Reversing physics law in classical mechanics?

Question

Leonard Susskind in one of the video's mentions: Link - It includes the timestamp so you don't have to wind.

simply starting some place letting it evolve for a long period of time and then letting it evolve with the reverse law of physics if you come back to the same place every time then your law physics is deterministic.

He says this for the example of 3 state system(Head, Tail, Feet). The laws of physics is H->T->F, but in a very small probability, one in a million, at some instance of time, if the current state is H, it can just say: "I will hold still", but probability of this is 1 in million.

Due to this, What could happen is the following sequence: HTFHTFHHTF...(note two H in the sequence). Now, what I actually understand is if you let the system evolve for 10 million of unit of time, and decide to reverse, there's a chance you won't get back to the original starting point(H). You might end up at F or T. In this case, we can't call the system or law deterministic.

but now, read the quote from Leonard again. In our example, there's still chance that the reversing the system could get back to the original configuration(H). If so, would it be deterministic as Leonard says ? I don't think so, because if you reverse, you go backwards one step at a time and there will be instances of time that you will think it's F for example, but in reality it was T. The funny thing is though, you won't be able to detect each value of each instance of time. Going backwards, you will write down the sequence: FTHFTHFTH because you don't know where the fluctuation happened, so you won't get every value/state correctly.

I thought determinism was if you got a system and you let it evolve for 10 instances of time, when you get backwards, you should be able to derive each state at each instance of time(in our case 10). but Leonard only says: just getting back to the original is enough. What don't I understand ?

Answer 1

I think you got the essential ideas correctly, but Professor Susskind didn't mean to imply that just getting back to the original state is enough as you say, to be certain that a law of physics is deterministic. Rather, evolving the present state with the reverse law, and comparing the final result to the original state, is a way to test whether the system is deterministic, it is a necessary condition but not a sufficient one.

I think that the reason he chose to focus on a situation where we let a system evolve for a long time and then apply the reverse law for that length of time, is that in reality the state of a system becomes more and more complex and "drifts" apart farther from its initial state, which wouldn't really apply to the three states system of $\text{HTF}$ he describes. The space of states of most physical systems, even simple ones like atoms, is immensely larger. So as he mentions, if we only allow evolution in a very small time interval, it isn't a great way to test reversibility because it may be that just by accident the system didn't have a chance to interact with its environment and hence change state, so in such a case it may appear to be reversible, and hence possibly also deterministic.

As an aside, what Prof. Susskind means by a "deterministic" law in this lecture is really one which is both deterministic & reversible, or in other words determinstic both into the past and future. One can imagine a law that is future determinstic but irreversible.