Quantum statistics and model equilibrium preparation

For a declared QSM model, let $I$ be a specified finite or countably infinite mode set with a fixed enumeration and let $\mathfrak{h}={\mathrm{\ell }}^2(I)$. The model assigns a nonzero separable complex Hilbert space $\mathcal{H}$ and declares whether it is a fixed-particle-number sector or the Fock space ${\mathcal{F}}_{\pm }(\mathfrak{h})$ of Symmetric and antisymmetric identical-particle Fock spaces. By that definition, ${\mathcal{F}}_{\pm }(\mathfrak{h})$ is the Hilbert direct sum of the symmetric or antisymmetric $n$-particle tensor sectors for $n\ge 0$, including the vacuum sector $\mathbb{C}$. The assignment is a physical model assumption. The symmetric or antisymmetric sector is selected independently for bosons or fermions; this selection is not derived from the Fock construction or from a nonrelativistic spin-statistics theorem.

The words pure preparation and state have the exact meanings in clauses (i)–(iii) of State and composition postulate P1: Hilbert spaces, preparations and composite systems: a pure preparation is a unit ray in $\mathcal{H}$, and a general state is a self-adjoint positive trace-class operator $\rho$ with $\mathrm{tr}\rho =1$. These mathematical state definitions are used for the QSM Hilbert spaces declared above; the reference does not extend that NRQM postulate's fixed-particle-number physical scope to a variable-number grand-canonical system. The countable-choice hypothesis is The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$).

A sharp measurement is specified by a densely defined self-adjoint operator $A:D(A)\subseteq \mathcal{H}\to \mathcal{H}$, its declared physical unit, and its spectral projection-valued measure $E_A$ on the Borel sets of $\mathbb{R}$, as in clauses (i)–(iii) of Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes. Here a projection-valued measure is a map from Borel sets to orthogonal projections with $E_A(\mathrm{\varnothing })=0$, $E_A(\mathbb{R})=I$, $E_A(B\cap C)=E_A(B)E_A(C)$, and strong countable additivity. The adopted Born law is ${\mu }_{\rho }^A(B):=\mathrm{tr}(\rho E_A(B))$ for Borel $B\subseteq \mathbb{R}$; clauses (a)–(b) of Density operators and scalar spectral probability give a probability measure under ${\mathrm{A}\mathrm{C}}_{\omega }$. For a finite unsharp measurement, a specified finite family of bounded effects $0\le F_a\le I$ with ${\sum }_a{F}_a=I$ has outcome probabilities $\mathrm{tr}(\rho F_a)$, as in clause (iv) of Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes. The arbitrary self-adjoint spectral construction in that postulate also assumes the Axiom of Choice; any use inherits that exact mathematical premise.

An Axiom of Choice hypothesis The Axiom of Choice accompanies the general self-adjoint spectral construction in P2. A model Hamiltonian is a densely defined self-adjoint operator $H:D(H)\subseteq \mathcal{H}\to \mathcal{H}$ representing energy, with its domain and energy unit declared. For a canonical preparation, write $H_N$ for the Hamiltonian on the declared fixed-number sector and adopt $\rho =Z^{-1}{e}^{-\beta H_N}$ only when the spectral-calculus exponential has $0<Z=\mathrm{tr}(e^{-\beta H_N})<\mathrm{\infty }$. For a grand-canonical preparation on the declared Fock space, let $\mathrm{\mid }n\mathrm{\mid }:={\sum }_j{n}_j$ for a finite-support occupation configuration $n$ of Occupation bases and actual CCR/CAR domains and define the particle-number operator by $N\mathrm{\mid }n⟩=\mathrm{\mid }n\mathrm{\mid }\mathrm{\mid }n⟩$ on $D(N)=\{c\in {\mathrm{\ell }}^2({\mathrm{\Omega }}_{\pm }):{\sum }_n\mathrm{\mid }n{\mathrm{\mid }}^2\mathrm{\mid }{c}_n{\mathrm{\mid }}^2<\mathrm{\infty }\}$, as in Explicit diagonal Fock domains and occupation trace. Require $K_{\mu }$ to be a specified self-adjoint realization of $H-\mu N$ on a stated domain; self-adjointness of $H$ and $N$ separately does not by itself specify a self-adjoint domain for their difference. For $T>0$ in K, $k_B>0$ in J K${}^{-1}$, $\beta =(k_{B}T{)}^{-1}$ in J${}^{-1}$, and chemical potential $\mu$ in J, adopt $\rho ={\mathrm{\Xi }}^{-1}{e}^{-\beta K_{\mu }}$ only when $0<\mathrm{\Xi }=\mathrm{tr}(e^{-\beta K_{\mu }})<\mathrm{\infty }$.

For the additive diagonal Fock model, the trace and occupation statements of Explicit diagonal Fock domains and occupation trace hold only under that lemma's mode-summability hypotheses and ${\mathrm{A}\mathrm{C}}_{\omega }$. Under those hypotheses its finite collections of grand-canonical mode occupations are independent. This independence is not asserted for interacting Hamiltonians; conditioning a product grand law on a fixed total number generally couples the modes.

If the spectral law ${\mu }_{\rho }^H$ has finite first absolute moment, define the model mean energy by $U(\rho ;H):={\int }_{\mathbb{R}}\lambda d{\mu }_{\rho }^H(\lambda )$. If the nonzero eigenvalues $r_j$ of $\rho$, repeated with multiplicity, satisfy ${\sum }_j-r_j\log r_j<\mathrm{\infty }$, define its von Neumann entropy by $S(\rho ):=-k_B{\sum }_j{r}_j\log r_j$, with $0\log 0:=0$; where finite, this is $-k_B\mathrm{tr}(\rho \log \rho )$. This postulate does not define Helmholtz free energy, grand potential, or other thermodynamic potentials; model-specific items define them with their own hypotheses. Identifying these quantities with measured macroscopic thermodynamic variables requires a separate physical bridge. No thermalization mechanism, empirical value, or law for a particular material is postulated here.