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.
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.