Appendix A. Partial derivatives with respect to k for the positive and negative branches of Eq. 13.
Demonstrating that for the positive branch
, for all x, and for the negative branch
,
for all x, is easier if we rescale k in Eq. 13 as
.
Now Ptot can be written as:
,
where
.
This (x) ³0 and (x) does not depend on k, since H(x) does not. The partial derivative of the negative branch is:
,
which has to be negative, that is
,
which is true since 2 > 0 and > . The latter is a necessary condition for the bitrophic system to exist. For the positive branch we need
, which is true since > .