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Connected Gibbs jump generators satisfy detailed balance

Statement

Let $d \geq 2$ , let $E = {1, \dots, d}$ , let $E_{a} \in R$ be finite energies in joules, and let $β > 0$ have units joule $^{- 1}$ . Define

$Z_{β} : = \sum_{a \in E} e^{- β E_{a}}, π_{a} : = Z_{β}^{- 1} e^{- β E_{a}} .$

For each $a \neq b$ , let $w_{a b} \geq 0$ be a rate in seconds $^{- 1}$ satisfying Gibbs detailed balance $π_{a} w_{a b} = π_{b} w_{b a}$ , and assume the undirected graph with edge ${a, b}$ whenever $w_{a b} > 0$ is connected. On $H = C^{d}$ , fix an orthonormal energy basis $(∣ a ⟩)$ , set $H_{s} = \sum_{a} E_{a} ∣ a ⟩ ⟨ a ∣$ , and define $L_{a b} = \sqrt{w_{a b}} ∣ b ⟩ ⟨ a ∣$ for $a \neq b$ . Fix $ℏ > 0$ in joule-seconds. For real functions $f, g : E \to R$ , define $⟨ f, g ⟩_{π} : = \sum_{a} π_{a} f_{a} g_{a}$ , and define the row-generator action on column observables by $(W f)_{a} : = \sum_{b \neq a} w_{a b} (f_{b} - f_{a})$ . Let

$σ : = \sum_{a} π_{a} ∣ a ⟩ ⟨ a ∣, r_{a} : = \sum_{b \neq a} w_{a b} .$

Use the finite GKSL generator with Hamiltonian $H_{s}$ and the jump matrices $L_{a b}$ . Its observable dissipator $D$ is self-adjoint for the $σ$ -GNS inner product of Specified finite GNS detailed balance, $H_{s}$ commutes with $σ$ , and $\ker D = C I$ . On diagonal observables, $W$ is self-adjoint on $L^{2} (E, π)$ with constants as its kernel; for $d \geq 2$ its restriction to $π^{⊥} : = {f : E \to R : \sum_{a} π_{a} f_{a} = 0}$ has a positive spectral gap $γ_{c l}$ . Thus the generator satisfies the specified finite GNS detailed-balance convention and the generic gap conclusion of Faithful scalar-kernel detailed balance gives a relaxation gap applies. This claim does not assert a microscopic bath limit.

Facts & Assumptions

Given: $d \geq 2$ , the energy basis, $E_{a}$ , $β > 0$ , and rates $w_{a b}$ satisfying the finite connected Gibbs detailed-balance hypotheses above.

[F1]

Specified finite GNS detailed balance: the observable dissipator is $D (A) = \sum_{j} (L_{j}^{*} A L_{j} - \frac{1}{2} {L_{j}^{*} L_{j}, A})$ ; GNS detailed balance means $D$ is self-adjoint for $Tr (σ B^{*} A)$ and the Hamiltonian commutes with $σ$ .

[F2]

Finite GKSL generates CPTP semigroups: the finite GKSL generator determined by $H_{s}$ and the finite family $L_{a b}$ generates CPTP maps.

[F3]

Faithful scalar-kernel detailed balance gives a relaxation gap: if a faithful finite GNS detailed-balance model has $\ker D = C I$ , then $D ∣_{M_{0}}$ has a strictly positive gap and its semigroup has the stated exponential relaxation and correlation bounds.

[F4]

Finite matrix spectral and state calculus: finite Hermitian matrices have an orthonormal eigenbasis and the weighted state form is an inner product when the density is faithful.

[F5]

Finite matrix trace cyclicity and nondegeneracy: finite traces are cyclic.

Proof

1.1givenF2algebra

The finite sum $Z_{β}$ is strictly positive, so each $π_{a} > 0$ and $\sum_{a} π_{a} = 1$ . Hence $σ > 0$ , $Tr σ = 1$ , and $[H_{s}, σ] = 0$ . The finite GKSL semigroup exists by [F2].

1.2givenF1F5algebra

Put $R = \sum_{x \neq y} L_{x y}^{*} L_{x y} = \sum_{a} r_{a} ∣ a ⟩ ⟨ a ∣$ . For a diagonal density $ρ = \sum_{a} p_{a} ∣ a ⟩ ⟨ a ∣$ , $L_{x y} ρ L_{x y}^{*} = w_{x y} p_{x} ∣ y ⟩ ⟨ y ∣$ , while the anticommutator contributes $- r_{a} p_{a} ∣ a ⟩ ⟨ a ∣$ . Summing entries gives the Schrödinger population equation

{\dot{p}}_{a} = \sum_{b \neq a} (p_{b} w_{b a} - p_{a} w_{a b}),

For a diagonal observable $f = \sum_{a} f_{a} ∣ a ⟩ ⟨ a ∣$ , direct multiplication gives

D (f) = \sum_{a} (\sum_{b \neq a} w_{a b} (f_{b} - f_{a})) ∣ a ⟩ ⟨ a ∣,

so its diagonal action is exactly $W$ . For $a \neq b$ , every gain term $L_{x y} ∣ a ⟩ ⟨ b ∣ L_{x y}^{*}$ vanishes; the $R$ anticommutator contributes $- (r_{a} + r_{b}) ∣ a ⟩ ⟨ b ∣ / 2$ , and the Hamiltonian commutator contributes $- i (E_{a} - E_{b}) ∣ a ⟩ ⟨ b ∣ / ℏ$ . Thus the coherence equation is

{\dot{ρ}}_{a b} = (- \frac{i (E_{a} - E_{b})}{ℏ} - \frac{r_{a} + r_{b}}{2}) ρ_{a b} .

For $c_{a b} : = π_{a} w_{a b} = c_{b a}$ and sums over $a \neq b$ , the cross terms satisfy $\sum_{a \neq b} c_{a b} f_{b} g_{a} = \sum_{a \neq b} c_{a b} f_{a} g_{b}$ after swapping $a, b$ , so $⟨ W f, g ⟩_{π} = ⟨ f, W g ⟩_{π}$ . Symmetrizing the quadratic form gives

- ⟨ W f, f ⟩_{π} = \frac{1}{2} \sum_{a} \sum_{b \neq a} π_{a} w_{a b} ∣ f_{b} - f_{a} ∣^{2},

so $W$ is self-adjoint on the diagonal subspace.

2.1F1F4step 1.2algebra

For $a \neq b$ , direct multiplication with diagonal $σ$ gives $⟨ ∣ a ⟩ ⟨ b ∣, ∣ c ⟩ ⟨ d ∣ ⟩_{σ} = 0$ unless $(a, b) = (c, d)$ , and $∥ ∣ a ⟩ ⟨ b ∣ ∥_{σ}^{2} = π_{b}$ . Also each jump gain $L_{x y}^{*} ∣ a ⟩ ⟨ b ∣ L_{x y}$ is zero when $a \neq b$ , and the anticommutator with $R$ gives $D (∣ a ⟩ ⟨ b ∣) = - (r_{a} + r_{b}) ∣ a ⟩ ⟨ b ∣ / 2$ . Diagonal and off-diagonal subspaces are orthogonal. Together with the diagonal self-adjointness from step 1.2, these real eigenvalues prove that $D$ is self-adjoint on all of $M_{d} (C)$ .

3.1givenF1F4step 1.2step 2.1algebra

Connectivity and $d \geq 2$ imply $r_{a} > 0$ for each $a$ , so every off-diagonal matrix unit has a strictly negative $D$ -eigenvalue. The Dirichlet identity in step 1.2 vanishes only when $f_{a} = f_{b}$ along every positive-rate edge; connectedness then forces $f$ to be constant. Therefore the diagonal kernel is $C I$ , the off-diagonal kernel is zero, and $\ker D = C I$ . Applying the finite spectral theorem [F4] to the self-adjoint $W ∣_{π^{⊥}}$ shows its finite list of eigenvalues is strictly negative; their minimum absolute value is $γ_{c l} > 0$ .

4.1F1F3step 1.1step 1.2step 2.1step 3.1∎

Steps 1.1, 1.2, 2.1, and 3.1 verify every hypothesis of Specified finite GNS detailed balance and [F3]. Applying [F3] yields the positive gap, exponential relaxation, trace-norm bound, and absolutely integrable centered correlations for this Gibbs jump generator.

Remarks

This is a finite mathematical rate model. Gibbs detailed balance specifies a stationary state and a symmetric generator; it does not derive the rates from a microscopic reservoir or establish that a physical system realizes this model.

Connected Gibbs jump generators satisfy detailed balance

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources