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Finite-bath canonical error bound

Statement

Let $I$ be a finite nonempty subsystem state set, let $E_{i} \in R$ satisfy $∣ E_{i} ∣ \leq L$ for $L \geq 0$ , and let the dimensionless bath entropy $s_{B}$ be $C^{2}$ on an open interval $U$ containing $[u_{0} - L, u_{0} + L]$ , with $∣ s_{B}^{''} (u) ∣ \leq M$ on that closed interval for $M \geq 0$ . Use one energy unit for $E_{i}, u_{0}, L$ , so $β = s_{B}^{'} (u_{0})$ has inverse-energy units and $M$ has inverse-energy-squared units; $δ = M L^{2} / 2$ is dimensionless. Put $Z : = \sum_{j \in I} e^{- β E_{j}}$ and $q_{i} : = e^{- β E_{i}} / Z$ . Suppose the joint shell weights induce the normalized subsystem law $p_{i} : = e^{s_{B} (u_{0} - E_{i})} / \sum_{j \in I} e^{s_{B} (u_{0} - E_{j})}$ . Then $∣ s_{B} (u_{0} - E_{i}) - s_{B} (u_{0}) + β E_{i} ∣ \leq δ$ for every $i$ , and $\sum_{i} ∣ p_{i} - q_{i} ∣ \leq e^{2 δ} - 1$ .

Facts & Assumptions

Given: As stated in the item.

[A1]

$I$ is finite and nonempty; $E_{i} \in R$ ; $L, M \geq 0$ ; $∣ E_{i} ∣ \leq L$ ; and $s_{B}$ is $C^{2}$ on an open interval U containing $[u_{0} - L, u_{0} + L]$ with the stated second-derivative bound. Both laws are normalized on the same state set.

[F1]

Countable canonical model: for a finite energy list and real $β$ , $q_{i} = e^{- β E_{i}} / Z$ is the normalized canonical law.

[F2]

A uniform derivative bound gives a uniform Taylor remainder bound: a bound $∣ s_{B}^{''} ∣ \leq M$ gives the first-order Taylor error at most $M ∣ E_{i} ∣^{2} / 2$ .

[F3]

The real exponential function and the number $e$ by a power series, The natural logarithm as the inverse of the exponential function, and Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm: $\exp$ is a positive bijection with inverse $\log$ , and $\log (x y) = \log x + \log y$ for positive $x, y$ . Thus $\exp (u + v) = \exp (u) \exp (v)$ : both positive sides have logarithm $u + v$ , and $\log$ is injective.

Proof

Given: The finite subsystem, the twice differentiable bath entropy with bounded second derivative, and the stated shell-weight rule.

1.1A1F2

Apply [F2] to $s_{B}$ at $u_{0}$ with increment $- E_{i}$ . It gives $s_{B} (u_{0} - E_{i}) = s_{B} (u_{0}) - β E_{i} + r_{i}$ where $∣ r_{i} ∣ \leq M E_{i}^{2} / 2 \leq δ$ .

2.1F1F3step 1.1

The marginal is therefore $p_{i} = q_{i} e^{r_{i}} / C$ , where $C : = \sum_{j} q_{j} e^{r_{j}}$ . The bounds $e^{- δ} \leq e^{r_{j}} \leq e^{δ}$ and $\sum_{j} q_{j} = 1$ imply $e^{- δ} \leq C \leq e^{δ}$ , so $e^{- 2 δ} \leq p_{i} / q_{i} \leq e^{2 δ}$ .

3.1F1F3step 2.1algebra∎

For $x \in [e^{- 2 δ}, e^{2 δ}]$ , $∣ x - 1 ∣ \leq e^{2 δ} - 1$ . Thus $\sum_{i} ∣ p_{i} - q_{i} ∣ = \sum_{i} q_{i} ∣ p_{i} / q_{i} - 1 ∣ \leq (e^{2 δ} - 1) \sum_{i} q_{i} = e^{2 δ} - 1$ . The case $δ = 0$ is included and gives equality of the laws.

Finite-bath canonical error bound

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources