Finite spectral periodic Bose/Fermi gas

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) and adopt the model preparation of Quantum statistics and model equilibrium preparation. Let $d\ge 1$, $L>0$ in m, $m>0$ in kg, $\mathrm{\hslash }>0$ in J s, $k_B>0$ in J K${}^{-1}$, and $T>0$ in K. Put $V=L^d$, $\alpha ={\mathrm{\hslash }}^2\mathrm{/}(2m)$, $\beta =(k_{B}T{)}^{-1}$, and let $g\in \mathbb{N}$ count distinct internal modes. Put $\mathrm{\Lambda }={\mathbb{Z}}^d\times \{1,\dots ,g\}$, enumerated by increasing $\mathrm{\parallel }k{\mathrm{\parallel }}_{\mathrm{\infty }}:={\max }_{1\le j\le d}\mathrm{\mid }{k}_j\mathrm{\mid }$, then lexicographically in $k$ and increasingly in $s$. Define the one-particle Hilbert space $\mathfrak{h}={\mathrm{\ell }}^2(\mathrm{\Lambda })$ with coordinate basis $e_{k,s}$. Its spectral energy multiplier is $h_1{e}_{k,s}={ϵ}_k{e}_{k,s}$ for ${ϵ}_k=\alpha \mathrm{\mid }2\pi k\mathrm{/}L{\mathrm{\mid }}^2$, on the maximal domain $D(h_1)=\{c\in {\mathrm{\ell }}^2(\mathrm{\Lambda }):{\sum }_{k,s}{ϵ}_k^2\mathrm{\mid }{c}_{k,s}{\mathrm{\mid }}^2<\mathrm{\infty }\}$; the real diagonal multiplier is self-adjoint on this domain by the coordinate adjoint calculation in Step 2.1 of Explicit diagonal Fock domains and occupation trace.

Let ${\mathrm{\Omega }}_B$ be the finite-support maps $n:\mathrm{\Lambda }\to {\mathbb{N}}_0$, and let ${\mathrm{\Omega }}_F$ be the finite-support maps $n:\mathrm{\Lambda }\to \{0,1\}$, with occupation bases $\mathrm{\mid }n⟩$ of the Bose or Fermi Fock space as in Occupation bases and actual CCR/CAR domains. Use ${\mathrm{\Omega }}_{\pm }={\mathrm{\Omega }}_B$ for bosons and ${\mathrm{\Omega }}_{\pm }={\mathrm{\Omega }}_F$ for fermions. Define the finite sums $E(n):={\sum }_{(k,s)\in \mathrm{\Lambda }}{ϵ}_k{n}_{k,s}$ and $\mathrm{\mid }n\mathrm{\mid }:={\sum }_{(k,s)\in \mathrm{\Lambda }}{n}_{k,s}$, and the mode occupation random variables $N_{k,s}(n):=n_{k,s}$. The diagonal many-body Hamiltonian and total number operators act by $H\mathrm{\mid }n⟩=E(n)\mathrm{\mid }n⟩$ and $N\mathrm{\mid }n⟩=\mathrm{\mid }n\mathrm{\mid }\mathrm{\mid }n⟩$ on their maximal domains $D(H)=\{c\in {\mathrm{\ell }}^2({\mathrm{\Omega }}_{\pm }):{\sum }_{n\in {\mathrm{\Omega }}_{\pm }}E(n{)}^2\mathrm{\mid }{c}_n{\mathrm{\mid }}^2<\mathrm{\infty }\},D(N)=\{c\in {\mathrm{\ell }}^2({\mathrm{\Omega }}_{\pm }):{\sum }_{n\in {\mathrm{\Omega }}_{\pm }}\mathrm{\mid }n{\mathrm{\mid }}^2\mathrm{\mid }{c}_n{\mathrm{\mid }}^2<\mathrm{\infty }\},$ as self-adjoint operators on those domains by Explicit diagonal Fock domains and occupation trace.

For bosons take $\mu <0$; for fermions take any finite $\mu \in \mathbb{R}$. In either case the grand trace ${\mathrm{\Xi }}_{\beta ,\mu ,L}:={\sum }_{n\in {\mathrm{\Omega }}_{\pm }}{e}^{-\beta (E(n)-\mu \mathrm{\mid }n\mathrm{\mid })}$ is finite and positive, and its normalized occupation law is $P_{\beta ,\mu ,L}(n):={\mathrm{\Xi }}_{\beta ,\mu ,L}^{-1}{e}^{-\beta (E(n)-\mu \mathrm{\mid }n\mathrm{\mid })}$. For a $P_{\beta ,\mu ,L}$-integrable real random variable $X$, write ${\mathbb{E}}_{\beta ,\mu ,L}X:={\sum }_{n\in {\mathrm{\Omega }}_{\pm }}{P}_{\beta ,\mu ,L}(n)X(n)$; if $X$ has finite second moment, set ${\mathrm{Var}}_{\beta ,\mu ,L}(X):={\mathbb{E}}_{\beta ,\mu ,L}[(X-{\mathbb{E}}_{\beta ,\mu ,L}X{)}^2]$. In particular, ${\mathbb{E}}_{\beta ,\mu ,L}N={\sum }_n{P}_{\beta ,\mu ,L}(n)\mathrm{\mid }n\mathrm{\mid }$ and ${\mathbb{E}}_{\beta ,\mu ,L}H={\sum }_n{P}_{\beta ,\mu ,L}(n)E(n)$. Under this additive diagonal Hamiltonian its mode occupations are independent, and ${\bar{n}}_{k,s}:={\mathbb{E}}_{\beta ,\mu ,L}{N}_{k,s}=\frac{1}{e^{\beta ({ϵ}_k-\mu )}\mp 1},{\mathrm{Var}}_{\beta ,\mu ,L}(N_{k,s})={\bar{n}}_{k,s}(1\pm {\bar{n}}_{k,s}),$ with the upper sign choice in the denominator interpreted as $-$ for bosons and $+$ for fermions (equivalently the variance is $\bar{n}(1+\bar{n})$ for bosons and $\bar{n}(1-\bar{n})$ for fermions). For fixed $\beta ,\mu$ in these regimes, as $L\to \mathrm{\infty }$, $\frac{{\mathbb{E}}_{\beta ,\mu ,L}N}{V}\to \frac{g}{(2\pi {)}^d}{\int }_{{\mathbb{R}}^d}\frac{dp}{e^{\beta (\alpha \mathrm{\mid }p{\mathrm{\mid }}^2-\mu )}\mp 1},\frac{{\mathbb{E}}_{\beta ,\mu ,L}H}{V}\to \frac{g}{(2\pi {)}^d}{\int }_{{\mathbb{R}}^d}\frac{\alpha \mathrm{\mid }p{\mathrm{\mid }}^{2}dp}{e^{\beta (\alpha \mathrm{\mid }p{\mathrm{\mid }}^2-\mu )}\mp 1},$ The continuum limits are conditional model deductions; they assert no thermalization or measured density.