Maxwell’s demon and the physical nature of space

Maxwell’s demon is a thought experiment to examine the second law of thermodynamics. But is not only about a special part of physics, but about some unspoken features of our Universe as well.
It is a small, intelligent entity who can detect the speed of individual molecules and opening or closing a door that divides a box into two parts, he/she is able to separate the fast molecules into one while the slow ones into the other part of the box. The result is the decrease of entropy in a closed system – that is contradicts the second law.
Or not, since, according to an argument based on information theory, the main problem is that if Maxwell’s demon’s memory isn’t infinitely large, then sooner or later information should be erased form it.This process emits heat into the box, because information erasing necessarily causes heat [Charles Seife: Decoding the Universe, p. 85.]. So the second law remains valid, since the entropy grows in the closed system.
This answer is partly based on the presupposition that only finitely large memories are possible and includes another presupposition, too, about the nature of space–after all, if you would be able to divide the space into infinitely small amounts, then it would be possible to store an infinitely large amount of information in a finite storage (at least, theoretically). I.e. If you have a two square cm surface of data storage, then you could use the first square cm to store the first piece of data; a half square cm to store the second piece of data; etc. ad infinitum.
An infinitely large memory perhaps not as unreal as it seems to be for the first sight, since there are plausible theories about hypercomputing based on relativistic spacetime of blackhole physics (see the details here). In this case manipulating an infinitely huge amount of data in an finite period of time is possible thanks to the nature: since it is presupposed that time is divisible infinitely many pieces, the result is that we have enough time to perform infinitely many operations in a finite period of time.
Ad analogiam: We can hypothesize that the space’s nature is similar and it is divisible infinitely, so it is possible to build a spatially finite storage to store an infinite amount of information. So we never should erase a bit of information–and the entropy wouldn’t rise in the box.
Obviously, nobody knows whether the space continuous, but it has been shown by this example how the laws of thermodynamics is embedded into the "environment" of the existing physical laws. And we cannot exclude the existence of Maxwell’s demon if the space we live in is not discrete, but continuous.

Where is every time traveler?

Perhaps nowhere, since they simply don’t exist, as time travel is impossible–but Einstein’s twin paradox (time dilatation, if you prefer) is nonetheless real. Notice that the time dilatation is not a proof of time travel; and that time travel and time dilatation are different questions. The first one is about whether time is similar to space in a certain sense, the second one is about what happens at very high speeds.
Obviously, time is a problematic field of modern physics. As far as we know, “the second law is the only fundamental law of physics that distinguishes between past and future” [Melanie Mitchell: A Guided Tour to the Complexity, p. 43]. Obviously, it has a kind of physical background, as the rise of entropy wouldn’t be possible without an early phase of high order (=Big Bang) but entropy’s concept is purely mathematical. I.e. in the case of the law of gravity, the gravitational force is inversely proportional to the square of distance–and this proportion cannot be deduced solely from the mathematical equations. But to understand the second law is enough to know that there are more ways to make disorder than order. What is to say, in this case there is no an additional physical law to determine the results over the logic of mathematics (opposite to gravitational law where there is a second “layer” over the mathematical description to determine that the connection between the distance and force is not, say, linear. By the way: if Newton’s law is two-layered (physics over math), then it is interesting question whether exist laws with tree, four etc. layer).
The second law creates the “time of arrow”. So since there is no any physical effect to modify its mathematics, time can be regarded as a result of the mathematics which is the basis of our time description.
British historian Arnold J. Toynbee said that history was regarded to be nothing but "one damned thing after another.” According to the logic of our argumentation, it is defendable that although time, using a kind of mathematical abstraction, can be represented as a dimension, in reality it is nothing more than “one damned thing after another”. So the meaning of travel in time is simply uninterpretable: It has no meaning at all, but it doesn’t exclude the time dilatation where according to the observers of a different frame, events follow slower each other if your speed is close to the speed of light.
Of course, it can be argued that it is a natural law that there is no an additional natural law over the level of mathematics in connection with entropy, and it can be asked why.