In my research on how the central nervous system controls human movement, I stepped into the Uncontrolled Manifold hypothesis proposed by John Scholz and Gregor Schöner, and later popularised by Mark Latash. I always explained this hypothesis to my students and coworkers by positing that controlling a quantity would entail a metabolic cost, and thus the parsimonious view of evolution led humans to keep control only of quantities that directly affect the motor goal. Control only what is necessary.
But I never had a true theoretical basis for this. Recently, while reading “Chance and Necessity: Essay on the Natural Philosophy of Modern Biology” by Jacques Monod, I found something that could help. It is related to the so-called Maxwell’s demon.
Maxwell imagined a tiny being that could open and close a small door between two gas-filled containers. It lets only fast molecules pass one way, and only slow molecules pass the other way. This seems to create a temperature difference without doing work, decreasing entropy and apparently violating the second law of thermodynamics.
The modern resolution is this: the demon must measure the molecules’ speeds and use that information to decide when to open the door. The measurement itself can, in principle, be performed with very little cost, but the demon has finite memory. Sooner or later, it must erase the information it has collected in order to continue operating.
Here comes a long train of concepts that I cannot detail, as they are too complex. In What is Life, Schrodinger proposes that living organisms produce negentropy, e.g. consume energy to keep their internal entropy low, at the cost of increasing the entropy of their environment. In 2009, Mahulikar & Herwig redefined thermodynamic negentropy as the specific entropy deficit of the dynamically ordered sub-system relative to its surroundings.
Shannon proposed that a Gaussian distribution has the highest entropy among all distributions with the same mean and variance (makes sense: if an event is totally random, it is normally distributed). He uses negentropy as the measure of distance, for a given distribution, from the normal one.
In 1929, Leó Szilárd suggested that the apparent paradox of Maxwell’s demon experiment could be solved if one accepted that the demon's information about the molecule's velocity cost the same amount of entropy that was missing. More specifically, the so-called Szilárd engine considers Maxwell's set-up, but with only a single gas particle in a box. If the demon knows which half of the box the particle is in (equivalent to a single bit of information), it can close a shutter between the two halves of the box, close a piston unopposed into the empty half of the box, and then extract kBTln2 joules of useful work if the shutter is opened again. The particle can then be left to expand isothermally back to its original equilibrium volume. In just the right circumstances, therefore, the possession of a single bit of Shannon information really does correspond to a reduction in the entropy of the physical system. The global entropy is not decreased, but information-to-free-energy conversion is possible.
In 1953, Léon Brillouin derived a general equation stating that changing the value of an information bit requires at least kBTln2 energy. This is the same energy that Leó Szilárd's engine produces in the idealised case. In his book, he further explored this problem, concluding that any cause of this bit-value change (measurement, a decision about a yes/no question, erasure, display, etc.) will require the same amount of energy.
In fact, one can generalise: any information that has a physical representation must somehow be embedded in the statistical mechanical degrees of freedom of a physical system. Thus, Rolf Landauer argued in 1961, if one were to imagine starting with those degrees of freedom in a thermalised state, there would be a real reduction in thermodynamic entropy if they were then reset to a known state. This can only be achieved under information-preserving microscopically deterministic dynamics if the uncertainty is somehow dumped somewhere else – i.e. if the entropy of the environment (or the non-information-bearing degrees of freedom) is increased by at least an equivalent amount, as required by the Second Law, by gaining an appropriate quantity of heat: specifically, kBTln2 of heat for every 1 bit of randomness erased.
On the other hand, Landauer argued, there is no thermodynamic objection to a logically reversible operation potentially being achieved in a physically reversible way in the system. It is only logically irreversible operations – for example, the erasing of a bit to a known state, or the merging of two computation paths – which must be accompanied by a corresponding entropy increase. When information is physical, all processing of its representations, i.e., generation, encoding, transmission, decoding, and interpretation, are natural processes in which entropy increases through the consumption of free energy.
Applied to the Maxwell's demon/Szilard engine scenario, this suggests that it might be possible to "read" the state of the particle into a computing apparatus with no entropy cost, but only if the apparatus has already been SET into a known state, rather than being in a thermalised state of uncertainty. To SET (or RESET) the apparatus into this state will cost all the entropy that can be saved by knowing the state of Szilard's particle.
I think all this provides the theoretical background to my postulate that control costs metabolic energy.
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