The two-site bond model: exact canonical weights, mean energy, entropy and free energy of one particle on two sites

Adopt P1, P2, P4 and P7 in the readings of Assigned sharp observables and the outcome law of the framework and Entropy, the Boltzmann constant and the free-energy functional of the declared finite-sector model, and work in the finite-mode realization of the declared fixed-$N$ model of The declared fixed-N many-body model: species sector, one-particle data, interaction and the sector energy observable with: one-particle space $K={\mathbb{C}}^2$, declared sites $\mathrm{\mid }1⟩,\mathrm{\mid }2⟩$ an orthonormal basis of $K$, particle number $N=1$, single site-occupation sector, and the declared one-particle operator (the bond Hamiltonian) $h:=-t(T_{\mathrm{\mid }1⟩,\mathrm{\mid }2⟩}+T_{\mathrm{\mid }2⟩,\mathrm{\mid }1⟩}),t>0\text{ in joules}\mathrm{.}$ Adopt the canonical equilibrium postulate The canonical equilibrium-ensemble assumption: the Gibbs preparation of a declared model at assigned temperature at assigned temperature $T>0$ and inverse temperature $\beta =1\mathrm{/}(k_{B}T)$, and read $\epsilon :=2t$ as the tunnelling splitting. Then:

(a) $h$ has the orthonormal eigenbasis $u_{-}:=\frac{\mathrm{\mid }1⟩+\mathrm{\mid }2⟩}{\sqrt{2}},u_{+}:=\frac{\mathrm{\mid }1⟩-\mathrm{\mid }2⟩}{\sqrt{2}},h{u}_{\mp }=\mp t{u}_{\mp },$ so its eigenvalues are $-t$ and $+t$; equivalently $h+t\mathrm{i}\mathrm{d}$ is the two-level Hamiltonian of Closed formulas for the two-level canonical weights, mean energy, entropy and free energy with levels $0$ and $\epsilon =2t$ and eigenvectors $u_{-},u_{+}$;

(b) the canonical preparation of $h+t\mathrm{i}\mathrm{d}$ and hence of $h$ is ${\rho }_{\beta }=\frac{T_{u_{-},u_{-}}+e^{-2\beta t}{T}_{u_{+},u_{+}}}{1+e^{-2\beta t}},$ with shifted partition sum $Z_0=1+e^{-2\beta t}$ and physical partition trace $Z_h=\mathrm{tr}{e}^{-\beta h}=e^{\beta t}{Z}_0$;

(c) the exact means, entropy and free energy are $⟨h{⟩}_{\beta }=\frac{2t}{e^{2\beta t}+1}-t=-t\tanh (\beta t),s({\rho }_{\beta })=\log (1+e^{-2\beta t})+\frac{2\beta t{e}^{-2\beta t}}{1+e^{-2\beta t}},F=-\frac{1}{\beta }\log (1+e^{-2\beta t})-t,$ and for every $\beta >0$ one has $-t<⟨h{⟩}_{\beta }<0$ and $0\le s({\rho }_{\beta })<\log 2$;

(d) scope: the computation is an exact statement about the declared two-dimensional bond model at the assigned temperature; it identifies no physical double-well or molecular system, asserts no measured splitting and no transition, and the temperature assignment is the tested identification of P7.

• The shift by $t\mathrm{i}\mathrm{d}$. Adding a constant to the Hamiltonian multiplies $Z$ by an exponential and leaves the canonical preparation and the entropy unchanged while shifting the free energy by the same constant; the example uses this to instantiate the supplied two-level formulas, whose levels are $0$ and $\epsilon$.
• Low- and high-temperature values. The formulas give $s\to 0$ and $⟨h{⟩}_{\beta }\to -t$ as the exponential factor is suppressed, and $s\to \log 2$, $⟨h{⟩}_{\beta }\to 0$ as it approaches one; the example states the exact formulas and the bounds, not a limit theorem.