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Packing boundary and exact truncated virial error

Physical scope

Conditional on ACω for the Euclidean Lebesgue product reference; finite one-dimensional Tonks gas with common positive rod length, positive mass, inverse temperature and action reference at half the close-packing density; exact force-conjugate pressure and geometric virial remainder only.

Example

Assume ${A C}_{ω}$ as required by Exact Tonks hard-rod pressure and convergent virial series. Use the one-species hard-rod model of Exact Tonks hard-rod pressure and convergent virial series with $N \geq 1$ , rod length $a > 0$ in metres, mass $m > 0$ in kg, inverse temperature $β > 0$ in J $^{- 1}$ , and fixed action reference $h_{0} > 0$ in J s. Set box length $L = 2 N a$ in metres. Then $L > N a$ and number density $ρ = N / L = 1 / (2 a)$ in m $^{- 1}$ , so $a ρ = 1 / 2$ . The exact force-conjugate pressure is $β p_{1 D} = 2 ρ$ . Truncating its virial series after the first two terms gives $ρ + a ρ^{2} = 3 ρ / 2$ , with exact remainder $ρ / 2$ .

Facts & Assumptions

Given: ${A C}_{ω}$ and the one-dimensional hard-rod model from Exact Tonks hard-rod pressure and convergent virial series with $N \geq 1$ , $a > 0$ metres, $L > N a$ metres, $m > 0$ kg, $β > 0$ J $^{- 1}$ , $h_{0} > 0$ J s, and $ρ = N / L$ in m $^{- 1}$ .

[F1]

Exact Tonks hard-rod pressure and convergent virial series: under those full hard-rod hypotheses, $β p_{1 D} = ρ / (1 - a ρ)$ and the remainder after $M$ terms is $ρ (a ρ)^{M} / (1 - a ρ)$ .

[A1]

Choose $L = 2 N a$ and truncate after $M = 2$ terms.

Verification

1.1A1algebra

The choices give $ρ = N / (2 N a) = 1 / (2 a)$ and therefore $a ρ = 1 / 2$ .

2.1F1step 1.1A1algebra∎

The exact equation in [F1] gives $β p_{1 D} = ρ / (1 - 1 / 2) = 2 ρ$ . The first two series terms are $ρ + a ρ^{2} = ρ + ρ / 2 = 3 ρ / 2$ . Applying the remainder formula in [F1] with $M = 2$ gives $ρ (1 / 2)^{2} / (1 - 1 / 2) = ρ / 2$ , exactly the difference.

Remarks

This is a finite-parameter illustration of the exact one-dimensional model. It is not a numerical measurement or an all-fluid virial expansion.

Depends on

Dependency tree · 2 levels

Sources

University of Warwick, PX439/CO904 Statistical Mechanics of Complex Systems, lecture notes, §3.5 The Tonks gas