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Can we argue based on Landauer's principle that if one bit information is changed inside a blackbody, the total radiated energy should be at least or in order of $kT\ln2$? If it is so, can we also argue that this energy should be distributed over all the modes of the cavity? Furthermore, can it also be argued that this contradicts with the Rayleigh–Jeans law which says the total energy should be infinite?

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The reason the Rayleigh–Jeans law is contradicted is just that it is wrong, for the reasons usually brought up in discussions of how the blackbody law was derived. The source of the heat energy is irrelevant.

If you erase a bit that will have a thermodynamic cost. However, it does not have to be emitted as waste heat, it is just the most common case. For example, one can put the entropy into other conserved quantities like angular momentum or randomizing a zero bit in a computer memory. Even when you erase it thermally the initial change may not be equally distributed over all modes. However, in normal macroscopic bodies it will equilibrate quickly to become general thermal noise.

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