Actually constructed Gibbs/KMS preparation

Adopt the selected stationary bath of Selected stationary thermal equilibrium on its connected smooth Lorentzian domain. A finite-cell preparation in this item means exactly the setup of Constructed independent cell Gibbs states: finitely many $r\ge 1$ noninteracting cells with finite-dimensional complex Hilbert spaces ${\mathcal{H}}_i$, Hermitian proper Hamiltonians $H_i$ in joules, and dimensionless Hermitian number matrices $Q_i$ with nonnegative integer eigenvalues and $[H_i,Q_i]=0$. Their tensor product is $\mathcal{H}={⨂}_{i=1}^r{\mathcal{H}}_i$; each cell operator acts as the identity on the other factors.

Choose a placement map $p:\{1,\dots ,r\}\to O$ assigning each cell to a spacetime event, and write $p_i:=p(i)$. Let $N_{\xi }(x):=\sqrt{-g_x({\xi }_x,{\xi }_x)}$ be the dimensionless lapse field, and let $T:O\to (0,\mathrm{\infty })$ and $\mu :O\to \mathbb{R}$ be the smooth proper-temperature and chemical-potential fields in the stationarity hypotheses. Evaluate them at the assigned event: $N_i:=N_{\xi }(p_i),T_i:=T(p_i),{\mu }_i:=\mu (p_i)\mathrm{.}$ Assume the same connected stationary geometry, selected future timelike Killing field $\xi$, common charge $q$ in coulombs, global smooth potential $A$, global compensator ${\mathrm{\Lambda }}_{\xi }$ satisfying ${\mathcal{L}}_{\xi }A=d{\mathrm{\Lambda }}_{\xi }$, and charged whole-future-mass-shell stationarity hypotheses as in that theorem; no patchwise potential is substituted. Using the length-coordinate convention and $c>0$ in metres per second from Thermodynamics-GR fields, currents, reference normalization and units, define the global gauge-compensated potential scalar ${\mathrm{\Phi }}_{\xi }:O\to \mathbb{R}$ by ${\mathrm{\Phi }}_{\xi }(x):=-c({\iota }_{\xi }A-{\mathrm{\Lambda }}_{\xi })(x)$ in volts, and set ${\mathrm{\Phi }}_i:={\mathrm{\Phi }}_{\xi }(p_i)$. The common parameters satisfy $N_i{T}_i=T_K>0$ and $N_i{\mu }_i+q{\mathrm{\Phi }}_i={\mu }_K$. The theorem's geometric stationarity argument is conditional on the Axiom of Countable Choice, ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)); no stronger choice axiom is used here.

Put ${\beta }_K=(k_B{T}_K{)}^{-1}$ in joules${}^{-1}$, ${\beta }_i=N_i{\beta }_K=(k_B{T}_i{)}^{-1}$, $H_K={\sum }_i(N_i{H}_i+q{\mathrm{\Phi }}_i{Q}_i)$, $Q={\sum }_i{Q}_i$, and $G_K=H_K-{\mu }_{K}Q={\sum }_i{N}_i(H_i-{\mu }_i{Q}_i)$. The finite-cell theorem constructs

$Z_i={\mathrm{Tr}}_{{\mathcal{H}}_i}{e}^{-{\beta }_i(H_i-{\mu }_i{Q}_i)},Z={\mathrm{Tr}}_{\mathcal{H}}{e}^{-{\beta }_K{G}_K}={\prod }_i{Z}_i,$

${\rho }_i=Z_i^{-1}{e}^{-{\beta }_i(H_i-{\mu }_i{Q}_i)},\rho =Z^{-1}{e}^{-{\beta }_K{G}_K}={⨂}_i{\rho }_i\mathrm{.}$

For $X\in \mathcal{B}(\mathcal{H})$, write ${\omega }_{\rho }(X)=\mathrm{Tr}(\rho X)$ and define the Killing-time grand-generator dynamics, with $t_K$ in seconds, by

${\alpha }_{t_K}^G(X)=e^{i{t}_K{G}_K\mathrm{/}\mathrm{\hslash }}X{e}^{-i{t}_K{G}_K\mathrm{/}\mathrm{\hslash }},\mathrm{\hslash }>0\text{ in joule-seconds}\mathrm{.}$

The referenced finite-dimensional theorem proves that $\rho$ is a normalized positive state and is ${\beta }_K$-KMS for ${\alpha }^G$; the physical-time strip has width $\mathrm{\hslash }{\beta }_K$ seconds under Physical-time KMS strip equilibrium condition. Its restriction to observables commuting with $Q$ is also KMS for the physical $H_K$ dynamics. This postulate adopts that actually constructed finite state as the state assignment for this specified model. It asserts no state on an infinite-volume algebra, no trace-class stress exponential in generic curved quantum field theory, no approach to equilibrium, and no observed preparation or material realization.

Fix one cell $i$ of the finite preparation in Actually constructed Gibbs/KMS preparation, at its assigned event $p_i\in O$, with $N_i=N_{\xi }(p_i)$, $T_i=T(p_i)$, and ${\mu }_i=\mu (p_i)$. Set ${\beta }_K:=(k_B{T}_K{)}^{-1}>0$ in joules${}^{-1}$. Let ${\mathcal{A}}_i=\mathcal{B}({\mathcal{H}}_i)$, let $G_i:=H_i-{\mu }_i{Q}_i$ be its proper grand generator in joules, let $G_{K,i}:=N_i{G}_i$ be its Killing-time grand generator, and let ${\omega }_i(X):=\mathrm{Tr}({\rho }_{i}X)$. Use length coordinates $x^0=ct$, with $c>0$ the speed of light in metres per second and the reference-normalized $\xi$ dimensionless in these coordinates. Choose an open interval $I_{K,i}\subseteq \mathbb{R}$ containing $0$ on which the integral curve through $p_i$ remains in $O$. Define the unique smooth curve ${\gamma }_i:I_{K,i}\to O$ by ${\gamma }_i(0)=p_i,\frac{d{\gamma }_i}{d{t}_K}(t_K)=c{\xi }_{{\gamma }_i(t_K)},$ where $t_K$ is physical Killing time in seconds; the equation states its parameterization by the reference-normalized Killing field. This curve is future-directed timelike. The stationary normalization used in KMS orbit clock rescaling gives $\xi (N_{\xi })=0$, so $N_{\xi }({\gamma }_i(t_K))=N_i$ throughout this interval. Its proper-time parametrization is ${\mathrm{\Gamma }}_i:N_i{I}_{K,i}\to O,{\mathrm{\Gamma }}_i(\tau ):={\gamma }_i(\tau \mathrm{/}{N}_i),$ which satisfies $g(d{\mathrm{\Gamma }}_i\mathrm{/}d\tau ,d{\mathrm{\Gamma }}_i\mathrm{/}d\tau )=-c^2$. Thus $\tau$ is proper time in seconds and $\tau =N_i{t}_K$ on the geometric orbit. No completeness of $\xi$ is assumed: the geometric curve is defined on its flow interval, while the finite-matrix dynamics below are defined for all real parameters by their generators and use the same constant rescaling.

Define

${\alpha }_{t_K}^{K,i}(X):=e^{i{t}_K{G}_{K,i}\mathrm{/}\mathrm{\hslash }}X{e}^{-i{t}_K{G}_{K,i}\mathrm{/}\mathrm{\hslash }},{\alpha }_{\tau }^{\mathrm{l}\mathrm{o}\mathrm{c},i}(X):=e^{i\tau G_i\mathrm{/}\mathrm{\hslash }}X{e}^{-i\tau G_i\mathrm{/}\mathrm{\hslash }},$

where both $t_K$ and $\tau$ are measured in seconds, and $N_i>0$ is dimensionless. Write ${\mathcal{A}}_i^0:=\{X\in {\mathcal{A}}_i:[X,Q_i]=0\}$ for the neutral observable algebra. Then ${\omega }_i$ is ${\beta }_K$-KMS for ${\alpha }^{K,i}$ and is ${\beta }_{\mathrm{l}\mathrm{o}\mathrm{c},i}$-KMS for ${\alpha }^{\mathrm{l}\mathrm{o}\mathrm{c},i}$, where

${\beta }_{\mathrm{l}\mathrm{o}\mathrm{c},i}=N_i{\beta }_K=\frac{1}{k_B{T}_i},\mathrm{\hslash }{\beta }_{\mathrm{l}\mathrm{o}\mathrm{c},i}\text{ is the strip width in seconds}\mathrm{.}$

On ${\mathcal{A}}_i^0$, the same proper-time automorphism is generated by the physical Hamiltonian $H_i$, because its chemical-potential term commutes with every observable in that subalgebra. Thus the local temperature is $T_i=T_K\mathrm{/}{N}_i$ for this constructed state and selected orbit. The conclusion is only orbitwise; it does not make a position-dependent lapse into one global time-rescaled dynamics.

