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I recently started reading "The Theoretical Minimum: What you need to know to start doing Physics".

In the first chapter, the authors define the "Minus-First law", and state that reversible ".. laws are deterministic into the past as well as the future.". They go on to define a deterministic reversible law as one where, ".. every state has a single unique arrow leading into it, and a single arrow leading out of it, ..".

While pondering on this, it struck me that if a dynamical law contains cycles, we are losing information reg. the initial/starting state of the system.

To provide a more concrete example, let's consider dynamical law 1 (1->2->3->4->5->6->1) of the 6 sided die system described by the authors. If at a certain time n we are told that we are state 4, how can we infer the initial/starting state of the system? We may have ended up at state 4 starting from any one of the 6 states.

  • 1->2->3->4->..
  • 3->4->...
  • 6->1->2->3->4...
  • 4->5->6->1->2->3->4..

and so on.


QUESTION 1

Does that not mean that we are losing information reg. the "past", and that we cannot determine the past based on the future? Can such a law (with cycles) be categorized as truly deterministic and reversible (w.r.t. inferring the past from the future)?


QUESTION 2

Does this mean that to be able to predict the (past or future) behavior of a closed system, we need 2 pieces of information:

  • a set of deterministic and reversible laws that cover the entire state space
  • the initial state of the system

QUESTION 3

If we assume that the universe is a closed system governed by deterministic and reversible dynamical laws, does that mean we cannot explain how the universe began (because we don't know the initial state of the system)?

I guess we could assume the big bang (hypothesis-1), and set out to infer the current state of the universe using a set of laws/truths that we believe govern the universe (hypothesis-2).

If we're able to do so, then our hypotheses (start state + governing laws) are correct, and so we have a possible explanation for how we came to be? But does that not mean there are potentially other hypotheses out there that can also explain the state of our universe?

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When I saw the accompanying lecture series on Youtube, my first 2 questions were answered.

We need both the initial state and a deterministic set of laws (not lose information about the past) to be able to predict the future/re-discover the past.

As for question 3, my current stance is that it is possible to that there are multiple alternate hypotheses that can explain the way a system (in this case our universe) evolves, until proven otherwise. But if two sets of hypotheses lead to the same outcome, are those 2 sets really different? A tad philosophical, and maybe not an appropriate question for this particular site.

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