Obviously caffeine is a stimulant, and wakes us up.
Is there any research into how it affects fatigue: e.g. how different it is to rest (i.e. I'm not asking about the pathways by which caffeine achieves this).
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Can caffeine replace sleep? The simple answer here is no. The easiest way to explain why is to dip our toes into the mechanism by which caffeine works.
In a nutshell, a large part of what you perceive as drowsiness is caused by adenosine building up in adenosine receptors. The longer you stay awake (and the more active you are to a degree), the more adenosine builds up, the more drowsy you get, until finally you sleep and the adenosine is flushed out. Caffeine is a competitive antagonist of adenosine receptors, meaning that it fights adenosine for spots in the receptors and blocks adenosine from activating the receptor, without the caffeine itself activating them. The longer you stay awake on caffeine, the more adenosine fights caffeine for spots in the receptors, and the more other signs of sleep deprivation that aren't affected by caffeine accumulate. This is why you might sometimes experience that odd "worn out but not exactly tired" feeling late into an all-nighter. Caffeine doesn't actually provide rest or any equivalent, it basically creates the illusion of being well-rested, even as you continue to build sleep debt.
Caffeine doesn't seem to be able to replace sleep, simply delay it, and you can easily test this yourself. Also, as your body breaks down caffeine over a number of hours, and the impact of sleep deprivation increases the longer you force yourself to stay awake, you will need to continue to consume more and more caffeine to maintain the effect.