= Second-order thermal mismatch uncertainty
\b[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 gives
$$
\mathbb E[d]=La_0\theta_0,
\qquad
\operatorname{Var}(d)=L^2\left(a_0^2\sigma_\theta^2+\theta_0^2\sigma_a^2+\sigma_a^2\sigma_\theta^2\right).
$$
To 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.<Guide to the expression of uncertainty in measurement by JCGM>[§5.1.2, note on nonlinear functions and 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 give
$$
\mathbb E[d]=L(a_0\theta_0+c),
$$
$$
\operatorname{Var}(d)=L^2\left(a_0^2\sigma_\theta^2+\theta_0^2\sigma_a^2+2a_0\theta_0c+\sigma_a^2\sigma_\theta^2+c^2\right).
$$
The 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.<The Gauge Block Handbook by Ted Doiron and John S. Beers>[§5.4.3.1, Temperature Effects]{ref} Keeping the nonlinear contribution is useful only when the inputs describe that actual measurement.
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