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The canonical preparation as the unique free-energy minimiser of the declared finite-sector model

Physical scope

The conditional free-energy characterisation of the canonical preparation in the declared finite-sector many-body model at assigned temperature: the Gibbs operator is a declared preparation, its mean energy and entropy are the finite Gibbs sums, and it is the unique minimiser of the free-energy functional over all declared sector preparations, the minimum being the equilibrium free energy. The theorem is conditional on the adopted ensemble postulate and the declared model; it asserts no equilibration, no approach to equilibrium, no thermodynamic limit and no measured thermal quantity.

Statement

Adopt the canonical equilibrium-ensemble postulate The canonical equilibrium-ensemble assumption: the Gibbs preparation of a declared model at assigned temperature together with the state, probability, dynamics and preparation postulates P1, P2, P4, P7 (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, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions) in the readings of Entropy, the Boltzmann constant and the free-energy functional of the declared finite-sector model and Assigned sharp observables and the outcome law of the framework, and assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Work in the finite-dimensional sector class of the entropy definition: the declared fixed- $N$ model of The declared fixed-N many-body model: species sector, one-particle data, interaction and the sector energy observable has sector space $H_{N}^{σ}$ of finite dimension $D \geq 1$ with self-adjoint energy operator $H_{N}^{σ}$ , assigned temperature $T > 0$ , inverse temperature $β = 1 / (k_{B} T)$ and eigen-decomposition $H_{N}^{σ} = \sum_{j = 1}^{D} E_{j} T_{v_{j}, v_{j}}$ with orthonormal eigenbasis $(v_{1}, \dots, v_{D})$ and real eigenvalues $E_{1}, \dots, E_{D}$ . Write $q_{j} : = e^{- β E_{j}} / Z_{N}$ with $Z_{N} : = \sum_{j} e^{- β E_{j}}$ , $ρ_{β} : = \sum_{j} q_{j} T_{v_{j}, v_{j}}$ , and let $F (ρ) = tr (ρ H_{N}^{σ}) - k_{B} T s (ρ)$ and $F_{e q} : = - k_{B} T \log Z_{N}$ be the free-energy functional and equilibrium free energy of Entropy, the Boltzmann constant and the free-energy functional of the declared finite-sector model clauses (e). Then:

(a) the canonical preparation: $Z_{N} \in (0, \infty)$ is a finite sum, $q_{j} > 0$ , $\sum_{j} q_{j} = 1$ , and $ρ_{β}$ is a declared sector preparation of the model with $tr (ρ_{β} H_{N}^{σ}) = \frac{1}{Z_{N}} \sum_{j = 1}^{D} E_{j} e^{- β E_{j}}, s (ρ_{β}) = \log Z_{N} + β tr (ρ_{β} H_{N}^{σ});$

(b) free-energy minimum: for every declared sector preparation $ρ$ one has $F (ρ) \geq F_{e q} = - k_{B} T \log Z_{N}$ ;

(c) uniqueness: equality $F (ρ) = F_{e q}$ holds if and only if $ρ = ρ_{β}$ ; consequently the canonical preparation is the unique minimiser of the free-energy functional over the declared sector preparations, and $F (ρ_{β}) = F_{e q}$ ;

(d) degenerate case: if $H_{N}^{σ} = c i d$ for a real $c$ , then $Z_{N} = D e^{- β c}$ , $ρ_{β} = D^{- 1} i d$ , $s (ρ_{β}) = \log D$ and $F_{e q} = c - k_{B} T \log D$ ;

(e) scope: the statement is conditional on the adopted ensemble postulate and on the declared finite-dimensional model; it asserts no equilibration of any preparation towards $ρ_{β}$ , no rate of approach, no thermodynamic limit, no pressure or volume variable, and, for a laboratory sample, it inherits the tested identification of the preparation and temperature assignment of P7.

Facts & Assumptions

Given: the declared finite-dimensional sector model with energy operator $H_{N}^{σ}$ , eigen-decomposition and parameters $β$ , $T$ , $k_{B}$ ; the canonical preparation $ρ_{β}$ and the functional $F$ of the statement.

