First off, just clarify that I have a very, very superficial knowledge of Physics, so my apologies if my question is based on an obvious misunderstanding of the basic principles underlying the second law of thermodynamics or if it has a rather simple answer. The doubt that I have is related to the justification behind the entropy increase, in particular, the fact that I have read two different approaches that in my view appear to arrive at completely different conclusions. One of them is related to the idea of the complexity required to selectively control the dynamics of the different particles of the system in order to decrease the entropy, i.e. the Maxwell's demon thought experiment, and the fact that this would require to keep record of an increasingly long set of parameters regarding the location of each particle in the phase space and that in the long run bit of information of this long list would have to be deleted in order to make room for new data and that according to the Landauer's principle the bit erasure produces an entropy increase of the total system including the storage space for keeping these parameters. According to this justification, the apparent decrease of the entropy would be only an illusion, because actually there is an overall increase of entropy of the whole system, which must necessarily include the demon interacting and modifying the dynamics of the particles and the storage space used for keeping track of the dynamics of the system. The second justification is completely different. It is argued that the reversibility of the Laws of Physics and in particular the Liouville's Theorem prevents that the entropy increases at all, but due to the complexity character of the dynamics of most physical systems, each point gets separated according to the Lyapunov exponent in a fractal like manner and it gets progressively more difficult to track the exact position of the different particles in the phase space, and therefore in a real situation it becomes impossible, after a certain time has elapsed, to have accurate enough information of the system as to reverse the process and decrease the entropy. A coarse grained entropy is for this reason defined with a summation over the phase-space of all possible states with a unit size corresponding to the maximum resolution of the measurements and this is the entropy that is increased in all cases, whereas the fine grained entropy is left unvaried due to Liouville's Theorem. Therefore, according to the first justification the entropy is increased whereas according to the second, at least the fine grained entropy is kept unchanged. Any idea regarding how to make this two justification with this apparent contradiction consistent with each other? Thank you very much in advance for your help!