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We know from information theory that the entropy of a deterministic function of a random variable $X$ is less than or equal to the entropy of $X$.

Block diagram depicting a function of a random variable and its output

Does this phenomenon violate the second law of thermodynamics?

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  • $\begingroup$ What the equation intuitively tells us is that you cannot add entropy via a deterministic function (but you can surely take it away, for example by the trivial mapping g(x) = 1). Could you clarify in the question why you think this might break the second law? $\endgroup$ – alarge May 25 '14 at 12:17
  • $\begingroup$ second law say that the entropy of an isolated system never decrease . if we consider the system above as an isolated system, it goes from one equilibrium state to another and entropy decrease! thank you $\endgroup$ – Ang May 26 '14 at 6:37
  • $\begingroup$ To enact g(X) you have to do something, put energy in. You expressions describe that there is a change in a system, and that it is deterministic, but they don't say how it is enacted. (or describe physically what the system is). If you try to do this in a closed system (you cannot put energy in) then it cannot be the case that g(X) is a deterministic function. You can have a local increase in order, but the energy source will add a greater amount of disorder, and that will not be deterministic. $\endgroup$ – JMLCarter Dec 25 '16 at 3:55

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