[F1]

Supplied finite Gibbs data. For the finite-dimensional self-adjoint $H_{N}^{σ}$ with eigenvalues $E_{1}, \dots, E_{D}$ and orthonormal eigenbasis $(v_{1}, \dots, v_{D})$ , and $β > 0$ : $Z_{N} = \sum_{j} e^{- β E_{j}} \in (0, \infty)$ , $ρ_{β} \in D (H_{N}^{σ})$ , $tr (ρ_{β} H_{N}^{σ}) = Z_{N}^{- 1} \sum_{j} e^{- β E_{j}} E_{j}$ , $tr (ρ_{β} H_{N}^{σ}) - β^{- 1} s (ρ_{β}) = - β^{- 1} \log Z_{N}$ , and for every density operator $ρ$ on $H_{N}^{σ}$ one has $tr (ρ H_{N}^{σ}) - β^{- 1} s (ρ) \geq - β^{- 1} \log Z_{N}$ with $s (ρ) \leq \log D$ and equality $s (ρ) = \log D$ exactly for $ρ = D^{- 1} i d$ (Finite Gibbs free-energy minimum clauses (a), (b), (c)).

[F2]

Spectral decomposition of a preparation. A density operator $ρ$ on the finite-dimensional space $H_{N}^{σ}$ has the form $ρ = \sum_{i = 1}^{D} w_{i} T_{u_{i}, u_{i}}$ with $w_{i} \geq 0$ , $\sum_{i} w_{i} = 1$ and $(u_{i})$ an orthonormal eigenbasis, by Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely and Rank-one operator calculus applied as in Finite Gibbs free-energy minimum clause (a); with $p_{i j} : = ∣ ⟨ u_{i}, v_{j} ⟩ ∣^{2}$ one has $\sum_{j} p_{i j} = 1$ and $\sum_{i} p_{i j} = 1$ , because $(u_{i})$ and $(v_{j})$ are orthonormal bases.

[F3]

Exact scalar logarithm suppliers. For $x > 0$ , $\log^{'} (x) = 1 / x$ and $\log x = \int_{1}^{x} d t / t$ (The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t); $\log$ is continuous, $\log 1 = 0$ and $\log (x / y) = \log x - \log y$ for $x, y > 0$ (Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm). For a continuous real function on $[a, b]$ differentiable on $(a, b)$ , $a < b$ , there is $ξ \in (a, b)$ with $f (b) - f (a) = f^{'} (ξ) (b - a)$ (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ). The strict tangent bound and its equality case are derived below from these statements, rather than assumed from a strict Jensen theorem.

Proof

technique · direct; express the free-energy difference as a sum of nonnegative scalar defects over the two orthonormal bases, with the zero-weight convention made explicit; the strict logarithm tangent bound determines the equality case, including degenerate Gibbs eigenspaces

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

1.1F1algebra

[The canonical preparation and its data] By [F1], $Z_{N} = \sum_{j} e^{- β E_{j}}$ is a finite sum with $0 < Z_{N} < \infty$ ; since $D \geq 1$ each term is positive, so $q_{j} = e^{- β E_{j}} / Z_{N} > 0$ with $\sum_{j} q_{j} = 1$ . By [F1] the operator $ρ_{β} = \sum_{j} q_{j} T_{v_{j}, v_{j}}$ is a density operator on $H_{N}^{σ}$ , hence a declared sector preparation of the model, and the two displayed identities are the clauses of [F1] together with the entropy definition. This proves (a).

1.2F3algebra

[Strict scalar defects, including zero weights] By [F3] and the mean value theorem applied to $\log$ between $1$ and $t > 0$ , $t \neq 1$ , one has $\log t = (t - 1) / ξ$ with $ξ$ strictly between $1$ and $t$ ; for $t > 1$ this is less than $t - 1$ , and for $0 < t < 1$ it is also less than $t - 1$ because the numerator is negative and $ξ < 1$ . Thus $\log t \leq t - 1$ , with equality exactly at $t = 1$ . For $w \geq 0$ , $r > 0$ define $b (w, r) : = w \log (w / r)$ when $w > 0$ and $b (0, r) : = 0$ , and $d (w, r) : = b (w, r) - w + r$ . If $w > 0$ , apply the bound to $t = r / w$ and the quotient law to get $d (w, r) \geq 0$ , with equality exactly at $w = r$ ; if $w = 0$ , $d (0, r) = r > 0$ . Likewise $h (q, r) : = \log r + (q - r) / r - \log q \geq 0$ for $q, r > 0$ , with equality exactly at $q = r$ , by applying the bound to $t = q / r$ .

1.3F1algebra

[The degenerate case] If $H_{N}^{σ} = c i d$ then $E_{j} = c$ for all $j$ and [F1] gives $Z_{N} = D e^{- β c}$ , $q_{j} = D^{- 1}$ , $ρ_{β} = D^{- 1} i d$ ; by [F1] $s (ρ_{β}) = \log D$ and $F_{e q} = - k_{B} T \log (D e^{- β c}) = c - k_{B} T \log D$ . This proves (d).

