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A site-localised preparation of the two-site bond model never becomes canonical, and its readouts need not take their thermal means

Physical scope

A hypothetical exactly solved scenario in the declared two-site bond model: the site-localised pure preparation at a site, its unitary evolution, the stationary canonical preparation of the adopted ensemble, and two readouts - the site occupation and the energy - whose means either do or do not coincide with their canonical values at specified times. The scenario is a model deduction from the declared preparation and P4; it asserts no observed equilibration, no measured occupation or energy, and no statement about any laboratory system.

Statement

Adopt P1, P2, P4 and P7 in the readings of Assigned energy observable, unitary flow and the two pictures of autonomous evolution, 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, work in the two-site bond model of The two-site bond model: exact canonical weights, mean energy, entropy and free energy of one particle on two sites with coupling $t > 0$ , sites $∣ 1 ⟩, ∣ 2 ⟩$ , bond Hamiltonian $h = - t (T_{∣ 1 ⟩, ∣ 2 ⟩} + T_{∣ 2 ⟩, ∣ 1 ⟩})$ and eigenbasis $u_{-} = (∣ 1 ⟩ + ∣ 2 ⟩) / \sqrt{2}$ , $u_{+} = (∣ 1 ⟩ - ∣ 2 ⟩) / \sqrt{2}$ with $h u_{\mp} = \mp t u_{\mp}$ , and adopt the canonical equilibrium postulate The canonical equilibrium-ensemble assumption: the Gibbs preparation of a declared model at assigned temperature at assigned temperature $T > 0$ , $β = 1 / (k_{B} T)$ . Consider the hypothetical declared preparation $ρ (0) : = T_{∣ 1 ⟩, ∣ 1 ⟩}$ (the particle localised at site $1$ ) evolving by $ρ (τ) = U (τ) ρ (0) U (τ)^{*}$ with $U (τ) = \exp (- i τ h / ℏ)$ . Then:

(a) $ρ (0) \neq ρ_{β}$ , and for every $τ \in R$ one has $ρ (τ) \neq ρ_{β}$ : the unitary evolution of the model never produces the canonical preparation from this preparation;

(b) the site- $1$ occupation readout $(P_{1}, 1)$ with $P_{1} : = T_{∣ 1 ⟩, ∣ 1 ⟩}$ , assigned in the dimensionless particle-count unit, has the exact mean $⟨ P_{1} ⟩ (τ) = \cos^{2} ⁣ (\frac{t τ}{ℏ}),$ and $tr (ρ_{β} P_{1}) = 1 / 2$ ; in particular $⟨ P_{1} ⟩ (0) = 1 \neq 1 / 2$ , so at the initial time the readout does not take its canonical mean;

(c) the energy readout $h$ has the constant mean $tr (ρ (τ) h) = 0$ for every $τ$ , while its canonical mean is $tr (ρ_{β} h) = - t \tanh (β t) < 0$ ; hence the energy readout never takes its canonical thermal value;

(d) scope and interpretation: the scenario is a hypothetical, exactly-solved model deduction in one declared preparation; it shows that agreement or disagreement of individual readout means with their canonical values is not convergence of the preparation to the canonical one, and that the model's P4 dynamics does not produce the canonical ensemble. It asserts no observed equilibration or non-equilibration of a sample, no measured occupation or energy, and no claim about environments or open systems.

Facts & Assumptions

Given: the declared bond model with $t > 0$ , the canonical preparation $ρ_{β}$ , the site-localised preparation $ρ (0) = T_{∣ 1 ⟩, ∣ 1 ⟩}$ , the flow $U (τ)$ , and the times $τ \in R$ .

[F1]

Spectral flow of the bond Hamiltonian. For the assigned energy observable $h$ with distinct eigenvalues $\mp t$ and orthonormal eigenvectors $u_{\mp}$ , the spectral projection-valued measure has the atoms $E ({- t}) = T_{u_{-}, u_{-}}$ and $E ({t}) = T_{u_{+}, u_{+}}$ , so the flow of Assigned energy observable, unitary flow and the two pictures of autonomous evolution clause (b) is $U (τ) = e^{i t τ / ℏ} T_{u_{-}, u_{-}} + e^{- i t τ / ℏ} T_{u_{+}, u_{+}}$ and maps $ψ_{0} = c_{-} u_{-} + c_{+} u_{+}$ to $c_{-} e^{i t τ / ℏ} u_{-} + c_{+} e^{- i t τ / ℏ} u_{+}$ ; the mean of a bounded observable in a pure preparation is $⟨ A ψ, ψ ⟩$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure, Assigned energy observable, unitary flow and the two pictures of autonomous evolution, Mean, second moment and variance of an assigned observable in a preparation clause (a)).

[F2]

Euler formula and parity. For real $θ$ , $e^{i θ} = \cos θ + i \sin θ$ , $\cos (- θ) = \cos θ$ , $\sin (- θ) = - \sin θ$ , and $\cos^{2} θ + \sin^{2} θ = 1$ , so $(e^{i θ} + e^{- i θ}) / 2 = \cos θ$ and $(e^{i θ} - e^{- i θ}) / 2 = i \sin θ$ ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ , Parity and the Pythagorean identity for sine and cosine).

[F3]

Canonical data. The canonical preparation of the bond model is $ρ_{β} = Z^{- 1} (T_{u_{-}, u_{-}} + e^{- 2 β t} T_{u_{+}, u_{+}})$ with $Z = 1 + e^{- 2 β t}$ , its mean energy is $- t \tanh (β t) \in (- t, 0)$ for every $β > 0$ , and $tr (ρ_{β} T_{u_{-}, u_{-}}) \neq tr (ρ_{β} T_{u_{+}, u_{+}})$ (The two-site bond model: exact canonical weights, mean energy, entropy and free energy of one particle on two sites clauses (b), (c), Closed formulas for the two-level canonical weights, mean energy, entropy and free energy).

[F4]

Invariance statement. In the finite-sector class, if $ρ (0) \neq ρ_{β}$ then $ρ (τ) \neq ρ_{β}$ for every $τ$ (Unitary evolution preserves entropy and cannot produce the canonical preparation from a different one clause (c)).

Proof

technique · direct; express the initial site vector in the energy eigenbasis, insert the supplied two-level flow, and compare the readout means with the canonical data

Given: the data of the statement and [F1]-[F3].

1.1F1F2algebra

[The evolved preparation and the site readout] With $ψ_{0} = ∣ 1 ⟩ = (u_{-} + u_{+}) / \sqrt{2}$ , [F1] gives $ψ (τ) = U (τ) ψ_{0} = (e^{i θ} u_{-} + e^{- i θ} u_{+}) / \sqrt{2}$ , where $θ : = t τ / ℏ$ ; by [F2] and $u_{-} = (∣ 1 ⟩ + ∣ 2 ⟩) / \sqrt{2}$ , $u_{+} = (∣ 1 ⟩ - ∣ 2 ⟩) / \sqrt{2}$ this equals $\cos θ ∣ 1 ⟩ + i \sin θ ∣ 2 ⟩$ , a unit vector. Since $T_{∣ 1 ⟩, ∣ 1 ⟩} = \frac{1}{2} (I + σ_{z})$ with $σ_{z} = T_{∣ 1 ⟩, ∣ 1 ⟩} - T_{∣ 2 ⟩, ∣ 2 ⟩}$ , and $σ_{z} ψ (τ) = \cos θ ∣ 1 ⟩ - i \sin θ ∣ 2 ⟩$ , the site-occupation mean is $⟨ P_{1} ⟩ (τ) = ∣ \cos θ ∣^{2} = \cos^{2} θ = \cos^{2} (t τ / ℏ)$ , in particular $⟨ P_{1} ⟩ (0) = 1$ .

1.2F3algebra

[Canonical means and the initial comparison] By [F3], $tr (ρ_{β} h) = - t \tanh (β t) < 0$ ; moreover $ρ_{β}$ is diagonal in the basis $(u_{-}, u_{+})$ with strictly positive diagonal weights, so $tr (ρ_{β} σ_{z}) = 0$ because $σ_{z}$ exchanges the two basis vectors, and $tr (ρ_{β} P_{1}) = \frac{1}{2}$ ; and $tr (ρ (0) P_{1}) = 1 \neq \frac{1}{2}$ , so $ρ (0) \neq ρ_{β}$ .

2.1F4step 1.2algebra

[The preparation is never canonical] Step 1.2 shows $ρ (0) \neq ρ_{β}$ ; applying [F4] to the declared finite-sector model gives $ρ (τ) \neq ρ_{β}$ for every $τ \in R$ , which is (a).

2.2F3step 1.1step 1.2algebra

[The energy readout never takes its thermal value] Since $h$ commutes with $U (τ)$ , $tr (ρ (τ) h) = tr (ρ (0) h) = ⟨ h Ψ, Ψ ⟩$ for $Ψ : = ∣ 1 ⟩$ , and this vanishes because $h Ψ = - t ∣ 2 ⟩$ is orthogonal to $Ψ$ ; by step 1.2 the canonical mean is $- t \tanh (β t) < 0$ , so the two values never agree. This proves (c), and step 1.1 proves (b) together with the comparison of step 1.2.

3.1step 2.1step 2.2∎

[Conclusion and scope] Steps 2.1-2.2 prove (a)-(c) for the declared hypothetical preparation. The scenario is an exact model deduction from the declared preparation, the bond Hamiltonian and the canonical data; no observed equilibration, measured readout or laboratory system entered, which is (d).

Remarks

What the scenario shows. The preparation never equals the canonical one, yet the site-occupation readout has the same mean $1 / 2$ as the canonical preparation at the times when $\cos^{2} (t τ / ℏ) = 1 / 2$ ; the energy readout meanwhile never takes its canonical mean. Readout coincidence is therefore not state convergence, and different readouts behave differently under the same unitary flow.
Hypothetical status. No apparatus, sampling or detection model is specified; the scenario is a conditional statement about the declared model, and any comparison with an experiment would require the declared preparation, isolation, detection and sampling models of P7.

A site-localised preparation of the two-site bond model never becomes canonical, and its readouts need not take their thermal means

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources