[pcwelder]

So the law of increasing entropy is not a fundamental law of the reality because it can be derived from other fundamental equations.

Suppose I show you a snapshot of a random universe, would you be able to tell if the entropy of the universe is going to increase or decrease as the time progresses?

Let's assume that universe's entropy would increase. Consider another universe exactly the same as current universe, but all the particles' velocities reversed. Then this universe's entropy would decrease.

So you are equally like to select both the universe and hence the original assumption of increasing entropy is wrong.

Discarding quantum properties of the particles, is it then fair to say that time's direction is unrelated to whether entropy increases or decreases?

[rixed]

If by "random universe" you mean a universe in which all states of every particule are random, then my understanding is that we would probably conclude that entropy is neither increasing nor decreasing. Our universe is not random. We could spot local phenomenons where entropy is clearly going in one direction. We assume everywhere the entropy would go in the same direction (increasing), and deduce from this hypothesis that the universe started with a very low entropy.

[canjobear]

The expected increase in entropy can be derived from laws of mechanics plus the critical stipulation that, in the past, entropy was very low. Essentially, physical systems want to be in high-entropy states. So if you observe one to be in a very low-entropy state, then you can conclude that with high probability the future of that system will go to higher-entropy states.

> Suppose I show you a snapshot of a random universe, would you be able to tell if the entropy of the universe is going to increase or decrease as the time progresses?

Yes, if it has low entropy then entropy will probably increase; if it has high entropy then the entropy will probably fluctuate up and down statistically.

> Let's assume that universe's entropy would increase. Consider another universe exactly the same as current universe, but all the particles' velocities reversed. Then this universe's entropy would decrease.

The key is that you're exponentially unlikely to find yourself in a universe where all the particles' velocities are reversed. See this: https://en.wikipedia.org/wiki/Fluctuation_theorem

The probability that a system randomly evolves in a way that reduces entropy is very very small.

[cubefox]

> > Suppose I show you a snapshot of a random universe, would you be able to tell if the entropy of the universe is going to increase or decrease as the time progresses?

> Yes, if it has low entropy then entropy will probably increase

The problem is that it probably increases in both time directions, such that the state of minimum entropy is now. As you said, we have to stipulate that the entropy in the past is low, we can't (yet?) infer it from observation. Which raises the question what justifies us making this assumption in the first place.