From the always seductively intelligent
Sean Carrol at Discovery.com
"Okay, sticking to my desire to blog rather than just tweet (we’ll see how it goes): here’s
a great post by John Baez
with the forbidding title “Information Geometry, Part 11.” But if you
can stomach a few equations, there’s a great idea being explicated,
which connects evolutionary biology to entropy and information theory.
There are really two points. The first is a bit of technical
background you can ignore if you like, and skip to the next paragraph.
It’s the idea of “relative entropy” and its equivalent “information”
formulation. Information can be thought of as “minus the entropy,” or
even better “the maximum entropy possible minus the actual entropy.” If
you know that a system is in a low-entropy state, it’s in one of just a
few possible microstates, so you know a lot about it. If it’s
high-entropy, there are many states that look that way, so you don’t
have much information about it. (Aside to experts: I’m kind of
shamelessly mixing Boltzmann entropy and Gibbs entropy, but in this case
it’s okay, and if you’re an expert you understand this anyway.) John
explains that the information (and therefore also the entropy) of some
probability distribution is always
relative to some other
probability distribution, even if we often hide that fact by taking the
fiducial probability to be uniform (… in some variable). The relative
information between two distributions can be thought of as how much you
don’t know about one distribution if you know the other one; the relative information between a distribution and itself is zero.
The second point has to do with the evolution of populations in
biology (or in analogous fields where we study the evolution of
populations), following some ideas of
John Maynard Smith.
Make the natural assumption that the rate of change of a population is
proportional to the number of organisms in that population, where the
“constant” of proportionality is a function of all the other
populations. That is: imagine that every member of the population breeds
at some rate that depends on circumstances. Then there is something
called an
evolutionarily stable state, one in which the
relative populations (the fraction of the total number of organisms in
each species) is constant. An equilibrium configuration, we might say.
Then the take-home synthesis is this: if you are
not in an
evolutionarily stable state, then as your population evolves, the
relative information between the actual state and the stable one
decreases
with time. Since information is minus entropy, this is a
Second-Law-like behavior. But the interpretation is that the population
is “learning” more and more about the stable state, until it achieves
that state and knows all there is to know!
Okay, you can see why tweeting is seductive. Without the
140-character limit, it’s hard to stop typing, even if I try to just
link and give a very terse explanation. Hopefully I managed to get all
the various increasing/decreasing pointing in the right direction…"
