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Finite-bath canonical error under a smooth shell-entropy model

Physical scope

A finite subsystem weakly coupled additively to a bath whose dimensionless shell entropy is twice continuously differentiable with bounded curvature on the full exchanged-energy interval; this is a conditional model bound, not a universal finite-size estimate.

Statement

Assume AC for the regular-shell reading of Microcanonical sampling assumption. Let the subsystem have a finite nonempty state set $Ω$ and internal energies $E_{x}$ in joules with $∣ E_{x} ∣ \leq L$ , where $L$ is in joules. The bath model supplies an open interval $U$ containing $[u_{0} - L, u_{0} + L]$ , with bath energies $u, u_{0} \in U$ in joules, and a dimensionless shell-log-weight function $s_{B} : U \to R$ ; define the positive shell weight by $W_{B} (u) : = e^{s_{B} (u)}$ . Assume $s_{B} \in C^{2} (U)$ and $∣ s_{B}^{''} (u) ∣ \leq M$ on the displayed interval, where $M$ is in $J^{- 2}$ . The model adopts $β = s_{B}^{'} (u_{0}) > 0$ in $J^{- 1}$ , $k_{B}$ in $J K^{- 1}$ , and $T = (k_{B} β)^{- 1}$ in kelvins. Assume weak additive coupling and separately stipulate that microcanonical shell sampling induces subsystem weights $P_{B} (x) \propto W_{B} (u_{0} - E_{x}) = e^{s_{B} (u_{0} - E_{x})}$ ; this induced-marginal rule is an additional bath-model assumption, not a consequence of the microcanonical sampling postulate alone. Let $P_{C} (x) = e^{- β E_{x}} / Z$ be the corresponding canonical law. With $δ = M L^{2} / 2$ (dimensionless), the shell log-weight remainder obeys $∣ s_{B} (u_{0} - E_{x}) - s_{B} (u_{0}) + β E_{x} ∣ \leq δ,$ and the normalized laws satisfy $\sum_{x \in Ω} ∣ P_{B} (x) - P_{C} (x) ∣ \leq e^{2 δ} - 1.$ This is an $ℓ^{1}$ probability-vector bound; it does not assert a universal error or a thermodynamic limit.

Facts & Assumptions

Given: The finite subsystem, the supplied bath shell-log-weight function and curvature bound, the adopted energy-additive model, and the separately stipulated normalized bath-shell marginal and canonical laws.

[F1]

Finite statistical microstate model: $Ω$ is finite and nonempty and each $E_{x}$ is a fixed internal energy in joules.

[F2]

Canonical sampling assumption for finite and normalized statistical models: the canonical comparison law has weights $e^{- β E_{x}} / Z$ when $Z > 0$ is finite.

[F3]

Microcanonical sampling assumption: the postulate adopts uniform sampling on a nonempty finite energy window or the normalized coarea law on an admitted regular shell, with AC for the regular-shell branch. It does not derive the subsystem marginal [A2].

[L1]

Finite-bath canonical error bound (current batch 11 draft): for finite energies $∣ E_{x} ∣ \leq L$ , $s_{B} \in C^{2}$ on the stated interval, $∣ s_{B}^{''} ∣ \leq M$ , $β = s_{B}^{'} (u_{0})$ , and normalized weights proportional to $e^{s_{B} (u_{0} - E_{x})}$ , the Taylor remainder is at most $δ = M L^{2} / 2$ and the canonical comparison law has $ℓ^{1}$ error at most $e^{2 δ} - 1$ . Its full current proof was inspected; its draft status remains conditional.

[A1]

$E_{x}, u_{0}, L$ are measured in joules; $L, M \geq 0$ with $M$ in $J^{- 2}$ ; $β = s_{B}^{'} (u_{0}) > 0$ is in $J^{- 1}$ ; the bath interval contains every $u_{0} - E_{x}$ ; and $s_{B}$ and $δ$ are dimensionless.

[A2]

The bath-shell model separately assigns the normalized subsystem marginal $P_{B} (x) = e^{s_{B} (u_{0} - E_{x})} / \sum_{y} e^{s_{B} (u_{0} - E_{y})}$ . This is an explicit assumption, not a marginal formula supplied by the microcanonical postulate.

Proof

Given: The finite-system energies, the smooth bath shell-log-weight model, and the weak-additive microcanonical marginal separately stipulated in the Statement.

1.1F1F2F3L1A1A2

The adopted parent microcanonical model uses the finite-window or regular-shell sampling of [F3], with the stated AC assumption for the regular-shell reading; that premise does not determine the subsystem marginal. With $∣ E_{x} ∣ \leq L$ , every bath argument $u_{0} - E_{x}$ lies in the stated interval. The curvature bound and $β = s_{B}^{'} (u_{0})$ therefore match the hypotheses of [L1], including the dimensionless remainder scale $δ = M L^{2} / 2$ . The normalized weights required by [L1] are the additional subsystem-marginal assumption [A2], not a consequence of the microcanonical postulate; the finite canonical comparison is [F2].

2.1F2L1step 1.1∎

Applying [L1] to these matched inputs gives $∣ s_{B} (u_{0} - E_{x}) - s_{B} (u_{0}) + β E_{x} ∣ \leq δ$ for each state and $\sum_{x} ∣ P_{B} (x) - P_{C} (x) ∣ \leq e^{2 δ} - 1$ . The claim is conditional on the separately stipulated shell-weight rule and the draft mathematical supplier, not on bath size alone.

Finite-bath canonical error under a smooth shell-entropy model

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources