For example when using a mixer to make soup, the more you randomly shuffle all the ingredients together the more you get a nice mix of all the ingredients, instead of, for example, the parts of the broccoli on the left part of the pan and the parts of the carrot on the right side of the pan at some random moment you measure

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If you keep looking at any section of the mix (say, a certain side of the pan), mixing consists of two parts: stuff randomly coming out of this location and stuff randomly coming into this location.

Stuff randomly coming out of the location doesn’t really change the proportion: if there was twice more broccoli than carrots in a certain place, on average the mixer will “pick” twice more broccoli than carrots to move away and still twice more broccoli than carrots will remain here.

Stuff coming into this location moves the proportion towards an average of what was here and what was where the stuff came from. If we put stuff from a broccoli-heavy place to a carrot-heavy place, there will be a little more broccoli in the latter now.

So, it’s easy for the parts to move towards average and it happens naturally, but it’s almost impossible for the parts to move away from average, there’s no mechanism for that. So, the entire thing just averages out.

To restate what another commenter said: imagine you are mixing coffee and sugar. Lets pretend all the sugar starts off in one half of the cup. As you mix, any particle of sugar has some chance of moving to the other half, say 50% (number doesn’t matter for final outcome).

At first, sugar can only move to the empty half, or stay in the original half. It is impossible for sugar to travel to the full half because all the sugar is already there.

After some time, let’s say 10% has moved by random chance to the empty half, so 90% is still in the original half. It is possible for sugar to travel back to the original half, but there are 9x as many chances for a bit of sugar to move to the less sugary half, because each sugar piece has the same chance of switching and there is still 9x more in the original half.

In this way, on average there will be a net transfer of sugar to the less sugary half until there is the same amount of sugar in both halves. At this point, on average, the same amount of sugar will leave each half for the other, so the amount in each half will stay the same.

This works the same way if we look at smaller portions of the coffee: more sugar is likely to leave a region than enter if there is more sugar in that region than in its surroundings. It’s the same as saying if you flip a coin 10 times, you will probably get more heads than if you flip a coin 2 times, as long as the coin has the same chance of heads for all flips.

This process is called diffusion.

This concept is called “entropy” by people who study physics, and your example of mixing ingredients is actually a classic example used to describe the concept.

It takes more energy to move from something that is chaotic and disordered to something that is organized or “ordered”. It takes a few stirs to shuffle your ingredients together, but to return them back to being all on one side you’d have to pick through each individual piece and carefully move them from one side to the other.

Another way to look at this is that there are only a few combinations of locations of the parts that are an ‘ordered’ arrangement (all the broccoli on one side, all the carrot on the other), and comparatively huge number of combinations of locations of the parts where they aren’t. So, if you pick a random combination of all the part locations in the pan, there is a much higher chance of it being mixed together.

If one ingredient is heavier than another, then if left alone the heavier one will settle to the bottom, because there is a force (gravity) acting on different masses.

If you have all the ingredients separate, they won’t be mixed. If you are mixing them by hand or with machinery you are increasing the instances each ingredient will be beside a different ingredient. The more you mix, the more instances you will have. This is increased if you have emulsifying ingredients, but that’s a different thing all together.