Hi Stef 730,
Emergent properties of systems come forward when you go from a microscopic description of a system (i.e. a description in terms of atoms, molecules, momentum, energy ... ) to a macroscopic description (in terms of entropy, total volume, total pressure, temperature). Some examples of emergent properties are phase transitions and irreversibility.
From a microscopic point of view, phase transitions should not occur. There is no obvious reason why some H2O molecules at 2 degrees Celsius should behave like a liquid, and at -2 degrees Celsius should behave like a solid. This sort of phenomena can only be understood when looking at large systems, with many H2O molecules. By looking at a lot of molecules at once, you are looking at macroscopic (global) properties, and no longer at the properties of the individual molecules.
Irreversibility is another emergent phenomena: from a microscopic point of view, the laws of motion (Newton's third law, the Schrodinger equation, ...) are all time reversible: this means that when you let time decrease instead of letting ít increase, the solution remains the same. For macroscopic systems this is obviously not the case: if you smash a cup of coffee into billions of pieces, then you dont expect them all to crawl back together to form the original cup again: there seems to be an arrow of time for macroscopic systems (= the arrow of time points in the direction of increasing entropy).
Since simply adding all the properties of the individual molecules/atoms/particles in the system together does not reproduce phase transitions or time irreversibility, scientists say that "many particle systems are more than just the sum of their constituting particles". New effects arise only when looking at many particle systems, and these effects can not immediatelly be explained from a microscopic point of view.
Bye!
Crisp