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 is conserved, though not momentum, though I will also give the paper to see if you can verify the claim, though you will have to download it on your computer/phone.
It’s on page 4 when Luke Schaeffer starts to claim that particles, which is the analogue energy, is conserved in his Cellular Automaton.
if there are hypothetical time-reversible rules that don’t have the first law of conservation of energy due to not implying time symmetry
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.
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).
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.
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 is conserved, though not momentum, though I will also give the paper to see if you can verify the claim, though you will have to download it on your computer/phone.
It’s on page 4 when Luke Schaeffer starts to claim that particles, which is the analogue energy, is conserved in his Cellular Automaton.
https://scottaaronson.blog/?p=1896#comment-110879
https://eccc.weizmann.ac.il//report/2014/084/
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.
What does translation invariance mean here? Is it the same thing as time-symmetry? Or is it time-symmetry for all T?
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).
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.