For ex. if two particals close to each other require n bits of info to describe them, why does it take n bits to describe them when they are far apart?
It is not to do with any particular number of particles. Rather, it is how the observer chooses to group microstates into macrostates.So if you have 2 hydrogen atoms bouncing around in a otherwize empty box, at a certain point when adding more H atoms the system becomes a gas?
Suppose you have a box 1000 cells in each direction, and two particles that can independently be in any of the 109 cells.
In general, there are 1018 possible states (if the particles are distinct), but it takes less information to describe an arrangement in which the two particles are within 10 cells of each other, in each dimension, because you can choose to describe it in terms of that block of 1000 cells (109 possibilities) and their relative positions within the block (106 possibilities). [Yes, I know that's grossly overcounting - it's more like 2x1012 all up.]