For Lurkers who are interested in the subject, here's a great historical overview of how "entropy" developed: The Second Law of Thermodynamics
In biological systems, there are three different types of entropy involved:
Logical Entropy: Shannon probability-distribution entropy
Algorithmic Entropy: Kolmogorov-Solomonoff-Chaitin sequence/algorithmic complexity.
In molecular machinery, the uncertainty of the receiver before a message is received and decoded is the Shannon entropy in the before state. When the incoming message (along with noise) has been received and decoded, the uncertainty in the molecular machinery decreases to the after state, also a Shannon entropy.
The difference between the two (after state less before state) is the gain of information in bits. In Shannon, a bit is not binary. Each bit gained by this reduction of uncertainty (state change) has a corresponding release of energy into the local surroundings. That pays the 2nd Law tab of thermodynamic entropy.
voila - two different entropies, one organism.
The third kind of entropy - algorithmic entropy - is not yet well defined. Adami is a primary investigator in this particular quest. In the end, that form of entropy may supplant Shannon entropy. And because it is wrapped up in the principles of Komologrov complexity, Solomonoff induction and Chaitin randomness - it may reach to some other questions as well. But it is a work-in-progess and thus not relevant to this discussion other than to acknowledge that it is "out there".
Wolfram, in pursuing the von Neumann challenge of cellular automata (self-organizing complexity) believes that evolution has happened despite natural selection. Rocha's approach is clearly autonomous biological self-organizing complexity.
And then we have Gehring, Weiss and others who are recognizing (owing to the concurrent evolution of eyeness across phyla, including between vertebrates and invertebrates) - that a common ancestor who had no eyes would have to have master control genetic mechanisms which were not (as) subject to wholesale mutations.
All of this has developed long after Darwin and points to a direction of evolution rather than a happenstance of evolution. That does not speak to whether or not the direction was "designed" but rather that the "randomness" pillar of evolution is in dire peril.
Personally, I expect the "random mutations + natural selection > species" formulation to be replaced with "autonomous biological self-organizing complexity".
And fascinating, the observation that "In molecular machinery, the uncertainty of the receiver before a message is received and decoded is the Shannon entropy in the before state. When the incoming message (along with noise) has been received and decoded, the uncertainty in the molecular machinery decreases to the after state, also a Shannon entropy."
Perhaps Shannon entropy truly does refer to a "regularity" of nature -- since it would seem to hold for both the "before-case" and the "after-case" as well. If a true universal, it would be equally common to both, of course.
Fundamentally, therodynamic entropy is the only one that applies absolutely to biological systems. It is defined by:
S = k ln(Omega)
where Omega is the number of possible states available. It does not refer to order and disorder as some suggest.
Shannon's entropy is a pure construction used for engineering purposes. It is not very useful for that purpose. There was an example of signal transreceiving above that is better analyzed with rms noise and signal levels and picking an acceptable error rate. It has no general physical significance at all. It is also unrelated to S.
The same goes for algorithmic complexity theory, which contains Shannon theory as a simple case. This is used to look at the overall system as an engineering analysis, but does not indicate anything at all about how it came to be.
"Each bit gained by this reduction of uncertainty (state change) has a corresponding release of energy into the local surroundings. That pays the 2nd Law tab of thermodynamic entropy."
There is no relationship with the second law. The "bit(s)" gained is the signal and appears as power-volts and amps, depending on the impedance.
I is very clear that you "entropy" education was via the internet ...