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Conditional Hamiltonian ensemble and response predictions

Physical scope

Smooth finite-dimensional nonrelativistic Hamiltonian models on an actually complete invariant domain; canonical normalizability, regular compact-shell hypotheses, sampled-map ergodicity, and compact response conditions are explicit and separate. No measured equilibration or generic ergodicity is claimed.

Statement

Adopt the selected smooth nonrelativistic Hamiltonian model and preparation of Adopted classical Hamiltonian dynamics and preparation. The following are conditional model deductions, with no experimental premise.

If $β = (k_{B} T)^{- 1} > 0$ and $Z_{β} = \int_{R} e^{- β H} d m_{0}$ is finite and positive, the canonical probability $d μ_{β} = Z_{β}^{- 1} e^{- β H} d m_{0}$ is invariant under the unforced Hamiltonian flow. For any absolutely continuous initial law covered by Liouville probability transport and constant fine entropy, its Liouville-transported density preserves the reference-measure fine entropy whenever that entropy is finite.
Under full AC, let $n \geq 1$ , let the canonical phase be open $O \subseteq R^{2 n}$ , let $H \in C^{2} (O, R)$ , and let $J \subseteq R$ be open. Suppose $\nabla H \neq 0$ on $H^{- 1} (J)$ and $H^{- 1} (K)$ is compact in $O$ for every compact $K \subset J$ . For each $E \in J$ with nonempty shell $Σ_{E} = H^{- 1} ({E})$ , if the flow preserves a full compact regular band containing $Σ_{E}$ , that shell has a finite positive weighted area law invariant under that flow. Critical levels and noncompact bands are outside this prediction.
If $μ$ is a Borel probability on $R$ invariant under every flow map $Φ_{s}$ and the selected time map $T = Φ_{Δ}$ is ergodic for $μ$ (every Borel $E$ with $T^{- 1} E = E$ has $μ (E) \in {0, 1}$ ), then for each bounded Borel observable $A$ , the model's continuous-time average of $A$ converges $μ$ -almost surely and in $L^{1}$ to its ensemble mean. This is a typical-initial-state prediction, not a claim for every trajectory or every finite observation window.
In the compact response regime and with the unperturbed canonical preparation, perturbing $H$ to $H - ϵ h (t) B$ gives the exact first-order response of Exact finite-time response on a compact Hamiltonian phase. It does not assert that the required correlation decays or that a noncompact physical system satisfies the compact hypotheses.

In SI mechanical units, $H$ is in J, absolute temperature $T$ is in K, $k_{B}$ is the adopted primitive energy-temperature conversion constant in J/K, $β$ is in J $^{- 1}$ , and the action scale $a_{0}$ is in J s. Observables and control fields retain the units declared in the response theorem. These deductions establish what follows inside the adopted model, not that nature obeys the model.

Facts & Assumptions

Given: The adopted Hamiltonian model and prepared law; where invoked, finite positive canonical normalization; ergodicity of the selected sampled map; and the compact smooth-control response hypotheses.

[A1]

Full AC is carried for the regular-shell branch of the mathematical supplier used below; ${A C}_{ω}$ is carried for its completed measure-theory suppliers. These are mathematical assumptions, not physical postulates (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[A2]

For the canonical clause, $β > 0$ and $0 < Z_{β} = \int_{R} e^{- β H} d m_{0} < \infty$ .

[A3]

For the time-average clause, $Δ > 0$ , $μ$ is invariant under every flow map, and $T = Φ_{Δ}$ is ergodic for $μ$ in the strict-invariance convention: $T^{- 1} E = E$ implies $μ (E) \in {0, 1}$ for Borel $E$ .

[A4]

For the response clause, $M$ is compact without boundary, $H, A, B$ are smooth, $h$ is smooth on the finite time slab, $μ_{β}$ is the unperturbed initial law, and $H - ϵ h (t) B$ has the units specified in the cited response theorem.

[A5]

For the shell clause, $n \geq 1$ , $O \subseteq R^{2 n}$ is open, $H \in C^{2} (O, R)$ , $J \subseteq R$ is open, $\nabla H \neq 0$ on $H^{- 1} (J)$ , and $H^{- 1} (K)$ is compact in $O$ for every compact $K \subset J$ . The chosen energy is $E \in J$ , its shell $Σ_{E}$ is nonempty, and the Hamiltonian flow is a diffeomorphism on and preserves a full compact band containing that shell.

[F1]

Adopted classical Hamiltonian dynamics and preparation: the flow and preparation are adopted model assumptions. This does not assert empirical validity.

[F2]

Invariant energy-weighted and regular-shell probabilities: the energy-weighted canonical law is invariant under the stated normalizability condition; regular shells at energies $E \in J$ require full AC, $n \geq 1$ , compact inverse images of compact subsets of $J$ , and nonzero gradient on $H^{- 1} (J)$ .

[F3]

Liouville probability transport and constant fine entropy: an absolutely continuous probability density is transported by $ρ_{t} = ρ_{0} \circ Φ_{- t}$ , and finite reference-measure fine entropy is constant.

[F4]

Conditional sampled and continuous-time averages: for the chosen ergodic time map, bounded-observable time averages converge almost surely and in $L^{1}$ to the invariant ensemble mean.

[F5]

Exact finite-time response on a compact Hamiltonian phase: in the specified compact unperturbed canonical model, the derivative with respect to the dimensionless perturbation is $\int_{0}^{t} h (s) β ⟨ (X_{H} B) (0) A (t - s) ⟩_{μ_{β}} d s$ .

Proof

Given: [A1]–[A5], the adopted postulate [F1], and the exact mathematical interfaces [F2]–[F5].

1.1A1A2A5F1F2F3

Under [A2], the first clause of [F2] makes $μ_{β}$ a normalized invariant probability on the actual complete domain. Under [A1] and [A5], the selected $E \in J$ and its nonempty shell meet the second clause of [F2], which gives the regular-shell prediction, with its finite positive normalization and invariance only on the specified band. For any initial density satisfying [F3], the transported law and its finite fine entropy have the conclusions stated in clause 1. Each conclusion is conditional on the normalization, shell, or integrability condition in its supplier.

1.2A1A3F1F4

Under [A3], the flow, probability, observable, and chosen time map meet the hypotheses of [F4]. Its limit is the ensemble mean for $μ$ and holds almost surely and in $L^{1}$ ; it does not turn the limit into a statement about every initial state or a finite sample.

2.1A1A4F1F5∎

Under [A4], all variables and the perturbation match the compact response theorem [F5]. Applying its exact conditional formula gives clause 4, with no further long-time limit unless a separate correlation-decay hypothesis is added. The result is a deduction within the adopted model, not a measurement or proof of physical applicability.

Remarks

The postulate fixes the model's physical formulation and preparation. Normalizability, ergodicity, and correlation decay are separate mathematical conditions; none is inferred from Hamiltonian form alone. Experiments may test predictions from a specified model, but this theorem contains no reported observations.

Conditional Hamiltonian ensemble and response predictions

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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