Manuscript Hawk02-D@1 · Hawk02-D.tex · theorem · e6e4b72f

Theorem: Exact energy ledger and asymptotic transfer

Energy transfer from the same dynamics · lines 1092–1144 · proof included or linked

Hawk02-D@1#thm:energy, Hawk02-D@1#eq:energy-ledger

For every allowed input,

⟨ H ⟩_{t} ⟨ H_{S} ⟩_{t} ⟨ H_{int} ⟩_{t} ⟨ H_{B} ⟩_{t} = ω_{0} k ˉ (ρ), = ω_{0} q (t) k ˉ (ρ), = 0, = ω_{0} [1 - q (t)] k ˉ (ρ) .

In particular,

⟨ H_{S} ⟩_{t} ⟶ 0, ⟨ H_{B} ⟩_{t} ⟶ ⟨ H ⟩_{0} .

The source-to-bath current, with positive sign toward the bath, is

J (t) = - \frac{d}{d t} ⟨ H_{S} ⟩_{t} = - ω_{0} q^{'} (t) k ˉ (ρ) .

Its time integral is the bath energy gain. The current is not asserted to be nonnegative at every time.

Proof. The evolved one-body correlation matrix has spatial factor $u_{j} (t) u_{k} (t)$ , multiplied by the input label correlation matrix. Summing the diagonal label entries gives $k ˉ (ρ)$ .

Equation (eq:amplitudes) shows that adjacent amplitudes differ by a factor $- i$ times real numbers after removing their common phase. Hence

ℜ (u_{j} (t) u_{j + 1} (t)) = 0

for every $j$ . Every hopping contribution to the displayed energy expectations therefore vanishes. The onsite contributions give (eq:energy-ledger). The total energy is also conserved by the autonomous unitary.

The limit follows from $q (t) \to 0$ . The current formula is differentiation of the source energy; its integral equals the difference of the bath energies.

Displayed formulas in this passage

Hawk02-D@1#eq:energy-ledger · line 1095
$⟨ H ⟩_{t} ⟨ H_{S} ⟩_{t} ⟨ H_{int} ⟩_{t} ⟨ H_{B} ⟩_{t} = ω_{0} k ˉ (ρ), = ω_{0} q (t) k ˉ (ρ), = 0, = ω_{0} [1 - q (t)] k ˉ (ρ) .$
unlabeled · line 1106
$⟨ H_{S} ⟩_{t} ⟶ 0, ⟨ H_{B} ⟩_{t} ⟶ ⟨ H ⟩_{0} .$
unlabeled · line 1113
$J (t) = - \frac{d}{d t} ⟨ H_{S} ⟩_{t} = - ω_{0} q^{'} (t) k ˉ (ρ) .$
unlabeled · line 1133
$ℜ (u_{j} (t) u_{j + 1} (t)) = 0$

Theorem: Exact energy ledger and asymptotic transfer

Displayed formulas in this passage

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