2.1F1F2F3step 1.2algebra

[The free-energy difference for every mixed preparation] Let $ρ = \sum_{i} w_{i} T_{u_{i}, u_{i}}$ be as in [F2] and put $p_{i j} = ∣ ⟨ u_{i}, v_{j} ⟩ ∣^{2}$ , $r_{i} = \sum_{j} p_{i j} q_{j}$ . Each $r_{i} > 0$ and $\sum_{i} r_{i} = \sum_{j} q_{j} \sum_{i} p_{i j} = 1$ . Use the entropy convention $η (w) = w \log w$ for $w > 0$ and $η (0) = 0$ (the notation $0 \log 0 = 0$ does not define $\log 0$ ). The trace pairing and the entropy definition give $tr (ρ H_{N}^{σ}) = \sum_{i, j} w_{i} p_{i j} E_{j}$ and $s (ρ) = - \sum_{i} η (w_{i})$ . Since $β E_{j} = - \log q_{j} - \log Z_{N}$ , $Δ : = β (F (ρ) - F_{e q}) = \sum_{i} (η (w_{i}) - w_{i} \sum_{j} p_{i j} \log q_{j})$ . Define $a_{i} : = \log r_{i} - \sum_{j} p_{i j} \log q_{j} = \sum_{j} p_{i j} h (q_{j}, r_{i})$ , the last equality using $\sum_{j} p_{i j} = 1$ and $\sum_{j} p_{i j} (q_{j} - r_{i}) = 0$ . Step 1.2 gives $a_{i} \geq 0$ , with equality exactly when $q_{j} = r_{i}$ for every $j$ with $p_{i j} > 0$ . By the quotient law for positive $w_{i}$ and by the explicit zero convention otherwise, $Δ = \sum_{i} b (w_{i}, r_{i}) + \sum_{i} w_{i} a_{i} = \sum_{i} d (w_{i}, r_{i}) + \sum_{i} w_{i} a_{i}$ , because $\sum_{i} (w_{i} - r_{i}) = 0$ . All summands are nonnegative, so $F (ρ) \geq F_{e q}$ , proving (b) for every density operator, including those with zero eigenvalues.

3.1F1F2step 1.2step 2.1algebra

[Equality, strict zero-weight exclusion and degeneracies] If $F (ρ) = F_{e q}$ , the finite sum of nonnegative defects in step 2.1 vanishes. Hence $d (w_{i}, r_{i}) = 0$ for every $i$ ; step 1.2 gives $w_{i} = r_{i} > 0$ , since $d (0, r_{i}) = r_{i} > 0$ excludes every zero weight. Thus $w_{i} a_{i} = 0$ forces $a_{i} = 0$ for every row, and step 2.1 gives $q_{j} = r_{i} = w_{i}$ whenever $p_{i j} > 0$ . In the expansion of $u_{i}$ in the orthonormal basis $(v_{j})$ , the coefficients at all $q_{j} \neq w_{i}$ therefore vanish, so $ρ_{β} u_{i} = w_{i} u_{i}$ . Since also $ρ u_{i} = w_{i} u_{i}$ and $(u_{i})$ is a basis, the two operators agree: $ρ = ρ_{β}$ . This argument allows arbitrary changes of basis within any repeated-eigenvalue Gibbs eigenspace. Conversely $ρ = ρ_{β}$ gives equality by [F1]. Together with step 2.1, this proves (c) with no full-rank assumption on the competing preparation.

4.1step 1.1step 1.2step 2.1step 1.3step 3.1∎

[Conclusion and scope] Step 1.1 proves (a), step 2.1 proves (b), step 3.1 proves (c), and step 1.3 proves (d). The argument is an optimisation inside the declared finite-dimensional sector at the adopted ensemble parameters; no equilibration, rate, thermodynamic limit or measured thermal quantity entered, which is (e).

Remarks

Why the minimiser is unique. The equality analysis shows that a minimiser must be diagonal in the same eigenbasis with the same weights; it excludes the block-diagonal counterexample in which $ρ$ is diagonal in a different basis, because equal weights on a degenerate eigenspace reassemble the same projector.
The adopted part. The minimum property is a theorem about the functional; that a physical sample is described by $ρ_{β}$ remains the adopted assumption of The canonical equilibrium-ensemble assumption: the Gibbs preparation of a declared model at assigned temperature, not a consequence of the optimisation.

The canonical preparation as the unique free-energy minimiser of the declared finite-sector model

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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