tuples with different constitution tags are not identified.

Fix two members $a=(c_A,x_A)$ and $b=(c_B,x_B)$, named constituents $A,B$ of those constitutions, and a declared contact-protocol convention. A thermal-contact protocol is an admitted preparation/process record $P$ of their compound system $A\bigsqcup B$, with the configuration curve $z_P:I_P\to Z_{A\bigsqcup B}$ and equilibrium endpoint configurations specified in Thermodynamic process records and transfer conventions. The contact model supplies constituent endpoint-coordinate maps

${\xi }_A:Z_{A\bigsqcup B}^{\mathrm{e}\mathrm{q}}\to D_{c_A},{\xi }_B:Z_{A\bigsqcup B}^{\mathrm{e}\mathrm{q}}\to D_{c_B},$

and requires ${\xi }_A(z_P(t_0))=x_A$ and ${\xi }_B(z_P(t_0))=x_B$. These constituent maps are additional declared model data; they are not inferred from the compound's coordinate tuple. Write $x_A^{+}={\xi }_A(z_P(t_1))$ and $x_B^{+}={\xi }_B(z_P(t_1))$ for the constituent final coordinates. The two constituents are separated by a declared diathermal contact: matter transfer and mechanical or composition work across that contact are forbidden, while the protocol permits thermal energy transfer between the constituents. The compound's external boundary and all retained constraints are specified as in Thermodynamic systems and permitted exchanges and Thermodynamic process records and transfer conventions. No continuity or differentiability of the configuration curve is added here.

Let ${\mathcal{P}}_T(A,B)$ be the set of contact records admitted by the fixed convention. The model supplies an internal transfer map $q_{A\to B}:{\mathcal{P}}_T(A,B)\to \mathbb{R}$, whose values are numerical magnitudes in joules: $q_{A\to B}[P]>0$ means net thermal energy transferred from $A$ to $B$ across this contact during $I_P$. Set $q_{B\to A}[P]:=-q_{A\to B}[P]$. Zero net internal heat transfer means exactly $q_{A\to B}[P]=0$; it does not assert zero transfer at every intermediate time. This map is additional protocol data, distinct from the compound's external-boundary map $Q$ in Thermodynamic process records and transfer conventions. In particular, an isolated compound may have nonzero $q_{A\to B}$ while its external $Q$ is zero. Identifying this internal map with constituent heat records requires separately supplied constituent records and consistent boundary assignments.

For each tested ordered pair $(a,b)$, the convention specifies one admitted contact preparation $P_{a,b}\in {\mathcal{P}}_T(A,B)$ with the initial coordinates above, and a subset ${\mathcal{E}}_T(A,B)\subseteq {\mathcal{P}}_T(A,B)$ designated as equilibrium contact preparations. These are model inputs, not an equilibration theorem. The pair is in thermal contact equilibrium, written $a{\sim }_{T}b$, exactly when its selected record $P_{a,b}$ belongs to ${\mathcal{E}}_T(A,B)$, has $x_A^{+}=x_A$ and $x_B^{+}=x_B$, and satisfies $q_{A\to B}[P_{a,b}]=0$. Untested pairs are not in the relation. This criterion is protocol-dependent: changing the selected preparation or retained constraints can change the relation. For fixed constitutions, coordinate notation $x_A{\sim }_T{x}_B$ abbreviates $(c_A,x_A){\sim }_T(c_B,x_B)$ using the two specified tag embeddings. It does not define a relation on untagged coordinates across unspecified material types. The relation is defined only on the stated family $\mathcal{T}$ and for the declared contact protocols. This definition does not assert that ${\sim }_T$ is reflexive, symmetric, or transitive, and it does not assign a real-valued temperature to its equivalence classes. Lieb–Yngvason §IV.A instead defines thermal equilibrium using adiabatic equivalence with a thermal join; the present record-based criterion is a separately adopted convention.

For each admitted material constitution $c$, the model adopts a finite scalar function

$S_c:D_c⟶\mathbb{R}\text{ with values measured in }\mathrm{J}{\mathrm{K}}^{-1},$

on the equilibrium state domain $D_c$ of Equilibrium state domains. This framework treats $S_c$ as primitive constitutive data: its existence and form are assumed, not derived from the word entropy, statistical mechanics, or experimental evidence.

On every specified convex region $C\subseteq D_c$ in the dimensionless coordinates of Equilibrium state domains, $S_c$ is continuous and concave, meaning that for all $x,y\in C$ and every $\alpha \in [0,1]$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length),

$S_c((1-\alpha )x+\alpha y)\ge (1-\alpha )S_c(x)+\alpha S_c(y)\mathrm{.}$

The domain assumptions are local to the material and region named; no unlisted coordinate tuple is presumed physically realizable. The input of $S_c$ is the represented state ${\pi }_c(e)$; when the function is written using physical coordinates $(U,V,n_1,\dots ,n_r)$, those coordinates mean the SI values whose dimensionless numerical tuple is ${\pi }_c(e)$. For independent separated macroscopic systems $A,B$ admitted by Additive separated systems and tagged bulk scaling, entropy is additive under that postulate's composition map ${\kappa }_{AB}$:

$S_{A\bigsqcup B}({\kappa }_{AB}(x_A,x_B))=S_A(x_A)+S_B(x_B)\mathrm{.}$

On each separately designated open, single-phase chart in the physical coordinates $(U,V,n_1,\dots ,n_r)$, assume the numerical representative of $S_c$ is $C^2$ in the sense of $C^k$ Euclidean maps and diffeomorphisms: its first and second coordinate partial derivatives exist and are continuous. The partial derivative $S_U$ at fixed $V,n_1,\dots ,n_r$ is the ordinary partial derivative in the $U$ coordinate, as in Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$, and is assumed finite and strictly positive. The numerical chart input uses $\widehat{U}=U\mathrm{/}(1\mathrm{J})$; consequently $S_U=(1\mathrm{J}{)}^{-1}\mathrm{\partial }{S}_c\mathrm{/}\mathrm{\partial }\widehat{U}$ and has units ${\mathrm{K}}^{-1}$. The other physical coordinates are similarly represented using the fixed SI units of SI quantities and unit conventions. No differentiability, strict concavity, or positive $S_U$ is asserted outside those charts. This item does not itself define a thermometer calibration or claim a temperature law.

For the tagged bulk family with scaling maps ${\delta }_{\lambda }$ and domains $B_{c,\lambda }$ from Additive separated systems and tagged bulk scaling, also adopt, for every admitted $\lambda >0$ and $x\in B_{c,\lambda }$,

$S_{c^{(\lambda )}}({\delta }_{\lambda }x)=\lambda S_c(x)\mathrm{.}$

This is a bulk homogeneity assumption for tagged scaled systems, not exact finite-size scaling for every body. Lieb–Yngvason develop a different accessibility-based route in which entropy is represented from additional operational hypotheses; this framework does not claim that their derivation proves the present primitive entropy postulate.

In addition to the general tagged bulk family of Additive separated systems and tagged bulk scaling and its entropy law in Concave macroscopic fundamental entropy, designate a common-constitutive-law bulk branch as follows. Fix a material constitution $c$ with an ordered species list and fixed intensive constitutive parameters. Let $C\subseteq D_c$ be a nonempty open single-phase chart in the fixed dimensionless SI numerical coordinates of Equilibrium state domains and SI quantities and unit conventions, and assume it is a positive cone: $\lambda x\in C$ for all $x\in C$ and $\lambda >0$. Use the adopted $C^2$ entropy with $S_{c,U}>0$ on this chart.

For every $\lambda >0$, admit scaling of every $x\in C$, so $C\subseteq B_{c,\lambda }$, and retain the distinct scaled-system tag $c^{(\lambda )}$. The image $C^{(\lambda )}:={\delta }_{\lambda }(C)$ is a chart in $D_{c^{(\lambda )}}$. Adopt a coordinate-preserving material-law transport $j_{c,\lambda }:C^{(\lambda )}\to C$ defined in numerical coordinates by $j_{c,\lambda }(y)=y$, and adopt entropy preservation under that transport:

$S_{c^{(\lambda )}}(y)=S_c(j_{c,\lambda }(y))(y\in C^{(\lambda )})\mathrm{.}$

The transport identifies the constitutive numerical law used for bodies in this branch; it does not equate their physical identities or material tags. It presumes a common ordered species list, units, internal-energy reference and entropy calibration. Fixed material parameters in this branch exclude a hidden size parameter whose independent variation changes the law. Combined with the existing tagged entropy law, this adopted transport gives the same-function relation

$S_c(\lambda x)=S_{c^{(\lambda )}}({\delta }_{\lambda }x)=\lambda S_c(x)(x\in C,\lambda >0)\mathrm{.}$

This is additional idealized constitutive information for the designated bulk branch. It is not implied by coordinate additivity, SI units, the general tagged relation, or the existence of local equilibrium charts. No empirical proof, finite-size error bound, or assertion for surface-sensitive, interacting long-range or self-gravitating bodies is supplied. Euler and Gibbs–Duhem identities are conditional deductions on this branch, not part of the postulate's source review.

Fix a finite list of named subsystems $s_1,\dots ,s_m$, where $m\in \mathbb{N}$ and $m\ge 1$ (The natural numbers $\mathbb{N}$ (von Neumann)), each having its own equilibrium state domain $D_{c_i}$ as defined in Equilibrium state domains. Their separated endpoint state space is the Cartesian product (The Cartesian product $A\times B:=\{z\in \mathcal{P}(\mathcal{P}(A\cup B)):\mathrm{\exists }a\in A\mathrm{\exists }b\in Bz=(a,b)\}$)

$\mathcal{D}:=D_{c_1}\times \cdots \times D_{c_m}\mathrm{.}$

If the boundary also includes named auxiliaries or work devices $d_1,\dots ,d_k$ not already among the $s_i$, where $k\in \mathbb{N}$, supply endpoint state sets $Y_j$ and energy maps $E_j:Y_j\to \mathbb{R}$ with numerical values interpreted in joules. Supply also every coordinate map needed for their matter, volume, retained constraints and separately tracked mechanical or external-field energy. The complete endpoint space is

${\mathcal{D}}_{\mathrm{a}\mathrm{l}\mathrm{l}}:=\mathcal{D}\times Y_1\times \cdots \times Y_k\mathrm{.}$

For $k=0$ use ${\mathcal{D}}_{\mathrm{a}\mathrm{l}\mathrm{l}}=\mathcal{D}$. Write an endpoint as $(\mathbf{x},\mathbf{y})$ and obtain constituent states by coordinate projection. In the separated additive energy model, declare $E_{\mathrm{t}\mathrm{o}\mathrm{t}}(\mathbf{x},\mathbf{y})={\sum }_{i=1}^m{U}_{c_i}(x_i)+{\sum }_{j=1}^k{E}_j(y_j)\mathrm{J}$, with finite sums as in Finite sums and finite products, by recursion and the empty auxiliary sum zero. Every tracked endpoint energy contribution must be included in these maps or in an additional explicitly supplied contribution; an interacting model must supply its total-energy map on ${\mathcal{D}}_{\mathrm{a}\mathrm{l}\mathrm{l}}$. For each entropy-bearing auxiliary supply $S_j^{\mathrm{a}\mathrm{u}\mathrm{x}}:Y_j\to \mathbb{R}$ with values in $\mathrm{J}{\mathrm{K}}^{-1}$. If the complete separated split has an adopted additive entropy model, its total includes those maps as well as the $S_{c_i}$; omitting an auxiliary requires an explicitly constant entropy contribution or a supplied reduced total-entropy function.

An isolated compound consists of all these constituents inside a declared outer boundary, with no matter or energy transfer across that boundary during the admitted process, in the sense of Thermodynamic systems and permitted exchanges. Internal transfers may change auxiliary energy and entropy. On pairs admitted to the simple aggregate composition map of Additive separated systems and tagged bulk scaling, totals and entropy satisfy that postulate and Concave macroscopic fundamental entropy. A retained partition endpoint in ${\mathcal{D}}_{\mathrm{a}\mathrm{l}\mathrm{l}}$ is not automatically such an admitted aggregate state: summing its coordinates can discard partition information. For endpoints outside that composition's admitted set, total-energy and total-entropy maps must be supplied separately; no aggregate entropy identity is inferred.

For specified totals and retained barriers, an equilibrium feasible set is the model-declared subset $\mathcal{F}\subseteq {\mathcal{D}}_{\mathrm{a}\mathrm{l}\mathrm{l}}$ of complete endpoint tuples satisfying those totals and exactly the exchange permissions in the declared protocol. Its thermodynamic projection is $\mathrm{\Pi }(\mathcal{F}):=\{\mathbf{x}\in \mathcal{D}:\mathrm{\exists }\mathbf{y}\text{ with }(\mathbf{x},\mathbf{y})\in \mathcal{F}\}$; that projection alone does not record variable auxiliary states, energy or entropy. When $k=0$ the original convention $\mathcal{F}\subseteq \mathcal{D}$ applies. Totals use all supplied endpoint maps and the dimensioned coordinate maps $U_c,V_c,n_{c,j}$ of Equilibrium state domains, rather than treating numerical coordinates as quantities with units. For two constituents without auxiliaries, thermal contact through Thermal contact family and constraints at fixed volumes and mole amounts gives the model-declared allowed energy partitions with prescribed total energy and any further retained constraints. Volume or species exchange changes $\mathcal{F}$ only when explicitly permitted; retained walls fix the corresponding coordinates.

The set $\mathcal{F}$ may be empty, noncompact, or otherwise fail to contain an entropy-maximizing state. This definition imposes no existence, compactness, convexity, uniqueness, or attainment assumption.

Let $\mathcal{F}$ be the explicitly declared feasible endpoint set of an isolated compound with the chosen independent separated macroscopic subsystem split and any explicitly modeled named auxiliaries, as defined in Isolated compound and feasible equilibrium set. For this route, adopt the physical criterion that any endpoint tuple the model classifies as a stable equilibrium under these fixed totals, retained partitions, and permitted exchanges satisfies

$S_{\mathrm{t}\mathrm{o}\mathrm{t}}(x^{\mathrm{e}\mathrm{q}})={\max }_{x\in \mathcal{F}}{S}_{\mathrm{t}\mathrm{o}\mathrm{t}}(x),S_{\mathrm{t}\mathrm{o}\mathrm{t}}(x):={\sum }_{i=1}^m{S}_{c_i}(x_i)+{\sum }_{j=1}^k{S}_j^{\mathrm{a}\mathrm{u}\mathrm{x}}(y_j),$

Here $\max$ means that $x^{\mathrm{e}\mathrm{q}}\in \mathcal{F}$ and $S_{\mathrm{t}\mathrm{o}\mathrm{t}}(x^{\mathrm{e}\mathrm{q}})\ge S_{\mathrm{t}\mathrm{o}\mathrm{t}}(x)$ for every $x\in \mathcal{F}$, using maximum in the sense of Maximum and minimum of a set. Here $x=(\mathbf{x},\mathbf{y})$ is a complete endpoint tuple from Isolated compound and feasible equilibrium set; $k=0$ gives the original thermodynamic-only formula. The finite sums are as in Finite sums and finite products, by recursion. The macroscopic summands are those of the separated constituents under Concave macroscopic fundamental entropy, and any auxiliary summands are the explicitly adopted endpoint entropy maps of the same complete compound model. This route assumes that these contributions add for the complete admitted split. A constant auxiliary entropy may be dropped as an additive constant when comparing maxima, whereas variable auxiliary entropy must be retained. Whenever the stable-equilibrium label is used, attainment is assumed: at least one maximizing endpoint lies in $\mathcal{F}$. If $\mathcal{F}$ is empty or the entropy supremum is not attained, this item supplies no equilibrium state.

“Global” refers to every endpoint in this exact $\mathcal{F}$, not just nearby states, states reachable by a restricted short-time dynamics, or states on a selected branch. A local maximum that is not global, or a long-lived preparation trapped by kinetic barriers, is not declared stable equilibrium by this postulate. No uniqueness or relaxation dynamics is assumed.

For every admitted closed process record $P\in {\mathcal{P}}_{\mathrm{c}\mathrm{l}}(s)$ of an isolated compound, assume its declared outer boundary is isolated throughout, its material constitution is fixed, its center-of-mass kinetic and separately tracked external-field potential energies have zero endpoint change, and both endpoint preparations are equilibrium macrostates. Include every named auxiliary and work device in the compound. Fix the finite constituent split $s_1,\dots ,s_m$ used for the endpoint bookkeeping, with $m\in \mathbb{N}$, $m\ge 1$, and constitutions $c_1,\dots ,c_m$. Every additional named auxiliary or work device has an endpoint set $Y_j$, energy map, and adopted entropy map $S_j^{\mathrm{a}\mathrm{u}\mathrm{x}}$ as in Isolated compound and feasible equilibrium set. Every constituent whose entropy can change must occur either in the thermodynamic split or through one of these explicit auxiliary maps. Every auxiliary is retained in the complete endpoint tuples and energy constraints. An explicitly entropy-neutral device may be assigned $S_j^{\mathrm{a}\mathrm{u}\mathrm{x}}=0$; more generally an auxiliary with declared zero endpoint entropy change cancels in the displayed inequality, while its variable state and energy remain in $\mathcal{F}$. Its endpoint product is

