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Are there are any known formulas or equations that can calculate information from energy or energy from information?

One of them, I believe, is the Bekenstein bound which is the maximum information that can be in a given space before becoming a black hole.

https://en.wikipedia.org/wiki/Bekenstein_bound

The equations involved can be converted for both mass and energy.

So I am wondering if there are other equations involving this kind of conversion from energy to information.

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  • $\begingroup$ This is a bit like asking for an equation to calculate an objects speed given its mass. The concepts are not totally unrelated, but there is a lot more going on and so it not possible to reduce it to a single formula. The relationship is very context dependent (and in many contexts essentially non-existent) and is too broad a subject to cover in a single answer $\endgroup$
    – By Symmetry
    Sep 23, 2018 at 8:47
  • $\begingroup$ I am aware that such conversions are not simple, I mean look at the Bekenstein Limit, I am simply asking, just like the Bekenstein Limit, is there any other equations similar to that $\endgroup$
    – C. Jordan
    Sep 23, 2018 at 20:09

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The key bound is the Landauer bound: a process that erases 1 bit of information has to spend $k_B T \ln(2)$ J of energy as waste heat to carry away the entropy.

Beside that link and the Bekenstein bound, the other main links between energy and information are the various bounds on how fast quantum states can change, such as the Margolus-Levitin bound (an "operation" must take at least time $\pi\hbar/2E$ where $E$ is the system energy), the Mandelstam-Tamm bound (the time is at least $\pi \hbar/2\sqrt{\sigma^2}$ where $\sigma^2$ is the variance of system energy) and their many relatives.

The speed bounds and the Bekenstein bounds are related (see the 12-step argument in this essay on what is going on). While the speed and Bekenstein bounds deal with how information-storing fields can change (and gravitate), the Landauer bound links information processing to thermodynamics.

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  • $\begingroup$ Cool, thanks, just one last additional question, is there an equation somewhat similar where you can calculate mass to information? $\endgroup$
    – C. Jordan
    Sep 25, 2018 at 13:58
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    $\begingroup$ @C.Jordan - You can always use mass-energy equivalence, $E=mc^2$. $\endgroup$
    – Anders Sandberg
    Sep 25, 2018 at 19:54
  • $\begingroup$ Thanks, that will help. $\endgroup$
    – C. Jordan
    Jul 19, 2019 at 4:05

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