Stability with respect to unilateral perturbations

Before embarking on a study of stability in general, it is instructive to begin by finding the condition of equilibrium of a system with respect to a parameter which can only vary unilaterally. Let us consider, to fix our ideas, S 0. To find the condition of equilibrium we must find the conditions under which the rate of reaction is zero in the state P. Because of the nature of , the rate of reaction d jdt must either be zero or positive. From De Bonder s inequality (3.21) 

This equilibrium condition is much less restrictive than that obtained 

In the first case i.e. Ap O) combination of (15.9) with (15.6) leads back to 

In the particular case of a perturbation from the state P at constant X and y (15.13) may be written 

These conditions of stability with respect to unilateral perturbations may also be expressed in terms of the thermodynamic potentials U, H, F, G. For example, in terms of U we have from (4.30) and (15.12) 

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