Stability of single-variable systems

Expanding F in a Taylor series to the first-order gives the approximation 

Because the initial condition of equilibrium requires that the first term on the right-hand side of the first equality vanishes, this equation simply becomes 

The reader is referred to textbooks on differential equations and applied mathematics for a more rigorous and general proof of the stability criterion (e.g., Logan, 1987). 

The first-order non-isothermal (FONI) reactor. A continuous, well-stirred magmatic reservoir similar to those discussed above is supposed to be thermally insulated. A dissolved element i precipitates with a temperature-dependent rate of crystallization. Crystallization rate is assumed to obey first-order kinetics with Boltzmann temperature dependence such as 

The heat balance follows a similar relationship with the rate of latent heat release in proportion with the amount of element crystallized 

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