It is often quoted that entropy only increases due to the 2nd law of thermodynamics, this isn’t really true in the general case, and the real behaviour is, I think, more interesting and revealing.
The 2nd law is concerned with what happens to certain measurements over time, such as the temperature of an object or the dispersion of a liquid, these measurements are called macroscopic measurements as they measure the collective state, as opposed to ‘microscopic’ measurements which measure the actual exact state of every atom in the object.
Take for example a tin of water-based paint which is white paint in the top half and black in the bottom half. If you leave it for a while it will begin to disperse and eventually the whole tin will reach a grey colour where it will be in equilibrium. The macroscopic measurement is the amount of separation, we could measure it as the vertical distance of the average particle of white paint from half way up the tin:
The 2nd law states that the amount of disorder (called the entropy) increases over time, i.e. the separation level (s) in this tin of paint will go down, which as you can see above, it does. This is because the number of microscopic states (position of the black and white paint particles) for a low amount of separation is far far greater than for a high separation. It is simply far more likely that the set of particles will find their way into the grey state.
To study this more carefully you can grid the paint tin as say 100*100*200 particle positions, and allow each particle to swap positions with one of its neighbours randomly, if you then plot a graph of the dispersion level with time, from any random start point, you actually find that the overall behaviour follows a shape like so:
The usual case for this tin of paint is that its s is 0, it is grey. Occasionally it will raise to a higher level (more ordered) and then come back down again making a little hill. The larger hills are far less frequent than the smaller hills, the resulting shape is a fractal pattern quite different to a single gradient as the 2nd law is often envisaged. This pattern remains pretty much regardless of the actual equations of motion used.
The reason why we usually think of order as always increasing with time is that are examples almost always involve something exceptionally ordered, such as the tin of paint on the left.
The way to think of this on the graph is that, if you searched the graph for a point high enough to represent the separated black and white paint, then you would find it at the top of a very large hill, and as such the projection forwards in time (the slope to the right of the hill top) would have to be decreasing.
Because each larger peak is so much less frequent than smaller peaks, if you find any point of a particular height it is overwhelmingly likely to be the top of a peak as in the above graph. The interesting thing is that this means that, if for instance you search the graph and find a separation level equal to the fairly separated paint 2nd from left in the picture, then if you simulate it backwards in time, the order also decreases. This is perhaps surprising at first, it means that an almost separated tin of paint is more likely to have arrived there ‘by accident’ from a tin of grey paint than to have arrived by being the dispersal of the separated paint on the left tin.
This goes against common thinking, which is that each level of order has arrived from a higher level of order. It is the common thinking that everything has a good reason for its existence, and something or other created it. Because separated paint will inevitably show a dispersion in the middle then a tin with dispersion in the middle must have come from separated paint left for a few hours. This chain of events from order to disorder is common thinking, and it isn’t actually true.
Does this mean that if I go to my garden shed and find a slightly separated tin of paint, that it just got their by accident from an unseparated state? No, because it is still far more likely that the paint arrived in that state by a person putting it there, than by accident.
Does it mean that our history in the universe is actually less ordered than it is now? did we all just appear from some random dust in the last year? Well we have historical records showing order in the past (e.g. photos prior to last year), and even though it is relatively unlikely to have come from a more ordered state, it is much more unlikely to have come from a disordered state and somehow have photos that show it being ordered. In other words, correlating evidence shows us that things have been more ordered in the past, but that doesn’t mean they have to continue to be more ordered in the deeper past.
So my conclusion is that, when we look deeper and deeper into the past, we are looking up to the top of a very big hill like in the graph above. The closer we get to the top, the more difficult the reasoning will become in how events unfolded. That is because they will become less and less causational (downwards slope) and more and more emergent (upwards slope), the other side of this slope will seem like a miracle, as chaos will seem to come together in just the right way, is this miracle actually just time running backwards?
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