Entropy, the Boltzmann constant and the free-energy functional of the declared finite-sector model

(a) Adopted readings and restricted sector class. Adopt the observable and probability postulate Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes (P2) and the autonomous-dynamics postulate Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution (P4), in the readings of Assigned sharp observables and the outcome law of the framework and Assigned energy observable, unitary flow and the two pictures of autonomous evolution, for the declared fixed-$N$ model of The declared fixed-N many-body model: species sector, one-particle data, interaction and the sector energy observable. This definition applies to the class of declared models whose sector space ${\mathfrak{H}}_N^{\sigma }$ is finite-dimensional, in particular to every finite-mode realization (The declared finite-mode model and its occupation readouts), where $D:=\dim {\mathfrak{H}}_N^{\sigma }$ is finite (and may be zero) by Finite-mode occupation configurations: fermionic exclusion counts, bosonic counts and truncated dimensions clauses (a), (b). Write $\mathcal{D}({\mathfrak{H}}_N^{\sigma })$ for the set of declared sector preparations, that is, the density operators on ${\mathfrak{H}}_N^{\sigma }$ (Density operators and scalar spectral probability clause (a)). If $D=0$, the only operator has trace zero, so $\mathcal{D}({\mathfrak{H}}_N^{\sigma })=\mathrm{\varnothing }$. The partition trace is $Z_N=0$, and there is no normalized canonical density or equilibrium free energy; $\log D$, $D^{-1}\mathrm{i}\mathrm{d}$ and $\log Z_N$ are not used in this branch. Clauses (b), (d) and (e) below apply when $D\ge 1$; the assigned parameters of (c) can still be declared when $D=0$. Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

(b) Dimensionless entropy. For $\rho \in \mathcal{D}({\mathfrak{H}}_N^{\sigma })$ with eigenvalues $w_1,\dots ,w_D\ge 0$ (counted with multiplicity, in any order), the entropy of $\rho$ is $s(\rho ):=-{\sum }_{i:w_i>0}{w}_i\log w_i\in [0,\log D],$ the quantity of Finite Gibbs free-energy minimum clause (a) with $\log$ the natural logarithm (The natural logarithm as the inverse of the exponential function). It is dimensionless, is a function of the eigenvalue multiset of $\rho$ alone, and equals zero exactly for pure preparations; by clause (c) of the cited theorem $s(\rho )=\log D$ holds exactly when $\rho =D^{-1}\mathrm{i}\mathrm{d}$.

(c) Boltzmann constant, temperature and inverse temperature. The model assigns a positive Boltzmann constant $k_B$ and a positive temperature $T$; in the SI conventions of Dimensions, SI unit conventions and quantity types of the framework the unit of $k_B$ is $\mathrm{J}{\mathrm{K}}^{-1}$, the unit of $T$ is the kelvin, and $k_B=1.380649\times {10}^{-23}\mathrm{J}{\mathrm{K}}^{-1}$ fixed exactly by the BIPM SI definition. The model declares the inverse temperature $\beta :=1\mathrm{/}(k_{B}T)$, a positive real number in ${\mathrm{J}}^{-1}$. The value and exactness of $k_B$ are unit-convention facts; the assignment of the parameter $T$ to a physical sample, and the identification of $\beta =1\mathrm{/}(k_{B}T)$ with an inverse thermodynamic temperature, are adopted assumptions of the ensemble model of this pair, not consequences of P1-P4, and are tested identifications with uncertainty under the preparation and apparatus postulate Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions (P7), whose clauses (i), (iii) and (iv) require a declared comparison and calibration model and qualifications for measured inputs. No thermometer, calibration or measured temperature is supplied here.

(d) Physical entropy. The physical entropy of $\rho$ is $S(\rho ):=k_{B}s(\rho ),$ read in $\mathrm{J}{\mathrm{K}}^{-1}$. It is nonnegative, is at most $k_B\log D$, and vanishes exactly for pure preparations.

(e) Free-energy functional. For $\rho \in \mathcal{D}({\mathfrak{H}}_N^{\sigma })$, let $\mathrm{tr}(\rho H_N^{\sigma })\in \mathbb{R}$ be the mean of the declared energy observable in $\rho$ (finite because $D<\mathrm{\infty }$, in the moment reading of Assigned sharp observables and the outcome law of the framework clause (e)). The Helmholtz free-energy reading of $\rho$ is $F(\rho ):=\mathrm{tr}(\rho H_N^{\sigma })-k_{B}Ts(\rho ),$ read in joules; it is the quantity $\mathrm{tr}(\rho H)-{\beta }^{-1}s(\rho )$ of Finite Gibbs free-energy minimum clause (a) at the identified $\beta$. For the canonical operator ${\rho }_{\beta }:=e^{-\beta H_N^{\sigma }}\mathrm{/}{Z}_N$ and $Z_N:=\mathrm{tr}{e}^{-\beta H_N^{\sigma }}\in (0,\mathrm{\infty })$ (positive finite sum in the nonzero finite-dimensional sector), the number $F_{\mathrm{e}\mathrm{q}}:=-{\beta }^{-1}\log Z_N$ is called the equilibrium free energy; it is the value of the functional at ${\rho }_{\beta }$ by the cited theorem.

(f) Scope. This definition covers only declared models with finite-dimensional sector space, with the zero-sector branch of (a), and preparations in $\mathcal{D}({\mathfrak{H}}_N^{\sigma })$. For a declared spatial realization, whose sector space is infinite-dimensional, the operator $\rho \log \rho$ need not be trace class; no entropy, physical entropy or free energy is defined here for that case, and a separate trace-class statement would be required. The definition asserts no increase of entropy in time, no equilibration to ${\rho }_{\beta }$, no thermodynamic limit, no pressure or volume variable, and no measured temperature, entropy or free energy.

• Why the finite case first. In the finite-dimensional sector the entropy is an ordinary finite sum over $D$ eigenvalues, and the comparison theorem of Finite Gibbs free-energy minimum is available without any trace-class convergence hypothesis on $\rho \log \rho$. The infinite-dimensional questions are deferred rather than assumed.
• Units of $\beta$. $\beta =1\mathrm{/}(k_{B}T)$ has the unit ${\mathrm{J}}^{-1}\mathrm{K}{\mathrm{K}}^{-1}={\mathrm{J}}^{-1}$, so the exponent $\beta H_N^{\sigma }$ is dimensionless, as the spectral calculus requires.