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Unitary evolution preserves entropy and cannot produce the canonical preparation from a different one

Physical scope

The conditional conservation statement of the declared finite-sector model: under the unitary flow of the model's self-adjoint energy operator, every declared sector preparation keeps its entropy, the adopted canonical preparation is stationary, and a preparation different from it is never mapped to it at any time, so the model's unitary dynamics alone does not produce the canonical preparation. The theorem is about the exact closed-system model and asserts nothing about open systems, about equilibration of individual readouts, about time averages, or about any laboratory sample.

Statement

Adopt the autonomous-dynamics postulate P4 (Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution) in the reading of Assigned energy observable, unitary flow and the two pictures of autonomous evolution, the state postulate P1 and the observable postulate P2 (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes) in the readings of The declared fixed-N many-body model: species sector, one-particle data, interaction and the sector energy observable and Entropy, the Boltzmann constant and the free-energy functional of the declared finite-sector model, and adopt the canonical equilibrium-ensemble postulate The canonical equilibrium-ensemble assumption: the Gibbs preparation of a declared model at assigned temperature for the canonical preparation $ρ_{β}$ ; assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Work in the finite-sector class of the entropy definition: the declared model has finite-dimensional sector space $H_{N}^{σ}$ , self-adjoint energy operator $H_{N}^{σ}$ with flow $U (t) = \exp (- i t H_{N}^{σ} / ℏ)$ , and every declared sector preparation evolves as $ρ (t) = U (t) ρ (0) U (t)^{*}$ under P4 clause (iii). Then:

(a) entropy is conserved: for every declared sector preparation $ρ (0)$ of the model and every $t \in R$ , $s (ρ (t)) = s (ρ (0));$

(b) the canonical preparation is stationary: $U (t) ρ_{β} U (t)^{*} = ρ_{β}$ for every $t$ , and every thermal mean $tr (ρ_{β} A)$ of an assigned observable is time independent;

(c) no unitary production of the canonical preparation: if $ρ (0) \neq ρ_{β}$ then $ρ (t) \neq ρ_{β}$ for every $t \in R$ ; equivalently, no evolution of this isolated model maps a non-canonical declared preparation to the canonical one at any time;

(d) scope: clauses (a)-(c) concern the model's exact unitary flow in the finite-sector class and the adopted canonical preparation. They assert nothing about open systems or environment coupling, about the time behaviour of individual readouts or their time averages, about an approach of any readout to a thermal value in a limit, about environments or measurement models, and about any laboratory sample; in particular clause (c) is not a claim that no physical system ever approaches thermal equilibrium, but the statement that the model's P4 dynamics alone does not do so.

Facts & Assumptions

Given: the declared finite-sector model with self-adjoint $H_{N}^{σ}$ , flow $U (t)$ , canonical preparation $ρ_{β}$ of the ensemble postulate, and a declared sector preparation $ρ (0)$ with $ρ (t) = U (t) ρ (0) U (t)^{*}$ .

[F1]

Unitary flow and density evolution. $U (t) = \exp (- i t H_{N}^{σ} / ℏ)$ is a strongly continuous unitary group with $U (t)^{*} = U (t)^{- 1}$ and $U (t + s) = U (t) U (s)$ , and $ρ (t) = U (t) ρ (0) U (t)^{*}$ is a density operator for every $t$ (Assigned energy observable, unitary flow and the two pictures of autonomous evolution clauses (b), (c), Density operators and scalar spectral probability).

[F2]

Spectral decomposition and entropy. Every density operator $ρ$ on the finite-dimensional $H_{N}^{σ}$ has an orthonormal eigenbasis $(u_{i})$ with eigenvalues $w_{i} \geq 0$ , and $s (ρ) = - \sum_{i : w_{i} > 0} w_{i} \log w_{i}$ depends only on the eigenvalue multiset (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely, Rank-one operator calculus, Entropy, the Boltzmann constant and the free-energy functional of the declared finite-sector model clause (b), The natural logarithm as the inverse of the exponential function).

[F3]

Canonical preparation. $ρ_{β} = Z_{N}^{- 1} e^{- β H_{N}^{σ}}$ with $Z_{N} = \sum_{j} e^{- β E_{j}}$ is diagonal in an orthonormal eigenbasis $(v_{j})$ of $H_{N}^{σ}$ with strictly positive weights $q_{j} = Z_{N}^{- 1} e^{- β E_{j}}$ , and it is a declared sector preparation (Finite Gibbs free-energy minimum clause (a), The canonical equilibrium-ensemble assumption: the Gibbs preparation of a declared model at assigned temperature clause (b)).

Proof

technique · direct; compare eigenvalue multisets under unitary conjugation and use that $\rho_\beta$ is a spectral function of the generator

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

1.1F1F2algebra

[Entropy conservation] Let $ρ (0) = \sum_{i} w_{i} T_{u_{i}, u_{i}}$ be the spectral decomposition of [F2]. By [F1], $U (t)$ is unitary, so $(U (t) u_{i})$ is an orthonormal basis and $ρ (t) = \sum_{i} w_{i} T_{U (t) u_{i}, U (t) u_{i}}$ : the eigenvalues of $ρ (t)$ are the same $w_{i}$ with the same multiplicities. By [F2] the entropy depends only on that multiset, so $s (ρ (t)) = s (ρ (0))$ for every $t$ . This proves (a).

1.2F1F3algebra

[Stationarity of the canonical preparation] By [F3] $ρ_{β}$ is diagonal in the eigenbasis $(v_{j})$ of $H_{N}^{σ}$ ; the same basis diagonalizes $U (t)$ with eigenvalues $e^{- i t E_{j} / ℏ}$ , so $U (t) T_{v_{j}, v_{j}} U (t)^{*} = T_{v_{j}, v_{j}}$ for every $j$ and hence $U (t) ρ_{β} U (t)^{*} = ρ_{β}$ . Every thermal mean $tr (ρ_{β} A)$ is therefore time independent by [F1]. This proves (b).

2.1F1step 1.2algebra

[No unitary production of the canonical preparation] Suppose $ρ (t) = ρ_{β}$ for some $t$ . Applying the unitary $U (t)^{*}$ on both sides and using $U (t)^{*} = U (t)^{- 1}$ from [F1], $ρ (0) = U (t)^{*} ρ_{β} U (t) = ρ_{β}$ by step 1.2 applied to $- t$ . Hence $ρ (0) \neq ρ_{β}$ implies $ρ (t) \neq ρ_{β}$ for every $t$ , which is (c).

3.1step 1.1step 1.2step 2.1∎

[Conclusion and scope] Steps 1.1-2.1 prove (a)-(c). The argument used only the unitary flow, the eigenvalue multiset of a preparation and the spectral form of $ρ_{β}$ ; no environment, measurement model, readout equilibration or laboratory sample entered, which is (d).

Remarks

Entropy versus readout equilibration. Clause (a) constrains the entropy of the full sector preparation; it does not prevent the outcome law of a chosen readout from approaching a stationary average under the flow. Clause (c) is the sharp statement about the preparation itself.
Open systems. The theorem does not apply to a reduced description obtained by tracing out an environment; such a description requires its own model assumption, as P4 clause (v) records.

Unitary evolution preserves entropy and cannot produce the canonical preparation from a different one

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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