 Second-order thermal mismatch uncertainty

Second-order thermal mismatch uncertainty retains product terms that a linear uncertainty propagation can miss when calculating differential thermal-expansion uncertainty. For known nominal length $L$, expansion-coefficient difference $a$ and common temperature offset $\theta$, the correction is $d=La\theta$. Both first derivatives vanish at $a=\theta =0$, but the product of uncertain inputs need not vanish.

An original exact calculation assumes independent random variables with means $a_0,{\theta }_0$ and variances ${\sigma }_a^2,{\sigma }_{\theta }^2$. Independence givesTo derive the variance, use $\mathbb{E}[a^2{\theta }^2]=\mathbb{E}[a^2]\mathbb{E}[{\theta }^2]$ and subtract $a_0^2{\theta }_0^2$. No normal-distribution assumption is required for this independent-input result. The last term is absent from first-order propagation.

For stipulated $L=100\mathrm{mm}$, $a_0={\theta }_0=0$, ${\sigma }_a=10^{-6}{\mathrm{K}}^{-1}$ and ${\sigma }_{\theta }=0.1\mathrm{K}$, the exact standard deviation is 10 nm. A zero central correction therefore does not establish zero uncertainty. The GUM specifically warns that significant nonlinearity requires higher-order terms.^[ref]

Correlated inputs introduce a further distinction between the central product and the output mean. If $(a,\theta )$ has a multivariate normal distribution with covariance $c$, expansion about its means and the Gaussian fourth-moment identity giveThe covariance term in the mean must not be silently discarded. With the same zero means and standard deviations as above, but correlation coefficient $1/2$, the mean correction is 5 nm and its standard deviation about 11.18 nm. These are probability-model calculations, not measured block specifications.

The formulas assume known $L$ and a justified joint input model. A thermal-product input-distribution check must address shared temperature calibration and coefficient estimation before assigning a correlation. NIST's physical comparison procedure treats thermal corrections through measured temperatures and material expansion properties.^[ref] Keeping the nonlinear contribution is useful only when the inputs describe that actual measurement.