${\mathcal{D}}_{\mathrm{a}\mathrm{l}\mathrm{l}}:=(D_{c_1}\times \cdots \times D_{c_m})\times Y_1\times \cdots \times Y_k,\mathcal{F}\subseteq {\mathcal{D}}_{\mathrm{a}\mathrm{l}\mathrm{l}},$

where $\mathcal{F}$ is the model-declared feasible endpoint set for the fixed totals, retained barriers, and permitted exchanges, as in Isolated compound and feasible equilibrium set. The process model supplies an endpoint association on the two formal endpoint labels $-$ and $+$:

${\rho }_P:\{-,+\}⟶\mathcal{F},{\rho }_P(-)={\mathbf{x}}_P^{-}=(x_{1,P}^{-},\dots ,x_{m,P}^{-},y_{1,P}^{-},\dots ,y_{k,P}^{-}),{\rho }_P(+)={\mathbf{x}}_P^{+}=(x_{1,P}^{+},\dots ,x_{m,P}^{+},y_{1,P}^{+},\dots ,y_{k,P}^{+}),$

where ${\mathbf{x}}_P^{-}$ and ${\mathbf{x}}_P^{+}$ are respectively the declared constituent-state tuples including all auxiliary endpoint states and representing the aggregate equilibrium endpoints $x_P^{-}$ and $x_P^{+}$ of $P$. Both tuples must belong to the same $\mathcal{F}$. Restrict to endpoint splits admitted by Concave macroscopic fundamental entropy as a collection of independent separated macroscopic systems, so that its additive entropy rule applies to the macroscopic split; additionally adopt additivity of the explicitly modeled auxiliary entropy contributions for the complete compound. For $k=0$ this is the original thermodynamic-only split. On this admitted $\mathcal{F}$, define

$S_{\mathrm{t}\mathrm{o}\mathrm{t}}(\mathbf{x}):={\sum }_{i=1}^m{S}_{c_i}(x_i)+{\sum }_{j=1}^k{S}_j^{\mathrm{a}\mathrm{u}\mathrm{x}}(y_j)(\mathbf{x}=(x_1,\dots ,x_m,y_1,\dots ,y_k)\in \mathcal{F})\mathrm{.}$

This uses the finite-sum convention of Finite sums and finite products, by recursion.

The adopted macroscopic second law is

$S_{\mathrm{t}\mathrm{o}\mathrm{t}}({\rho }_P(+))-S_{\mathrm{t}\mathrm{o}\mathrm{t}}({\rho }_P(-))\ge 0.$

The process record and the zero-change conditions are those of Thermodynamic process records and transfer conventions. The process may pass through nonequilibrium configurations; this postulate compares endpoint entropies only and assigns no entropy function to those intermediate configurations.

For an admitted reversible operation in the sense of Reversible thermodynamic process record, adopt $S_{\mathrm{t}\mathrm{o}\mathrm{t}}({\rho }_P(+))-S_{\mathrm{t}\mathrm{o}\mathrm{t}}({\rho }_P(-))=0$. An admitted irreversible adiabatic operation may have a strictly positive entropy increase, but strict increase is not required for every irreversible operation. The postulate does not state monotonicity for an open compound, an externally driven process, a subsystem that exchanges energy with an omitted environment, or every microscopic trajectory.

For the declared thermal-contact family $\mathcal{T}$ and the contact relation ${\sim }_T$ of Thermal contact family and constraints, assume that ${\sim }_T$ is an equivalence relation: for all $x,y,z\in \mathcal{T}$,

$x{\sim }_{T}x,x{\sim }_{T}y⟹y{\sim }_{T}x,x{\sim }_{T}y\text{ and }y{\sim }_{T}z⟹x{\sim }_{T}z\mathrm{.}$

This is an adopted physical assumption about the stated contact protocols. It does not by itself construct a real-valued thermometer reading, an absolute temperature, a differentiable temperature state function, or a common calibration across different material families. Those require further operational, entropy, regularity, and calibration hypotheses.