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If time in systems moving with different speed goes differently, does speed of entropy change differ in these systems? (is "speed of entropy change" a valid term? can we compare them?)

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Entropy is an invariant $S \rightarrow S' = S$, whereas time is not, in general. Therefore the rate of change of entropy (this is the correct term) is a frame dependant quantity.

$$\frac{\mathrm{d} S}{\mathrm{d} t} = \gamma \frac{\mathrm{d} S'}{\mathrm{d} t'} = \gamma \frac{\mathrm{d} S}{\mathrm{d} t'}$$

with $\gamma$ the time-dilation factor.

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I'd like to add to juangra's answer an illustration I bet you are familiar with. Remember the twin paradox? One brother gets older while the other stays young. That's precisly means that one brother's entropy changed more then the others.

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Good point. Thank you! –  juanrga Nov 28 '12 at 11:50
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