> If you have a casual system, then statistically, things will tend to become more disordered over time, because there are just more ways to be disordered than ordered.
contains a tacit assumption that states are chosen at random. But assuming that is begging the question. Why are states chosen at random? What is the source of the randomness? Newtonian mechanics doesn't have any obvious source of randomness, and it's an open question whether quantum randomness is "really random". Bohmian mechanics is completely deterministic, and so is (obviously) superdeterminism.
It's true that "there are more ways to be disordered than ordered", but in any time-reversible dynamic there are exactly as many states where entropy decreases as there are states where entropy increases because for any entropy-increasing state, the time-reversed state has decreasing entropy.
Even a completely deterministic causal system will tend towards greater disorder (if it wasn't already maximally disordered I guess). There is no need for randomness, just statistics.
> but in any time-reversible dynamic
Now that is making an assumption that the dynamics of an entire system could be thrown into reverse. Everything flipped to its opposite, and then proceeding onwards causally from that point.
It also presumes that the system started in a state of order, got more disordered, and then you reversed it and then order "magically" appears. Your eggs unscramble themselves. If you merely take a disordered system and throw it into reverse, you will still just see a disordered system becoming more disordered, because there was never any surprising ordered state built in to be uncovered later.
> If you merely take a disordered system and throw it into reverse, you will still just see a disordered system becoming more disordered, because there was never any surprising ordered state built in to be uncovered later.
This is circular. If the system is becoming more disordered, it follows that playing it in reverse will make it less disordered. Sure, if the initial state wasn't very ordered to begin with, this won't look very different, but that's entirely irrelevant: as long as we accept that "disorder" is a measurable objective property of a system, then there is a quantifiable difference between moving forward and backward in time. And this doesn't match either a deterministic time-reversal symmetric theory like classical mechanics, nor a deterministic CPT-reversal symmetric theory like QFT.
> that is making an assumption that the dynamics of an entire system could be thrown into reverse
No, that's not an assumption, that's a mathematical feature of all known laws of physics. An it's not that "the dynamics could be thrown into reverse", it's that for every initial state, there is a corresponding initial state where the system runs in reverse, and hence, for every state from which entropy increases there is a corresponding state where it decreases. For a Newtonian system, it's a state where all the velocities have opposite sign. (For quantum systems it's a little trickier to describe.) So if you choose a state uniformly at random from among all possible states, the odds that you will end up with one where entropy is increasing is exactly 50%.
And it gets even worse than that. In a universe that obeys certain conservation laws (which as far as we can tell ours does) a time-reversible dynamic is unitary, which is to say, there is a one-to-one correspondence between an initial state and its successors. Therefore, for any initial state, the time evolution of that state must eventually loop back to its initial state [1], and so it must eventually enter an entropy-reducing state to get there.
Your theory:
> If you have a casual system, then statistically, things will tend to become more disordered over time, because there are just more ways to be disordered than ordered.
contains a tacit assumption that states are chosen at random. But assuming that is begging the question. Why are states chosen at random? What is the source of the randomness? Newtonian mechanics doesn't have any obvious source of randomness, and it's an open question whether quantum randomness is "really random". Bohmian mechanics is completely deterministic, and so is (obviously) superdeterminism.
It's true that "there are more ways to be disordered than ordered", but in any time-reversible dynamic there are exactly as many states where entropy decreases as there are states where entropy increases because for any entropy-increasing state, the time-reversed state has decreasing entropy.