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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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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 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 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 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 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 at 0:02

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