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An identity dynamics leaves a noncanonical finite preparation unchanged

Physical scope

A two-state finite model with explicitly frozen identity dynamics; it shows that a canonical preparation rule alone supplies no relaxation dynamics and is not an empirical statement.

Example

Let $Ω = {a, b}$ with energies $E_{a} = 0$ , $E_{b} = ε > 0$ joules, and fix $T > 0$ kelvins. The separately adopted canonical law has probabilities proportional to $(1, e^{- β ε})$ . For the dynamical comparison, use a dimensionless model-time parameter $τ \geq 0$ , stipulate the identity transition map $Φ_{τ} (a) = a$ , $Φ_{τ} (b) = b$ for all $τ$ , and start with the point mass $δ_{a}$ .

Facts & Assumptions

Given: A two-state energy model, positive temperature, the canonical preparation rule, and the explicitly stipulated identity dynamics.

[F1]

Finite statistical microstate model: the two microstates and their fixed finite energies define an admissible finite statistical model.

[F2]

Canonical sampling assumption for finite and normalized statistical models: the canonical probabilities are the normalized weights $e^{- β E_{x}}$ ; this is an adopted preparation rule and includes no time-evolution equation.

[A1]

The comparison dynamics is the deterministic identity map $Φ_{τ}$ for every dimensionless model time $τ \geq 0$ , and the initial law for that comparison is $δ_{a}$ .

Verification

Given: $ε > 0$ , $T > 0$ , and the stipulated dynamics.

1.1F1F2algebra

Put $q = e^{- β ε}$ , so $0 < q < 1$ . The canonical law is $P_{β} (a) = 1 / (1 + q)$ and $P_{β} (b) = q / (1 + q)$ ; in particular it is not $δ_{a}$ .

2.1A1step 1.1algebra∎

The identity map preserves the initial point mass at every model time: $(Φ_{τ})_{*} δ_{a} = δ_{a}$ . Hence the time-evolved law does not converge to the canonical law, whose probability of $b$ is strictly positive. This is a counterexample to deriving equilibration from the sampling rule alone.

An identity dynamics leaves a noncanonical finite preparation unchanged

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources