A practical question that arises in quantitative modeling is whether the results of a modeling study are unique. In other words, is it possible to arrive at results that differ, at least slightly, from the original ones but nonetheless satisfy the governing equations and honor the input constraints? In the broadest sense, of course, no model is unique (see, for example, Oreskes et al., 1994). A geochemical modeler could conceptualize the problem differently, choose a different compilation of thermodynamic data, include more or fewer species and minerals in the calculation, or employ a different method of estimating activity coefficients. The modeler might allow a mineral to form at equilibrium with the fluid or require it to precipitate according to any of a number of published kinetic rate laws and rate constants, and so on. Since a model is a simplified version of reality that is useful as a tool (Chapter 2), it follows that there is no“correct” model, only a model that is most useful for a given purpose. A more precise question (Bethke, 1992) is the subject of this chapter: in geochemical modeling is there but a single root to the set of governing equations that honors a given set of input constraints? We might call such a property mathematical uniqueness, to differentiate it from the broader aspects of uniqueness. The property of mathematical uniqueness is important because once the software has discovered a root to a problem, the modeler may abandon any search for further solutions. There is no concern that the choice of a starting point for iteration has affected the answer. In the absence of a demonstration of uniqueness, on the other hand, the modeler cannot be completely certain that another solution, perhaps a more realistic or useful one, remains undiscovered. Geochemists, following early theoretical work in other fields, have long considered the multicomponent equilibrium problem (as defined in Chapter 3) to be mathematically unique. In fact, however, this assumption is not correct. Although relatively uncommon, there are examples of geochemical models in which more than one root of the governing equations satisfy the modeling constraints equally well. In this chapter, we consider the question of uniqueness and pose three simple problems in geochemical modeling that have nonunique solutions.
Curved Nanotube Structures under Mechanical Loading