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Maxwell speed marginal and Poisson number example

Physical scope

Conditional on ACω for the Euclidean Lebesgue product reference; exact d=3 massive ideal-gas canonical speed law and one grand-canonical activity choice for one operationally indistinguishable species, including the vacuum sector; no measured gas data.

Example

Assume ${A C}_{ω}$ as required by Exact canonical and grand ideal-gas preparations. In a canonical sector $N \geq 1$ of the massive $d = 3$ ideal-gas model of one operationally indistinguishable species, choose any mass $m > 0$ in kg and inverse temperature $β > 0$ in J $^{- 1}$ . The most probable speed of a uniformly selected particle is $\sqrt{2 / (m β)}$ and its ensemble mean speed is $\sqrt{8 / (π m β)}$ . In the separate grand model, which includes the vacuum sector, choose the chemical potential $μ_{c h e m} = β^{- 1} \log (λ^{3} / V),$ so the mean activity parameter $ν = \exp (β μ_{c h e m}) V / λ^{3}$ equals $1$ , where $V$ is in m $^{3}$ , $λ$ in metres, and $μ_{c h e m}$ in joules. The particle count then has the Poisson $(1)$ law, including $P (N = 0) = P (N = 1) = \exp (- 1)$ .

Facts & Assumptions

Given: ${A C}_{ω}$ , the massive ideal-gas canonical and grand preparations of Exact canonical and grand ideal-gas preparations for one operationally indistinguishable species, with $d = 3$ , $m, β, V, λ > 0$ and the vacuum convention $Z_{0} = 1$ .

[F1]

Exact canonical and grand ideal-gas preparations: in $d = 3$ , the uniformly selected particle's speed mode and mean are $\sqrt{2 / (m β)}$ and $\sqrt{8 / (π m β)}$ ; with the defined vacuum sector, the grand count has Poisson mass $\exp (- ν) ν^{j} / j!$ for every $j \geq 0$ , where $ν = \exp (β μ_{c h e m}) V / λ^{3}$ .

[F2]

The real exponential function and the number $e$ by a power series, The natural logarithm as the inverse of the exponential function: $\exp$ and $\log$ are inverse functions on $R$ and $(0, \infty)$ , so $\exp (\log x) = x$ for $x > 0$ .

[A1]

Choose the grand model parameter $μ_{c h e m} = β^{- 1} \log (λ^{3} / V)$ ; this is a parameter selection, not an observation.

Verification

1.1F1

Specializing the $d = 3$ clauses of [F1] gives the stated mode and mean of the model speed distribution; these are distributional functionals, not values assigned to every individual particle.

2.1A1F1F2algebra∎

Substituting [A1] and applying [F2] gives $\exp (β μ_{c h e m}) = λ^{3} / V$ , hence $ν = 1$ in [F1]. The Poisson formula then gives $P (N = j) = \exp (- 1) / j!$ , so the $j = 0$ and $j = 1$ probabilities both equal $\exp (- 1)$ .

Remarks

The displayed values are exact consequences of the specified distributions. They are not measured speeds or particle counts.

Maxwell speed marginal and Poisson number example

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources