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.
and so on.
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)?
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
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?