Examples of nonunique solutions

To demonstrate nonuniqueness, we pose here three problems in geochemical modeling that each have two physically realistic solutions. In the first example, based on data from an aluminum solubility experiment, we assume equilibrium with an alumina mineral to fix the pH of a fluid of otherwise known composition. Setting pH by mineral equilibrium is a widespread practice in modeling the chemistry of 

In SPECE8, we prepare our calculation by swapping boehmite (AlOOH) for H+, 

We assume a 0.1 molal KC1 solution containing hypothetical amounts of silica, aluminum, and carbonate. We set temperature to 200 °C and run the calculation 

For simplicity, we do not allow supersaturated minerals to precipitate. The result is an acidic fluid in which A10H++ and Al(OH) predominate among aluminum species. 

We repeat the calculation, this time swapping Al(OH) into the basis to represent dissolved aluminum. 

1 Solubility diagram for aluminum species in aqueous solution as a function of pH at 200°C in the presence of boehmite (solid lines) and kaolinite plus quartz (dashed lines). Aluminum is soluble at a specific activity (horizontal line) either under acidic conditions as species A1(OH)2, AlOH, or Al+++ (e.g., point A), or under alkaline conditions as A1(OH)J (point B). 

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M. S. Mock, An example of nonuniqueness of stationary solutions in semiconductor device models, COMPEL, Internat. J. Comp. Math., 1 (1982), pp. 165-174. 

The examples in the previous section demonstrate that nonunique solutions to the equilibrium problem can occur when the modeler constrains the calculation by assuming equilibrium between the fluid and a mineral or gas phase. In each example, the nonuniqueness arises from the nature of the multicomponent equilibrium problem and the variety of species distributions that can exist in an aqueous fluid. When more than one root exists, the iteration method and its starting point control which root the software locates. 

A key problem is the evaluation of the relevance of observed seismicity patterns. First, it is important to decide whether an observed pattern has a physical origin or is an artifact, arising for example from inhomogeneous reporting or from man-made seismicity, like quarry blasts or explosions. Second, the non-artifical events have to be analyzed with respect to their underlying mechanisms. This leads to an inverse problem with a nonunique solution, which can be illustrated for the most pronounced observed seismicity pattern, the occurrence of aftershocks. It is empirically known... 

If substantial molecular displacements are involved in solid-state pol)niierization, the reaction has a type of nonuniqueness whose effect upon polymer crystal perfection depends upon the kinetics of reorganization processes in the solid state. For example, if the monomer-center to monomer-center separation in the reaction direction changes during polymerization, then the displacements in monomer centers can differ statistically within the pol3nnerizing monomer array. If reaction proceeds via formation of a random solid solution of polymer chains in the polymerizing phase, these shifts will be poorly correlated for essentially non-interacting (well separated) chains formed at low conversions. 

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