How do we approach order in a world whereby everything is both qualitatively the same (energy) but also qualitatively different (mass, time, space etc)? — Benj96
What you are drawing attention to is that “disorder” is a relative claim. The question becomes “disordered in relation to what kind of expectation, meaning, purpose or constraint?”
So a more general definition of entropy would be grounded in an information theoretic perspective. What about this world counts as a degree of uncertainty or surprise in relation to my simplest model of it as a system?
You can see that your first system - 20 identical balls - is already a highly constrained or ordered one as you have somehow managed to reduce all possible surprise as to the colour of the balls. Surprise is minimised. Your world is completely predictable on that score.
A truely entropic situation would be if the balls could randomly take on any colour at any time. Even as you grouped them, they could switch colour on you. Or split, merge, be in multiple places at once, etc.
So note how the standard mental image of an entropic system already smuggles in an atomising assumption - some stably countable degree of freedom like a particle that itself is already in a highly negentropic state of constraint. The particle and its qualities are made as homogenous as possible so that - by contrast - a chosen variable like location becomes maximally surprising. The thing you have the least information about, the least control over ... until you get grouping and impose order over that too.
Of course, treating physical systems as if they were systems of particles - an ideal gas confined in a container and sat in a heat sink - is a useful model. If you are doing practical thermodynamics here on the warm surface of a planet floating in a cosmic heat sink with a temperature of 2.7 degrees K, then the statistics of bags of marbles pitches things at a suitable level.
But once you want to apply the concept of entropy to the Universe itself as a system, then you have to recognise this habit of including negentropic assumptions in your metaphysical accounts.
Take the Big Bang to Heat Death story of a Universe that starts off hot and constrained and becomes cold and spread out. In a broad sense, nothing changes as the positive contribution to entropification in terms of a disordering of position is matched by a negative contribution in terms of an increase of resulting gravitational potential. If the universe was just a bunch of balls spilling out, then a gravitational gradient wanting to clump them all back becomes an ever swelling constraint on their apparently unconstrained kinetics.
Of course, that in itself is way too simplistic a model of the actual universe as it is presuming that the BIg Bang and Heat Death can be modeled in terms of countable degrees of freedom - definite material particles with a defined location and energetic state, so therefore a matchingly undefined degree of surprise as to the locations or energies they might have.
In the Big Bang, any such degree of freedom is maximally indeterminate. The quantum uncertainty of any claim for identity is as high as it could be. So - relative to that accountancy point of view - the Big Bang was a chaos that became increasingly ordered by a process of spatiotemporal expansion. What got constructed was a developing heat sink that started to make particles - as localised energy densities - countable elements. After a while, the chaos got sorted into collections of quarks and electrons with their identities constrained by fundamental symmetry breakings.
Then at the other end of the story, you have the Heat Death which - to our best knowledge - will be a state of immense order and uniformity ... measured from a relative point of view.
At the Heat Death, you will left with an empty vacuum that continues to radiate with only a zero point quantum energy. All particles will have been swallowed up by black holes that then themselves eventually evaporate. The contents of this world are black body photons with a wavelength of the width of the visible universe - the de Sitter horizon. Or an uncountable number of photons with a temperature within a Planck’s hairsbreath of absolute zero K.
So again, like the Big Bang, essentially a nothingness without a point of view. But still some kind of transition from a hot everythingness of an ur-potential to the chill emptiness of a generalised spatially structured void.
Thus using entropy models to describe the evolutionary trajectory of systems such as the universe is tricky and fraught. But for quite understandable reasons. We have to make three shifts in our point of view to arrive at a point of view that is actually “objectively” outside the totality of the thing we want to describe.
The first rung of the modelling is the standard entropy story. We have a bag of balls, a die with a fixed number of faces, an ideal gas with a defined number of identical particles. We are creating a world that is completely ordered or constrained in a way that, by contrast, leaves other aspects completely free or random. A world of degrees ... of freedoms. So this is an internalist dichotomy. We stand inside a world where this contrast is between what we are certain of - some number of balls - and what we are matchingly uncertain about - their possible location.
A second rung of modelling would be to recognise that this state of affairs is only relative to that constructed point of view. It could be otherwise. We could be certain about the location of the balls - clumped in this group - but uncertain as to their identity, So now your counting of entropy/surprise/disorder is relative to what you decide to fix vs what you leave to swing free. If you are imagining a system as a bag of balls spilling out freely, well what about the gravitational pull that is a countering quantity of negentropy?
Like cosmologists do, you would have to step up to a viewpoint where the creation of spacetime - as the great heat sink being manufactured to absorb what now looks to be so be some initiating Big Bang quantity of located energy - is also a thing to be counted in the final balance.
Then from there, you need to step up to a third rung that achieves a viewpoint completely outside the system in question. If the Universe isn’t just a messy dispersion of degrees of freedom, nor even the orderly construction of a vast heat sink void, then you have to have an evolutionary tale that combines the local and global scales of what is going on in holistic fashion.
Now you arrive at a picture where the very distinction you seek - order vs disorder - has to emerge into being. At the beginning of time - the Big Bang - order and disorder are radically indistinguishable as there is just an absolute (quantum/Planckian) potential. And at the end of time, you have the opposite of that. The Heat Death is final maximal dispersion of that potential into the ever lasting and unchanging definiteness that is an infinite void with a single temperature and undifferentiated holographic glow of de Sitter radiation. Both locally and globally, there is maximal uniformity across all possible locations along with a maximal number of those possible locations where something could have been different.
So at the beginning of time, nothing could be counted as distinctive variety - individual bits of information or degrees of freedom. Everything was a hot quantum blur of potential. A quantified account can only be imputed retrospectively by the countable variety - in terms of a quantity of energy/a quantity of space - that we observe around us now.
And at the end of time, the number of energy bits (Heat Death photons) and number of spatial bits (Planck scaled distances) will be matchingly infinite in number. So uncountable for the opposite reason of being in unlimited abundance and hence offering zero distinctiveness once more. A chill blandness of differences (radiation) that can’t make a difference (to the cosmically prevailing temperature).
Standing on the third rung right outside the system that is the universe, we now see a transition from unlimited potential to unlimited difference (that also, matchingly, makes no meaningful difference).
Each view of the situation can be correct. So the standard bag of marbles modelling works fine within its own limits. But also each enfolds the other as a succession of larger views. And the largest view is radically unlike the standard, or even the second tier relativistic models used mostly in cosmology.
It is only when you get to quantum holographic type models of the universe - de Sitter horizons, etc - that you start tracking everything that is emerging. Marbles with some countable identity (surprising or otherwise) to have, along with countable locations that give them some place (surprising or otherwise) to be.