Subject: CCL:100% parallel MD

From: Martyn Winn <[email-masked]>
Date: Mon, 31 Oct 94 10:52:27 MEZ

Two points:
1) My impression is that there is much work devoted to
chaotic trajectories for small molecules (i.e. with a
small no. of degrees of freedom) but that the issue is
ignored for MD simulations of bulk systems. This is, of
course, purely for reasons of convenience.
2) If one is in a chaotic regime, then I'm sure that
the -dt, dt trajectory will differ from the 2dt trajectory,
but taken individually each would be a perfectly valid
trajectory. There would be no reason to prefer one over
the other. In fact, a chaotic trajectory is surely ideal
to satisfy the ergodic requirements of most MD simulations.

I wonder if anyone is answering the original question
about efficiency?!

Martyn Winn

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From: [email-masked] (Jeyapandian Kottalam)
Date: Mon, 31 Oct 94 10:52:54 CST

> From: Martyn Winn <[email-masked]>
> 

  I think the efficiency question did not evoke much discussion
because everyone would agree with the theoretical 100% parallelism.
However, the time-parallelism is not very interesting because it
is strictly limited to a factor of 2.  From the Massively Parallel
point of view, we would like to have scalability to 500 to 1000
processors even with slightly less than 100% parallelism.
  This factor of 2 limitation is in addition to (or in spite of)
the chaotic nature of MD trajectories.

Kottalam                           Phone: (USA) 612 683 3622
Cray Research, Inc.

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From: Robert_Topper <[email-masked]>
Date: Mon, 31 Oct 1994 12:47:57 -0500

BANDWIDTH ALERT:
A LONG discussion on chaos in molecular systems follows.
I hope that some find it interesting. - robert
> course, purely for reasons of convenience.

Yes and no...certainly most work in nonlinear dynamics and chaos
of MD trajectories has concentrated on small molecular models at fixed E,
but not simply for "reasons of convenience."  The fact of the
matter is that there remains a lot to learn about nonlinear
dynamics in conservative Hamiltonian systems, esp. in high-dimensional
spaces. There are nonlinear phenonema that only occur in systems
with dimension >/= 3 (ex: Arnold diffusion)...plus the beautiful
visualization tools suggested by Poincare (Poincare maps, or
"surfaces of section") are not unique phase-space maps in
>/= 3 dimensions.  Progress into understanding chaos in "big"
systems has been slow (but see recent papers in Physica D and JCP
for counterexamples). I personally believe that the problems
are less numerical than fundamental, and more basic theory needs
to emerge to make some progress.  Song Ling has just authored a series
of interesting papers on HCN isomerization and chaos (although he's
too modest to tell you so). I did my thesis research in this area.
I'd be glad to steer those who are interested towards a few key papers.

> I wonder if anyone is answering the original question

Martyn has an excellent point. The "practical" consideration for those
interested in getting at the NVT or NpT ensembles (either directly
or otherwise) is: am I sampling "properly" and "completely" (i.e.,
does my sample accurately represent the statistical properties of
the distribution function I'm trying to mimic, and do I have enough
configurations)?  This doesn't necessarily mean that the trajectory
needs to be DYNAMICALLY correct...if you want only THERMODYNAMIC
properties. However, if you also want dynamic properties (autocorrelation
functions?), the distinction between a (-dt,dt) trajectory and a
(0,2t) trajectory could be relevant. I think this should be ironed
out before sweating out how to make the calculation efficient.  

That said, I personally don't believe that whether the trajectory 
is chaotic or not is relevant here. The Lyapunov exponent for the 
starting point in phase space will give the same "divergence rate" 
whether one integrates asymmetrically (0,2t) or symmetrically (-t, t)...
either numerically generated trajectory is just as "wrong" 
(or just as "right" depending on your point of view).  Chaos theory
tells us that it's fundamentally impossible to exactly GENERATE a
chaotic trajectory...unless you computer carries an infinite number
of significant figures.  Fundamentally, your ability to back-integrate 
on the interval (2t,0) is going to be just as poor (or good) as your 
ability to back-integrate (t,-t). If the trajectory is chaotic, it's 
chaotic. 

Chaos / nonlinear dynamics has some important things to say about
the calculation of thermodynamic properties using MD in terms
of satisfaction / nonsatisfaction of the Ergodic Hypothesis.
Constant-(NVE) computer experiments on many-oscillator systems carried out
by Fermi/Pasta/Ulam at the dawn of "computational experimentation"
showed that quasiperiodic volumes of phase space can be sizeable
no matter how many particles you have in the system. There are
volumes of phase space which may have non-zero "measure" and which are
(at least on the time scales of simulations) toroidal in shape.

Chaotic trajectories are what one hopes for in MD...they
come closest to satisfying the ergodic hypothesis, because they sample
a lot of phase space (in principle, they will sample all of
phase space given infinite time). However, even a chaotic
trajectory cannot sample the ENTIRE phase space on an MD time
scale IF there are "regular" regions of phase space...because such
regions are not accessible to a chaotic trajectory, they are excluded
>from it (until Arnold diffusion, a very very slow process, sets in).
So one needs ways to "skip" between regular and chaotic regions of
phase space in one's sampling...sometimes. This is really probably
only an issue IF the regular regions have big volumes, and EITHER
(1) These "excluded phase-space volumes" lock out significant portions
of CONFIGURATION space, and/or
(2) one is averaging something which depends on configurations AND/OR
MOMENTA (i.e., your averaged quantity is defined in the full phase space).
Usually in thermodynamics one looks at properties which are solely
configuration-space averages, so (2) is not often the case.

Of course the "masters of MD" (you know who you are) have devised
various sampling tricks over the years to ensure that (1) and (2)
are not a problem. But when calculating phase-space averages in
the microcanonical ensemble, one must be aware of the potential
for trouble if there are sizeable and/or numerous "islands" of
quasiperiodicity in the "chaotic sea."

I hope that this is helpful or interesting -
awaiting inevitable flames
robert

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