Classical Gibbs quantities and SI reference conventions

Use SI units. Let $k_B>0$ be Boltzmann's constant in joules per kelvin, let $T>0$ be absolute temperature in kelvin, and define $\beta =(k_{B}T{)}^{-1}$ in joules${}^{-1}$. For a finite-alphabet lattice model in dimension $d\ge 1$, the lattice spacing $a>0$ is in metres; a spin is a dimensionless label. In the nearest-neighbor Ising convention $H_{\mathrm{\Lambda }}=-J_{\mathrm{I}}{\sum }_{\{i,j\}\subset \mathrm{\Lambda },\mathrm{\mid }i-j{\mathrm{\mid }}_1=1}{s}_i{s}_j-h{\sum }_{i\in \mathrm{\Lambda }}{s}_i,$ the coupling $J_{\mathrm{I}}$ and field parameter $h$ are energies in joules, so $\beta J_{\mathrm{I}}$ and $\beta h$ are dimensionless. Here $h$ is an energy bias per spin, not by itself a magnetic-field strength.

For a finite nonempty region $\mathrm{\Lambda }\subset {\mathbb{Z}}^d$, set $H_{\mathrm{\Lambda }}:=H_{\mathrm{\Lambda }}^{\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}$ from Finite-range lattice interaction and conditional energy, with each physical interaction value ${\mathrm{\Phi }}_r$ and total energy measured in joules. Thus the partition below uses free boundary conditions: only interaction supports contained in $\mathrm{\Lambda }$ contribute. Exterior-conditioned energies $E_{\mathrm{\Lambda }}(\cdot \mid \eta )$ and their partitions carry a separate exterior label and are not denoted by this $H_{\mathrm{\Lambda }}$ or $Z_{\mathrm{\Lambda }}$. Let $Z_{\mathrm{\Lambda }}={\sum }_{\sigma \in A^{\mathrm{\Lambda }}}{e}^{-\beta H_{\mathrm{\Lambda }}(\sigma )}$ and define dimensionless pressure per site $p_{\mathrm{\Lambda }}=\mathrm{\mid }\mathrm{\Lambda }{\mathrm{\mid }}^{-1}\log Z_{\mathrm{\Lambda }}$. When $p={\lim }_{n\to \mathrm{\infty }}\mathrm{\mid }{B}_n{\mathrm{\mid }}^{-1}\log Z_{B_n}$ exists for cubes $B_n=[-n,n{]}^d\cap {\mathbb{Z}}^d$, define physical pressure $\mathsf{P}=k_{B}Tp\mathrm{/}{a}^d$ in J m${}^{-d}$ (pascals when $d=3$). For a stationary state $\mu$ on a finite alphabet (a probability invariant under every translation $({\theta }_j\sigma {)}_x={\sigma }_{x+j}$), let $H_n=-{\sum }_{\sigma \in A^{B_n}}{\mu }_{B_n}(\sigma )\log {\mu }_{B_n}(\sigma )$ with $0\log 0=0$; when the limit exists, its Shannon entropy density is $h_{\mathrm{S}\mathrm{h}}={\lim }_{n\to \mathrm{\infty }}{H}_n\mathrm{/}\mathrm{\mid }{B}_n\mathrm{\mid }$ in nats per site. The mean energy per site is $u={\sum }_{r\in \mathcal{R}}{\mathbb{E}}_{\mu }[{\mathrm{\Phi }}_r]$ in joules per site. Define reference spin entropy density $s=k_B{h}_{\mathrm{S}\mathrm{h}}\mathrm{/}{a}^d$ and free-energy density $f=u\mathrm{/}{a}^d-Ts$; both are per physical volume. Thus $f$ and $\mathsf{P}$ have units J m${}^{-d}$, while $s$ has units J K${}^{-1}$ m${}^{-d}$; for $d=3$ these are the usual per-cubic-metre units.

For a continuum model in dimension $d$, let $x,p\in {\mathbb{R}}^d$ be position and momentum, let $m_{\mathrm{p}}>0$ be particle mass in kilograms, and use kinetic energy $\mathrm{\mid }p{\mathrm{\mid }}^2\mathrm{/}(2m_{\mathrm{p}})$. Assume Countable Choice for the Lebesgue change-of-variables interface The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). Tonelli's theorem Tonelli's theorem for nonnegative measurable functions on a sigma-finite product and the one-dimensional Gaussian integral The Gaussian integral ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-x^2}dx=\sqrt{\pi }$ give ${\int }_{{\mathbb{R}}^d}{e}^{-\mathrm{\mid }q{\mathrm{\mid }}^2}{d}^{d}q={\pi }^{d\mathrm{/}2}$. Set $c=\sqrt{\beta \mathrm{/}(2m_{\mathrm{p}})}>0$ and apply the nonnegative change-of-variables theorem A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions to the $C^1$ diffeomorphism $q\mapsto cq$ of ${\mathbb{R}}^d$ and $f(q)=e^{-\mathrm{\mid }q{\mathrm{\mid }}^2}$. Since its Jacobian determinant is $c^d$, ${\int }_{{\mathbb{R}}^d}{e}^{-\beta \mathrm{\mid }p{\mathrm{\mid }}^2\mathrm{/}(2m_{\mathrm{p}})}{d}^{d}p=c^{-d}{\pi }^{d\mathrm{/}2}=(2\pi m_{\mathrm{p}}{k}_{B}T{)}^{d\mathrm{/}2}\mathrm{.}$ Under the phase-space reference measure $d^{d}x{d}^{d}p\mathrm{/}{h}_{\mathrm{r}\mathrm{e}\mathrm{f}}^d$, with reference action $h_{\mathrm{r}\mathrm{e}\mathrm{f}}>0$ in joule-seconds, the integrated momentum factor is $h_{\mathrm{r}\mathrm{e}\mathrm{f}}^{-d}{\int }_{{\mathbb{R}}^d}{e}^{-\beta \mathrm{\mid }p{\mathrm{\mid }}^2\mathrm{/}(2m_{\mathrm{p}})}{d}^{d}p={\left(\frac{\sqrt{2\pi m_{\mathrm{p}}{k}_{B}T}}{h_{\mathrm{r}\mathrm{e}\mathrm{f}}}\right)}^d={\lambda }_T^{-d},{\lambda }_T=\frac{h_{\mathrm{r}\mathrm{e}\mathrm{f}}}{\sqrt{2\pi m_{\mathrm{p}}{k}_{B}T}}\mathrm{.}$ For chemical potential $\mu$ in joules, define the classical activity $z_c=e^{\beta \mu }{\lambda }_T^{-d}$, with units ${\mathrm{m}}^{-d}$. Thus ${\lambda }_T$ is a length and $z_c$ is inverse volume. This momentum integration explains the one-particle thermal factor; it does not derive particle indistinguishability or a factorial convention. The choice of $h_{\mathrm{r}\mathrm{e}\mathrm{f}}$ is a reference convention, not a consequence of the classical equations of motion. For continuum interactions, the potential hypotheses and energy meaning are as in Stability, superstability and regular tails.

The parameter status in these formulas is model-relative. $T$, $a$, $J_{\mathrm{I}}$, and $h$ are inputs for the chosen lattice model; $m_{\mathrm{p}}$ and $\mu$ are inputs for the chosen continuum model. Here $h$ is an energy bias per dimensionless spin, not a magnetic-field strength. The reference action $h_{\mathrm{r}\mathrm{e}\mathrm{f}}>0$ is a primitive normalization convention, not a derived quantity or an assertion that it equals Planck's constant. In the current SI, $k_B$ has the exact defining value $1.380649\times {10}^{-23}\mathrm{J}{\mathrm{K}}^{-1}$; it is a fixed SI conversion constant, not a measured fit parameter in this model. The inverse temperature $\beta$, thermal wavelength ${\lambda }_T$, activity $z_c$, pressure and entropy densities, energy density $u$, entropy density $s$, and free-energy density $f$ are defined or derived from those inputs and the state as specified above; their formulas do not calibrate the inputs to a material or assert empirical accuracy.