Fix one cell $i$ of the preparation in Actually constructed Gibbs/KMS preparation at its assigned event $p_i\in O$, with $N_i=N_{\xi }(p_i)$, $T_i=T(p_i)$, and ${\mu }_i=\mu (p_i)$. Let $H_i=H_i^{\ast }$ be its proper Hamiltonian in joules, $Q_i$ its dimensionless number matrix, $G_i:=H_i-{\mu }_i{Q}_i$, and $\sigma :={\rho }_i$ its constructed density. Put ${\beta }_K:=(k_B{T}_K{)}^{-1}>0$ in joules${}^{-1}$ and ${\beta }_{\mathrm{l}\mathrm{o}\mathrm{c},i}:=N_i{\beta }_K=(k_B{T}_i{)}^{-1}$. Take Hermitian matrices $A,B\in \mathcal{B}({\mathcal{H}}_i)$ with $[A,Q_i]=[B,Q_i]=0$. Let $h:[0,S]\to \mathbb{R}$ be continuous for $S>0$ seconds, let $ϵ\in \mathbb{R}$ be dimensionless, and require $h(\tau )B$ to have units of joules. Define the local and Killing-time dynamics by

${\alpha }_{\tau }^{\mathrm{l}\mathrm{o}\mathrm{c}}(C)=e^{i{G}_i\tau \mathrm{/}\mathrm{\hslash }}C{e}^{-i{G}_i\tau \mathrm{/}\mathrm{\hslash }},{\alpha }_t^K(C)=e^{i{N}_i{G}_{i}t\mathrm{/}\mathrm{\hslash }}C{e}^{-i{N}_i{G}_{i}t\mathrm{/}\mathrm{\hslash }}\mathrm{.}$

Let $H_{K,i}:=N_i{H}_i+q{\mathrm{\Phi }}_i{Q}_i$ be the cell's physical Killing-energy operator, where $q$ is the common charge in coulombs and ${\mathrm{\Phi }}_i={\mathrm{\Phi }}_{\xi }(p_i)$ is in volts as in the preparation. Let $U_{\mathrm{l}\mathrm{o}\mathrm{c}}:[0,S]\to \mathcal{B}({\mathcal{H}}_i)$ and $U_K:[0,S\mathrm{/}{N}_i]\to \mathcal{B}({\mathcal{H}}_i)$ be the unique solutions of $i\mathrm{\hslash }\frac{d}{d\tau }{U}_{\mathrm{l}\mathrm{o}\mathrm{c}}(\tau )=(H_i-ϵh(\tau )B)U_{\mathrm{l}\mathrm{o}\mathrm{c}}(\tau ),U_{\mathrm{l}\mathrm{o}\mathrm{c}}(0)=I_{{\mathcal{H}}_i},$ $i\mathrm{\hslash }\frac{d}{dt}{U}_K(t)=(H_{K,i}-ϵN_{i}h(N_{i}t)B)U_K(t),U_K(0)=I_{{\mathcal{H}}_i}\mathrm{.}$ They are unitary propagators. Define the model expectations by $E_{\mathrm{l}\mathrm{o}\mathrm{c}}(\tau ):=\mathrm{Tr}\text{\hspace{0.17em}⁣}(\sigma U_{\mathrm{l}\mathrm{o}\mathrm{c}}(\tau {)}^{\ast }A{U}_{\mathrm{l}\mathrm{o}\mathrm{c}}(\tau )),E_K(t):=\mathrm{Tr}\text{\hspace{0.17em}⁣}(\sigma U_K(t{)}^{\ast }A{U}_K(t))\mathrm{.}$ Because $A$ and $B$ commute with $Q_i$, the second expectation is unchanged if $H_{K,i}$ is replaced by $N_i{H}_i$; that is the neutral-sector version of the Killing control. The exact finite-cell predictions are

