How about total energy? In tends to be constant in NVE simulations. I
don't know what the conserved property would be in an NVT simulation.
David.

For most algorithms that give correct ensembles, there are
pseudo-energy properties that are conserved.  For NVE, it's obviously
the energy.  For Nose-Hoover NPT, there's a term involving the "mass"
of the coupling to the heat reservoir that, when added to the scaled
energy term produces a quantity that will be conserved.  For
Parrinello-Rahman dynamics, there's a similar term involving the
coupling to the pressure bath.

It would.
Although I would still argue that when using a thermostat there
is in most cases no "useful" quantity that would be affected
by the current, very small, velocity inaccuracies.
Berk

I think algorithmic self-consistency could generally be justified as
useful.  If one is applying a certain algorithm, it's good to have a
measure of how well that algorithm is being implemented, so one can be
sure that any errors are negligible.  I agree with Berk that such
inaccuracies are small.  But it's good to have a measure of how small
they are built in, just to be sure! :)

Best,
Michael
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