# Landauer's principle vs Wien's displacement law

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 kTln2? 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 Wien's displacement law which says the total energy should be in the order of $k\frac{T^4}{h^4}$ (as integrated using Mathematica 8.0)?

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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 kTln2?

No, not really. Landauer's principle applies to erasing bits of a computer's memory, not changing bits. In general you do not need to emit heat to change bits, unless you want to do an irreversible operation such as erasing.

On the other hand, black bodies must always emit heat according to the Stefan-Boltzmann law, regardless of whether there is a computer inside the black body erasing bits of its memory or not.

This isn't to say that the two things are completely unrelated. It makes more physical sense to do the opposite calculation and say that if a black body is radiating heat at a rate $J=\sigma T^4$ then bits must be changing form at a rate $JkT\ln 2 = \sigma kT^3\ln 2$. These are bits of entropy that are being transferred from the motion of the molecules making up the body to degrees of freedom of the electromagnetic field.

The radiated energy is distributed equally over all the modes of a black body. This is a fundamental assumption in Planck's derivation of the black body spectrum, and can pretty much be seen as the definition of a black body.

Wien's displacement law is a property of the Planck spectrum, so it's hard to see how this can lead to a contradiction. But for anyone to give a good answer to this part of the question, I think you will need to give more details of how you arrived at your formula. For one thing it doesn't have the units of energy.

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I integrated the formula of Wien's displacement law. According to my text, it is $$u(\nu) d\nu \sim \nu^3 e^{\frac{-h\nu}{kT}}d\nu$$. I integrated it over $\nu$ from zero to infinity using Mathematica 8.0 command 'Integrate[x^3 E^(-h x/k T), {x, 0, Infinity}]'. It gave me the result, $$\frac{6 k^4}{h^4 T^4}$$. –  Omar Shehab Mar 26 '13 at 16:05
That's not Wien's displacement law, it's Wien's distribution, which is an approximation to the Planck spectrum. It's only a good approximation at high frequencies. It has units of energy per unit surface area per unit time per unit solid angle per unit frequency, so integrating it over frequency will give energy per surface area per time per solid angle, not total energy. But nevertheless the $T^4$ is the same as the Stefan-Boltzmann law, so there's no contradiction. –  Nathaniel Mar 27 '13 at 0:59