Macroscopic second law for isolated compounds

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.