Uncertainty Propagation Addendum #49
Description
I'm late to this discussion, but I'm a strong supporter of the topic and the project, and I have a couple of comments on Section 7.4.
I think that the section is a little "thin". Although the density example of Eqs. (20) and (21) is fine, it's a bit trivial, and more examples could be given. Here are 2.
- The example of the propagation of the uncertainty in enthalpy to that in Cp is given in our paper in JCP, 147, 034508(2017).
Over a given T range at fixed P, one can fit h(T) = a+bT+cT^2 and then differentiate to get Cp. The uncertainty in Cp arises from the uncertainties in the regression coefficients b and c. The linear approximation gives
s^2(Cp) = s^2(b) + 4T^2s^2(c) +4Tcov(b,c)
The covariance matrix is available from the output of a typical linear regression algorithm (e.g., in Excel).
- The example of the uncertainty in the solubility in an aqueous solution at a given (T,P), which is calculated by equating the solute solid chemical potential mu_s to its chemical potential in solution, mu(m), where m is the molality. This is described in Eqs,. (21)-(24) in our review paper in Molec. Phys., 114, 1665-1690 (2016).
This is an example of uncertainty propagation via the solution of a nonlinear equation. In this particular case, the goal is to estimate the uncertainty in the solution m_s of the nonlinear equation
mu(m) = mu_s
Both mu_s and mu(m) have simulation uncertainties in this case.