Following up from this post, there is a recent sequence of articles in an AGU journal on Water Resources Research under the heading: “Debates: Does Information Theory Provide a New Paradigm for Earth Science?”
By anticipating all these ideas, you can find plenty of examples and derivations (with many centered on the ideas of Maximum Entropy) in our book Mathematical Geoenergy.
Here is an excerpt from the “Emerging concepts” entry, which indirectly addresses negative entropy:
That is, there are likely many earth system behaviors that are highly ordered, but the complexity and non-linearity of their mechanisms makes them appear stochastic or chaotic (high positive entropy) yet the reality is that they are just a complicated deterministic model (negative entropy). We just aren’t looking hard enough to discover the underlying patterns on most of this stuff.
The objective is to apply negative entropy to find an optimal solution to a deterministically ordered pattern. To start, let us contrast the behavior of autonomous vs non-autonomous differential equations. One way to think about the distinction is that the transfer function for non-autonomous only depends on the presenting input. Thus, it acts like an op-amp with infinite bandwidth. Or below saturation it gives perfectly linear amplification, so that as shown on the graph to the right, the x-axis input produces an amplified y-axis output as long as the input is within reasonable limits.