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I know what a Markov chain is but what does it mean in physics when I say that I assume Markovian dynamics? For example in Quantum Mechanics, I read that it means that the time evolution can be described by a linear time-independent differential equation but I don't know how to interpret this statement. This phrase is taken from the paper "Controllability aspects of quantum dynamics: a unified approach." Could anybody elaborate on this?

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So Markovian dynamics usually arises in the field of open quantum systems, where you have some system coupled to a (much larger) reservoir. For instance, consider an atom coupled to the electromagnetic field. In these cases the atom (system) is much smaller than the electromagnetic field (reservoir), and they interact because the atom can absorb and emit photons into the electromagnetic field.

Up to this point there's no approximation--you can always split a system in two like this. The trick is, you want to talk about the system without describing the full reservoir, and so you attempt to integrate out/trace out the degrees of freedom associated with the reservoir. In reality, those exact details matter, because the reservoir stores some of the history of your system. If you've just emitted a photon, the field you're coupled to is not the same as a field with no photon near you, perhaps. All "Markovian dynamics" means is a limit or approximation in which the recent details don't matter, that is, the reservoir is "memoryless" and does not contain information about the history of the system. This is good because it allows you to write a master equation for $\dot{\rho(t)}$ that depends only on $\rho(t)$, and not on, say, $\rho(t - t')$. "Time local" equations like this are much easier to handle.

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