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It is said that the law of least action is that nature tries to convert potential energy into kinetic one as fast as possible.

Information can't be thought without a physical realisation, see here. It may be thought as a form of entropy. So information is physical and therefore it's connected to a certain kind of energy. But is it a potential or a kinetic kind of energy?

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"the law of least action is that nature tries to convert potential energy into kinetic one as fast as possible" - I'm not sure about this. Do you have a reference? –  innisfree Jul 22 at 20:50

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up vote 4 down vote accepted

Neither. Information isn't energy and can't be measured using energy units. As you pointed out, information is related to entropy which is about degrees of freedom in a system.

In physics, information is usually measured in nats but bits are common too.

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Provided a Szilard's engine a demon can extract $k_BT\ln 2$ out of 1 bit. Isn't this energy? –  draks ... Jan 8 '14 at 23:31
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@draks... no but it does take energy to change the state of a system: $E \geq \frac{\pi \hbar}{2 \Delta t}$ to change the state in $\Delta t$ time. The amount on information a system can store is related to how much energy is available. Check out arxiv.org/abs/quant-ph/9908043 for a good casual treatment of information as it relates to energy. –  Brandon Enright Jan 8 '14 at 23:35
    
It's been a while since I've seen this paper. I'll reread it. +1 for bringing it back to my mind... –  draks ... Jan 8 '14 at 23:39
    
Maybe my question is more philosophical: Can we measure the effort to create a mathematical theory in terms of energy? –  draks ... Jan 9 '14 at 0:00
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@draks... there is a really good book by James Gleick named The Information: A History, A Theory, A Flood which touches upon some of the history, some of the math, and some of the philosophy of information. I really enjoyed it. –  Brandon Enright Jan 9 '14 at 0:02

I like to think if it as connected to potential in the way described below, but transmitting or otherwise erasing it might be called kinetic energy. The one thing for sure is that there is a minimal entropy generated when a bit is erased.

Landauer's principle says at least kT ln(2) energy must be lost as heat when a bit is erased. At least k ln(2) entropy is generated when 1 bit is erased. It seems to me that a perfectly efficient (minimal) information storage system is holding this energy as a potential energy in chemical bonds. So I vote for "potential" energy. The maximum "shannon" informational bits a physical system holds is its physical entropy divided by ln(2). Divide the entropy by ln(2) to get the minimal number of informational bits someone would need to convey to you to get the exact state of the system.

The energy of kT ln(2) is barely equal to the breaking of a van der waals force at room temperature, so it's not a very stable bit. I believe it has only a 50.0001% chance of not being broken by a single thermal collision. Stronger storage bits increase the probability that the data is accurate. So I believe there might be some error in what I'm saying, but I am not in a position to overide Landuaer's exact statement.

k is boltzman's constant which is in units of energy per temperature. However, temperature is an exact measure of kinetic energy that has an exact distribution. In other words, it is a mistake to think temperature is fundamentally different from kinetic energy. So entropy is unitless (nits) which can be seen more directly in the more fundamental equations for entropy. Divide nits by ln(2) to convert to bits. Entropy is really and truly a measure of "disorder" in an informational sense.

I want to correct the premise. The principle of least action (the stable expression of the principle of stationary action**) states the average kinetic energy minus the average potential energy over any time period will be minimized (this exact, see Feynman's red books), which means mechanics prefers potential energy over kinetic energy. In separate systems, the bias against kinetic energies that would move "contrary" to each other implies a bias against heat and therefore entropy generation. Higher potential energy implies less entropy especially since the highest available bonds provides a cieling on what bonds will be used, which means copies of the highest energy bonds. Copies means less entropy, all other things equal. This might provide a physical bias for evolution.

More directly, Feynman mentioned least action's "cousins" that minimize heat and entropy generation. He stated he was not able to find the quantum mechanics equivalents. This does not mean entropy is reduced, but that there is a bias against it being created in the first place. It is a lot lower than if you are only aware of Newton's Law that erringly treats friction as a fundamental force (again, from Feynman).

** In the presence of tempreatures and QM, how can maximal action be stable? Is there real example instead of theoretical examples?

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If information is the same units as entropy, then it is not a potential energy. Note that losing energy to heat when you erase information does not make information a potential energy. The energy lost as heat reduces the efficiency of a process, but it is not the case that 1 bit of information is converted into heat energy. The lost energy comes from somewhere that needn't have stored the information itself. Nonetheless, you can't call information potential energy AND say it's the same dimensions as entropy. Also, information doesn't have to be stored in chemical bonds. –  Jim Jul 22 at 20:00
    
Good point. The physical expression of information requires energy to overcome thermal and quantum noise energies, so there should not be a discrepancy (the units should divide out). The amount of energy needed depends on the physical system's temperature, an energy, as well as the amount of information. I corrected the post to include the probability of correctness per bit, which I believe will be something like e raised to the power of the ratio of the energy per bit per kT ln(2). So this might be the real minimal energy per reliable bit instead of what I've stated. –  zawy Jul 22 at 20:15
    
Where reliable bit = 1 times its probability of being correct. The kT ln(2) energy would be "minimal energy needed to store a minimally trustworthy bit on a system at temperature T." –  zawy Jul 22 at 20:32

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