$E_K(t)=E_{\mathrm{l}\mathrm{o}\mathrm{c}}(N_{i}t)(0\le t\le S\mathrm{/}{N}_i),$

and

$E_{\mathrm{l}\mathrm{o}\mathrm{c}}(\tau )=\mathrm{Tr}(\sigma {\alpha }_{\tau }^{\mathrm{l}\mathrm{o}\mathrm{c}}(A))+ϵ{\int }_0^{\tau }h(s)R_{\mathrm{l}\mathrm{o}\mathrm{c}}(\tau -s)ds+r(\tau ),$

where

$R_{\mathrm{l}\mathrm{o}\mathrm{c}}(u)=\frac{i}{\mathrm{\hslash }}\mathrm{Tr}\text{\hspace{0.17em}⁣}(\sigma [{\alpha }_u^{\mathrm{l}\mathrm{o}\mathrm{c}}(A),B]),C_S=\frac{2}{\mathrm{\hslash }}{\int }_0^S\mathrm{\mid }h(s)\mathrm{\mid }\mathrm{\parallel }B\mathrm{\parallel }ds,\mathrm{\mid }r(\tau )\mathrm{\mid }\le \mathrm{\parallel }A\mathrm{\parallel }\frac{(\mathrm{\mid }ϵ\mathrm{\mid }{C}_S{)}^2{e}^{\mathrm{\mid }ϵ\mathrm{\mid }{C}_S}}{2}\mathrm{.}$

Here $C_S$ is dimensionless and $R_{\mathrm{l}\mathrm{o}\mathrm{c}}$ has units $[A][B]\mathrm{/}(\mathrm{J}\mathrm{s})$. For the equilibrium correlations $C_{\mathrm{l}\mathrm{o}\mathrm{c}}(u)=\mathrm{Tr}(\sigma {\alpha }_u^{\mathrm{l}\mathrm{o}\mathrm{c}}(A)A)$ and $C_K(t)=\mathrm{Tr}(\sigma {\alpha }_t^K(A)A)$, let $S_{\mathrm{l}\mathrm{o}\mathrm{c}}$ and $S_K$ be their Fourier measures in the distributional convention $S(\omega )={\int }_{\mathbb{R}}{e}^{i\omega t}C(t)dt$, where $\omega$ is in seconds${}^{-1}$. Set $D_{N_i}(\omega )=N_i\omega$. Then

$S_K=(D_{N_i}{)}_{\ast }{S}_{\mathrm{l}\mathrm{o}\mathrm{c}},S_K(\omega )=N_i^{-1}{S}_{\mathrm{l}\mathrm{o}\mathrm{c}}(\omega \mathrm{/}{N}_i),$

the detailed-balance factor is unchanged because ${\beta }_K\mathrm{\hslash }(N_i\omega )={\beta }_{\mathrm{l}\mathrm{o}\mathrm{c},i}\mathrm{\hslash }\omega$, and any zero-frequency atom remains at zero and is kept separate from the $\coth$ formula. Finally, for the separately re-prepared finite Gibbs family ${\sigma }_{\lambda }\propto e^{-{\beta }_{\mathrm{l}\mathrm{o}\mathrm{c},i}(G_i-\lambda B)}$, where $\lambda B$ has units of joules,

${\frac{d}{d\lambda }\mathrm{Tr}({\sigma }_{\lambda }B)\mid }_{\lambda =0}={\beta }_{\mathrm{l}\mathrm{o}\mathrm{c},i}{\mathrm{Cov}}_{\mathrm{K}\mathrm{M},\sigma }(B,B)\ge 0,$

with ${\mathrm{Cov}}_{\mathrm{K}\mathrm{M},\sigma }(B,B)={\int }_0^1\mathrm{Tr}({\sigma }^{1-u}\delta B{\sigma }^u\delta B)du$ and $\delta B=B-\mathrm{Tr}(\sigma B)I$. It equals the ordinary variance when $[B,G_i]=0$; otherwise no such reduction is asserted. These are conditional model predictions, not reported observations or a theorem of relaxation.