validation

Uncertainty Analysis and Error Propagation

How input uncertainties propagate through the product form of P(t): relative variances add, the largest relative uncertainty dominates, and correlated inputs add covariance terms, with an illustrative worked example.

Last updated: 2026-07-26

A prediction without an uncertainty is not a scientific statement. Once the Schubart Master Formula returns a number for $P(t)$ , the immediate question is how much that number can be trusted given that its inputs — the flux, the cross-section, the momentum transfer, the phonon velocity, the efficiency — are each known only to some finite precision. This chapter derives how those input uncertainties propagate to the output, and it arrives at a result that is unusually clean because of the formula’s structure: since $P$ is a product of factors, the relative variances of the factors simply add. That single fact organises the entire analysis and tells us at a glance which input is worth measuring more carefully.

The product structure and the log-differential

Write the formula, for the purpose of propagation, as a product of positive factors,

P = \prod_i x_i ,

where the $x_i$ stand for the effective values of $\Phi$ , $\sigma$ , $\Delta p$ , $v_{\text{ph}}$ , $\eta_{\text{conv}}$ and any measure factors that carry uncertainty. The cleanest route to the propagation law is to take the logarithm before differentiating. Because the logarithm of a product is a sum,

\ln P = \sum_i \ln x_i ,

the differential separates additively:

\mathrm{d}\ln P = \sum_i \mathrm{d}\ln x_i , \qquad\text{that is}\qquad \frac{\mathrm{d}P}{P} = \sum_i \frac{\mathrm{d}x_i}{x_i} .

This is the exact statement that a fractional change in the output is the sum of the fractional changes in the inputs — a direct consequence of the multiplicative form, and the seed of everything that follows.

First-order propagation: relative variances add

To turn the differential into a variance we take the first-order Taylor expansion of $P$ about the mean values $\bar x_i$ . Interpreting $\mathrm{d}x_i$ as a small random deviation $\delta x_i$ with zero mean and variance $\sigma_{x_i}^2$ , and assuming the factors are statistically independent, we square the relation $\delta P/P = \sum_i \delta x_i/x_i$ and take expectations. The cross-terms $\langle \delta x_i\,\delta x_j\rangle$ vanish for $i\neq j$ under independence, leaving only the diagonal:

\left(\frac{\sigma_P}{P}\right)^{2} = \sum_i \left(\frac{\sigma_{x_i}}{x_i}\right)^{2} .

This is the central result of the chapter. The squared relative uncertainty of the output is the sum of the squared relative uncertainties of the inputs. Equivalently, relative uncertainties add in quadrature. It holds to first order for any product of independent factors, regardless of how many there are or what units they carry — the relative form is dimensionless, so the flux (in $\mathrm{W\,m^{-2}\,sr^{-1}\,J^{-1}}$ ) and the efficiency (dimensionless) contribute on exactly the same footing.

Dominant-term behaviour

Because the contributions add in quadrature, they do not add democratically: the largest relative uncertainty dominates the sum. If one input is known to $30\%$ and the others to $5\%$ each, the output uncertainty is governed almost entirely by the $30\%$ term, since $0.30^2 = 0.09$ dwarfs $0.05^2 = 0.0025$ . Quadrature suppresses the small terms quadratically, so the total is close to the single worst factor:

\frac{\sigma_P}{P} \approx \max_i \frac{\sigma_{x_i}}{x_i} \quad\text{when one term dominates.}

The practical consequence is decisive. Effort spent tightening an already-precise factor is nearly wasted; the return comes from measuring the least well-known factor better. In this framework the usual suspect is the symmetry-breaking momentum transfer $\Delta p$ , discussed in the sigma analysis, which is typically the hardest quantity to pin down and therefore the one that sets the output precision.

Correlated inputs: the covariance terms

The clean quadrature law assumes independence. When two inputs are correlated — because they share a common measurement, a common calibration, or a genuine physical coupling — the cross-terms no longer vanish, and the propagation law acquires covariance contributions:

\left(\frac{\sigma_P}{P}\right)^{2} = \sum_i \left(\frac{\sigma_{x_i}}{x_i}\right)^{2} + 2\sum_{i<j} \frac{\mathrm{cov}(x_i,x_j)}{x_i\,x_j} .

The sign of the correction matters. Positively correlated factors (both drift up together) inflate the output variance beyond the independent estimate; negatively correlated factors partially cancel and reduce it. Ignoring a real positive correlation therefore produces a falsely optimistic error bar — one of the most common failure modes in uncertainty reporting. The relevant correlations here are exactly the factor couplings flagged as breakdowns of Assumption A in the assumptions and limitations: whenever the product form is threatened by saturation or feedback, the covariance terms are the quantitative fingerprint of that threat.

Illustrative worked example

Consider a purely illustrative set of relative uncertainties, chosen only to show the arithmetic — none of these are measured values for any system:

flux $\Phi$ : $10\%$ ,
cross-section $\sigma$ : $15\%$ ,
momentum transfer $\Delta p$ : $25\%$ ,
phonon velocity $v_{\text{ph}}$ : $5\%$ ,
efficiency $\eta_{\text{conv}}$ : $8\%$ .

Assuming independence, the output relative variance is the quadrature sum

\left(\frac{\sigma_P}{P}\right)^{2} = 0.10^2 + 0.15^2 + 0.25^2 + 0.05^2 + 0.08^2 = 0.01 + 0.0225 + 0.0625 + 0.0025 + 0.0064 = 0.1039 ,

so that

\frac{\sigma_P}{P} = \sqrt{0.1039} \approx 0.32 ,

about $32\%$ . Notice how the $\Delta p$ term alone ( $0.0625$ ) supplies more than half of the total, while the $5\%$ phonon-velocity term ( $0.0025$ ) is essentially negligible. This is the dominant-term behaviour made concrete: the output precision is set almost entirely by the least-known input. If a positive correlation of, say, $\mathrm{cov}(\sigma,\Delta p)/(\sigma\,\Delta p) = +0.02$ existed between the cross-section and the momentum transfer, the extra term $2\times0.02 = 0.04$ would push the variance to $0.144$ and the output uncertainty to $\approx 38\%$ — a substantial widening from a modest correlation. The numbers here are illustrative orders of magnitude, not claims.

Connection to Monte Carlo sampling

The first-order law above is a linearised approximation, exact only in the limit of small deviations and a locally linear response. Because $P$ is a product of positive factors, its output distribution is generally skewed and long-tailed rather than Gaussian, and a symmetric $\pm\sigma_P$ interval understates the upper tail. The rigorous way to capture the full distribution is to sample the inputs from their joint distribution and evaluate the formula for each draw — the forward Monte Carlo propagation described in the Monte Carlo validation. That procedure makes no linearity assumption and respects correlations automatically, provided the inputs are drawn from a joint distribution with the correct covariance structure. In the small-uncertainty limit the Monte Carlo spread reproduces the quadrature law above, which is why the two are consistent checks on one another: the analytic law gives the intuition and the dominant-term diagnosis, while the sampling gives the honest, possibly asymmetric, output distribution. The width of that distribution is the model-consistency sigma reported throughout the framework, and it can be explored interactively in the playground. A fully worked numerical instance is carried through in the worked example, and the apportionment of the total variance across factors — which term to measure first — is developed as a formal error budget in sensitivity and error budget.