Macroscopic and microscopic entropy quantities

Fix a material constitution $c$ and its equilibrium state domain $D_c$ from Equilibrium state domains. The adopted macroscopic entropy is the finite-valued state function $S_c:D_c\to \mathbb{R}$ of Concave macroscopic fundamental entropy: for every $x\in D_c$, $S_c(x)$ is a finite real value measured in $\mathrm{J}{\mathrm{K}}^{-1}$ under SI quantities and unit conventions. Its argument is an equilibrium macrostate. The definition does not assign $S_c$ to an arbitrary nonequilibrium configuration or to a microscopic trajectory.

For a finite microstate model $(\mathrm{\Omega },ϵ)$ of Finite statistical microstate model, $\mathrm{\Omega }$ is finite and nonempty and $ϵ:\mathrm{\Omega }\to \mathbb{R}$ assigns a finite numerical energy. The model's physical energy is $E(x)=ϵ(x)\mathrm{J}$ for $x\in \mathrm{\Omega }$, with its zero and any retained internal offsets fixed by the declared material model. By The cardinality $\mid A\mid$ of a finite set, $N:=\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }$ is the unique natural number equinumerous with $\mathrm{\Omega }$, and $N\ge 1$. Choose an enumeration $\mathrm{\Omega }=\{{\omega }_1,\dots ,{\omega }_N\}$. A chosen coarse-graining is a finite label set $\mathcal{M}$ and a function $\pi :\mathrm{\Omega }\to \mathcal{M}$ (A function is a relation $f$ with $(a,b)\in f$ and $(a,c)\in f$ implying $b=c$; $f:A\to B$, the value $f(a)$, domain and codomain). For $m\in \mathcal{M}$, let $M_m=\{x\in \mathrm{\Omega }:\pi (x)=m\}\subseteq \mathrm{\Omega }$ (Subset $x\subseteq y$, proper subset $x⊊y$, and the separation notation $\{z\in x:\phi (z)\}$). Each $M_m$ is finite: choose a bijection $b:N\to \mathrm{\Omega }$ and set $J_m=b^{-1}(M_m)\subseteq N$. Induction on $n$ (The principle of mathematical induction) proves every subset $J\subseteq n$ finite. For $n=0$, $J=\mathrm{\varnothing }$; for $n=\sigma (k)$, if $k\mathrm{\notin }J$, then $J\subseteq k$; if $k\in J$, the set $J_0:=\{j\in J:j\mathrm{\ne }k\}\subseteq k$ is finite by induction, say via a bijection $f:n_0\to J_0$, and extending $f$ by $f(n_0)=k$ gives a bijection $\sigma (n_0)\to J$. Thus $J_m$ is finite, and the restriction $b{\mathrm{\mid }}_{J_m}:J_m\to M_m$ is a bijection. By The cardinality $\mid A\mid$ of a finite set, $N_m:=\mathrm{\mid }{M}_m\mathrm{\mid }$ is therefore defined. Only labels with $N_m\ge 1$ are admitted here. The Boltzmann macrocell entropy of this specified coarse-graining is

$S_{\mathrm{B}}(m):=k_{\mathrm{B}}\log N_m\mathrm{.}$

Under this enumeration, a probability vector $p=(p_1,\dots ,p_N)$ assigns probability $p_i$ to microstate ${\omega }_i$, where $p_i\ge 0$ and ${\sum }_{i=1}^N{p}_i=1$. Define its Gibbs–Shannon entropy by

$S_{\mathrm{G}}(p):=-k_{\mathrm{B}}{\sum }_{i=1}^N{p}_i\log p_i,$

with the summand $p_i\log p_i$ defined to be $0$ when $p_i=0$. Both expressions have units $\mathrm{J}{\mathrm{K}}^{-1}$, because their logarithms are dimensionless and $k_{\mathrm{B}}$ has those units. The finite sum and logarithm have the meanings of Finite sums and finite products, by recursion and The natural logarithm as the inverse of the exponential function.

For a density operator $\rho$ in the declared finite-dimensional sector class of Entropy, the Boltzmann constant and the free-energy functional of the declared finite-sector model, its von Neumann physical entropy is $S_{\mathrm{v}\mathrm{N}}(\rho )=k_{\mathrm{B}}s(\rho )$, with $s(\rho )$ the dimensionless entropy defined there. That supplier defines no such quantity for its infinite-dimensional spatial sectors.

These are distinct definitions on different domains: $S_c$ acts on an equilibrium macrostate, $S_{\mathrm{B}}$ on a chosen macrocell, $S_{\mathrm{G}}$ on a specified probability vector, and $S_{\mathrm{v}\mathrm{N}}$ on a finite-sector density operator. A numerical equality among them requires an additional, explicit identification of model, coarse-graining, preparation and applicable limit. This item supplies no such identification and adopts no canonical or microcanonical sampling rule.

The finite microstate model and its numerical-to-physical energy assignment are those of Finite statistical microstate model. That definition supplies neither a sampling rule nor an identification of microscopic entropy with $S_c$; any such bridge requires separate assumptions and results.