Does a time-reversible physical law/Cellular Automaton always imply the First Law of Thermodynamics?

9ProgramCrafter

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1ProgramCrafter

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3Noosphere89

3jacob_drori

2Carl Feynman

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Do you happen to have definition of "energy" for cellular automata? I guess you could group states reachable via the reversible law (thus being on one loop) into equivalence classes, but that does not say anything about cells in any local area.

Physics is continuous and has Noether's theorem; for it, time shift symmetry (not even reversing time direction) implies conservation of energy.

For the physics case, I'm asking essentially whether there can be a physical rule (at least in a hypothetical universe different than our real one) that is time-reversible, but not having time shift symmetry, and thus not implying conservation of energy, or if time-reversible physical rules always imply time shift symmetry/time symmetry.

Another way to say it is I'm asking if there are hypothetical time-reversible rules that don't have the first law of conservation of energy due to not implying time symmetry.

Scott Aaronson claims that the analogue of energy...

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Yes, there are. For instance, take one-dimensional model with a particle (to avoid questions about origin, you might add two reference particles which would not move) having conserved quantity v and position x(t). The law could be x′(t)=v∗abs(t); it is symmetric around zero, it is reverse-simulatable, but not translation invariant.

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What does translation invariance mean here? Is it the same thing as time-symmetry? Or is it time-symmetry for all T?

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Translation-invariance means that law is invariant in t - that is, if you test a system twice at different times, you might obtain same results. It is kind of continuous symmetry.
There is discrete, mirror symmetry, which is weaker than continuous but very much stronger than allowing to reverse-simulate. Mirror symmetry with "axis" T means that if system evolved from state A to B in time interval [T-t; T], then it would evolve from B to A during interval [T; T+t].
Reverse-simulation is even weaker, it only requires to specify a law for calculating past states (no need for reverse law to be same as forward one).

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Thanks for answering the question, so if I'm correctly reading this, time-reversibility in simulation does not imply time symmetry or discrete mirror symmetry, and is strictly weaker than both of these, allowing you to have time-reversible rules that violate the first law of thermodynamics, allowing for creation or destruction of energy/heat.

I'll just answer the physics question, since I don't know anything about cellular automata.

When you say time-reversal symmetry, do you mean that t -> T-t is a symmetry *for any T*?

If so, the composition of two such transformations is a time-translation, so we automatically get time-translation symmetry, which implies the 1st law.

If not, then the 1st law needn't hold. E.g. take any time-dependent hamiltonian satisfying H(t) = H(-t). This has time-reversal symmetry about t=0, but H is not conserved.

“Time-Symmetric” and “reversible” mean the same thing to me: if you look at the system with reversed time, it obeys the same law. But apparently they don’t mean the same to OP, and I notice I am confused. In any event, as Mr Drori points out, symmetry/reversibility implies symmetry under time translation. If, further, the system can be described by a Hamiltonian (like all physical systems) then Noether’s Theorem applies, and energy is conserved.

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From my perspective, time reversibility means that you can always simulate the laws of physics backward in a well defined way, and time symmetry means the laws of physics remain the same if you simulate it backward in time.
I didn't see an obvious way of showing that "simulating the laws of physics backwards in a well defined way" which is reversibility, always implied symmetry, which means that the laws of physics remain unchanged when running the simulation backwards.
Also, he showed this:
Which I'm pretty sure doesn't mean that all physical systems that are described by a Hamiltonian satisfies Noether's theorem, only the ones that have time-symmetry for all T/any T.
For some context on the question, I'm not focused on our physical universe, but about hypothetical universes where time-reversible laws may not always imply time-symmetric laws, nor does such a time-symmetric universe have symmetry for all T.
I'm essentially asking if these are able to be picked apart in the general case, not the specific case like in our universe.

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I was right to seperate the 2 concepts, since @ProgramCrafter has provided an example where a time-reversible law doesn't imply a time symmetry, possibly avoiding the first law of thermodynamics in the process. Here's the link:
https://www.lesswrong.com/posts/NakKdgW4BJKdwXAe8/#yStS9fff4HzrtPZAC

Hm, I'm talking about time reversible physical laws, not necessarily time symmetric physical laws, so my question is do you always get time-symmetric physical laws that are symmetric for any T, out of time-reversible physical laws?

See also this question in another comment:

...For the physics case, I'm asking essentially whether there can be a physical rule (at least in a hypothetical universe different than our real one) that is time-reversible, but not having time shift symmetry, and thus not implying conservation of energy, or if time-reversible physical r

This question is kind of self-explanatory, but for people who are physicists, if a time reversible rule of physics/cellular automaton exists in a world, does this automatically imply the first law of thermodynamics, that is energy may not be created or destroyed?

Note I'm not talking about time-symmetry or the 2nd law of thermodynamics, which states that you can't have a 100% efficient machine, just time-reversible physical laws/cellular automatons and the first law of thermodynamics.

Edit: @jacob_drori has clarified what exactly I'm supposed to be asking, so the edited question is this:

Do you always get time-symmetric physical laws that are symmetric for any T, out of time-reversible physical laws?

The question of do you always get time-symmetric physical laws from time reversible laws is also a valid question to answer here, but the important part for the first law of thermodynamics to hold is that it's symmetric for all times T, and in principle, the question of time reversible laws of physics always implying time symmetry could have a positive answer while having a negative answer to the original question, because it doesn't imply time symmetric laws of physics for all T.