The question is subtle: Even if theoretically it is known that one
has ergodicity in the microcanonical ensemble, in practice one is
performing a numerical integration of Newton equations of motion, and
it is important to know which algorithms/integration schemes reproduce
better this features. Verlet algorithm preserves phase space volume
(which is a fundamental property of the hamiltonian evolution which in
turn implies ergodicity) far better than other higher order
integration schemes, such as higher order Runge-Kutta methods (this
gives you an idea of why Verlet algorithm is used although it is
low-order numerical integrator!)
I would recomend you to read about Liouville theorem, Poincare theorem
about microreversibility, and then Ergodic theory. This will you a
good background. You will easily find as well a lot of documents and
texts discussing the properties of Verlet integrator (not only the
basic definitions)
The question about NVT is very interesting and far from trivial. In
some thermostats, such as Nose-Hoover,ergodicity is inherited from
that of the "larger" system with which you fake the reservoir at
constant temperature (which is a microcanonical ensemble).
For Langevin dynamics, which is a more realistic model, ergodicity is
ALWAYS implicitly assumed when running MD simulations, but it is
something very complicated and subtle from the theoretical point of
view (I only know proofs for particular cases).
If anyone can provide references about ergodicity in Langevin
dynamics, I will also appreciate it very much!
Quoting Andreas Tosstorff <andreas.tosstorff.cup.uni-muenchen.de>:
> Dear Amber user,
>
> I have a question on MD theory. I am not a physicist, so please excuse
> me if I make any wrong statements.
>
> As far as I know ergodicity is defined for the microcanonical ensemble
> (NVE). For a simulation of such an ensemble this implies that the longer
> I run a simulation (or by changing starting conditions), the more
> portions of my phase space I will sample.
>
> Are MD simulations of an NVE ensemble ergodic? What about NVT and NPT
> ensembles then?
>
> If possible could you refer me to some further reading?
>
>
> Best,
>
> Andy
>
>
>
> --
> M.Sc. Andreas Tosstorff
> Lehrstuhl für Pharmazeutische Technologie und Biopharmazie
> Department Pharmazie
> LMU München
> Butenandtstr. 5-13 ( Haus B)
> 81377 München
> Germany
> Tel.: +49 89 2180 77059
>
>
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Received on Wed Jun 21 2017 - 04:30:02 PDT
