Conversely, under some conditions you can figure out knowing the measures and . This trick is called the Radon-Nikodym derivative because if you just follow your nose it looks like a derivative

It’s really more like division: the measure divided by the measure is the function .

If this is too abstract for you, imagine is the usual thing that shows up in integrals and define

for functions and . Then the Radon-Nikodym derivative at the point is

whenever this exists, that is, whenever you’re not dividing by zero.

I’m stating everything in a way that leaves out the technical fine print that’s needed for an actual theorem. For that, try Wikipedia. But you said you wanted intuition, and the intuition is a lot simpler than the Wikipedia article makes it seem.

Consider that when you take a stroll in the woods, metric tons of raw matter around you are acting in concert, there is a high degree of fittingness on many meta-levels, everything is mutually finding its proper angle of repose and surprisingly, for great deal of matter, that angle is nearly vertical. There is consensus, a congruous entwining of energy paths, and a multiplicity of ‘formal’ agreements that are both dynamic and enduring. You belong in this scene to the very level of complex DNA chemistry that is resonant within the world around you.

Now, in an ideal gas at maximal entropy, I would expect each molecule to be traveling on its own unique trajectory, each with its own information theoretic distinction. There would be no concerted action, no consensus as to path. Does the trend toward increasing entropy and the final state thereof allow us to roughly characterize the nature of energy? That is, if there is utility in the notion ‘nature abhors a gradient’ is there also some utility in the notion that energy eschews pathways?

If we tentatively accept that idea, then how is it possible to view the world about us as resulting from the evolving ‘chemistry’ and entwinement of energy pathways?

‘Into the Cool: Energy Flow, Thermodynamics and Life’ sounds very similar in scope to Howard T Odum’s book, ‘Environment, Power and Society’, published in 1971, both in search of general organizing principles.

General organizing principles are useful and in that regard I am wondering if it is useful to identify some grand counterpoise to energy as being catalytic in the emergence of path. Pathways emerge on gradients, not at equilibrium. On a deep level, could path be viewed as emergent between countervailing gradients of distinction?

I’ll try to explain it a bit to encourage you to read it.

Related to Jarzynski’s work and the fluctuation theorem is the idea of measuring equilibrium quantities by forcing a system out of equilibrium and observing the response, see: http://www.physics.berkeley.edu/research/liphardt/pdfs/JarzynskiTest.pdf

In Still’s paper they observe that much of the literature assumes that the driving signal is given/known explicitly, while in nature and biological systems this is most often not the case, hence they study stochastic driving signals.

The idea is that implicit in the dynamics of such a forced system is a model of its (stochastic) environment. They then ask how the quality of this model is related to thermodynamic efficiency.
How do you measure the quality of a model?
For them a good model should have predictive power and not be overly complicated. This comes down to a balance of a systems memory, namely you break it into two parts, one useful, predictive part, and the other “useless nostalgia”. It is this latter part that is related to dissipation, hence having less nostalgia is having less dissipation is having better thermodynamic efficiency.

Here is a whirlwind tour of how the paper makes this idea precise, taking many quotes straight from the paper:
“the dynamics of system are modeled by discrete time Markovian conditional state-to-state transition probabilities”
For the driving signal they assume only that its changes are governed by some probability density.
Their system is in contact with a heat bath and starts in equilibrium. It then goes through a bunch of steps where the environment forces it out of equilibrium and then it relaxes. The environment does work on the system in each driving step and heat flows at each relaxation step.
If you let it relax all the way to equilibrium any additional free energy gets dissipated as heat back to the environment. The additional free energy is given by the Kullback-Leibler divergence, or the relative entropy between the current state/distribution and the equilibrium one. The change in non-equilibrium free energy is the sum of the change in equilibrium free energy and this additional piece:
Dissipation work is given by the difference between work done on the system and the non-equilibrium change in free energy:
Excess work is given by the difference in work done on the system and the equilibrium free-energy (the work done for the quasistatic case):
The excess work minus the dissipation work gives you, , which is precisely the additional free energy (the KL divergence of the current distribution relative to the equilibrium one).

Now for the information/prediction half, which I understand even less!
They look at Shannon’s (symmetric) mutual information of the system state and the external driving signal, both at a certain time t, this is called the system’s ‘instantaneous memory’, where s is for system and x is for external signal. The ‘instantaneous predictive power’ is, , or the mutual information of a state at time t and the driving signal at t+1. The difference of the two is the ‘instantaneous nonpredictive information;’ “it represents useless nostalgia and provides a measure for the ineffectiveness of the implicit model.” (So memory-power=information, kidding)
The paper then shows that this instantaneous nostalgia is proportional to the average work dissipated as t goes to t+1.

They derive a lower bound on the total dissipation and use it to refine Landauer’s principle. They then discuss this in relation to biological systems, where the systems have adapted to their environments forcing, asking if minimizing nostalgia is a factor driving things towards energetic efficiency.

This paper is more about thermodynamics and prediction (as the title suggests) than about reversibility. I don’t understand it yet, but there are hints of connections here, not only among biology, chemistry and nature, but also information and complexity (algorithmic information theory). I guess that’s why there is all the talk of Kolmogorov above. I’m new at this, and I’m sure many of you reading this have a better big picture! It seems though that in Still’s paper, a good model is about balancing complexity with predictability, and that this is done by not having any nostalgia, which doesn’t sound practical for us nostalgic humans! I would like to learn more though about mutual information in biological systems, and how some systems we understand a little bit have some ‘memory’ built in, either about their environment, or even their own dynamics.

So read the paper, it does a better job explaining itself!

http://www.sekj.org/PDF/anzf40/anzf40-185.pdf

It has simulations of populations constantly adapting to a changing environment, and sometimes going extinct.

What happens in an oscillating environment, where the population keeps adapting, then adapting back?