Subject: 100% parallel MD (1st Summary)

From: [email-masked] (Noy)
Date: Mon, 31 Oct 1994 15:37:53 +0100 (NFT)

Dear fellows,
	Thanks a lot for your kind suggestions and comments.
I have summarized the response. All of the responses
will be posted on the next mail.
	Many responses  state that the connection of two MD trajectories
are in doubt, due to numerical error in integrating schemes. Some of
them believe that the reversibility of velocity is disturbed by the
coupling with Thermo-bath and the cut-off of the potential. However, a
few people agree on this idea. 
	My point of view is that the structural properties obtained
>from connecting two MD trajectories are the same as running one MD
double time.
                                t0------------->t1
             t2 <---------------t0

             and
                                t0--------------------------------->t1

	If we look at the phase space (p,q)

      p
         |
         |          ta \
         |              \
         |                 \
         |                    \                  ....... t1
         |                      \ t0  ......./
         |                     ....+ 
         |                   /
         |            ...../
         |          *'
         |         t2
         |
         |___________________________________________  q          

        Well, I start equilibrating the system from the time ta up to
the time t0, which I am sure it is an equilibrium state. Then at this
point, I traditionally perform a simulation from time t0 to the time t1.
At the time t0, I revert the sign of all velocities and simulate in
parallel up to the time t2. I believe that both of the trajectories
>from the time t0--->t1 and t0--->t2 are in equilibrium state. So, the
phase space should be expored correctly, isn't it ? The statistical
average shoud be, hence, conserved.
        If I revert all momentum back again from the time t2--->t0, then
two trajectories are connected to produce one longer run. My purpose
is to have longer time-correlation rather than having more statistical
samples. So, I must be sure that this is correct or not. Many people
are in doubt that the reconnection of two MD is invalid. I would like
to ask some comments if it is possible or not that ......

        Since the prediction of the next coordinates and velocities are
respective to the current (t) and the former (t-1) one. If I have this
in hand       t, t-1 and t+1        t = t0
        I integrate forwardly from  t0 ----> t1 using  t-1 as the previous
step and I integrate backwardly from t0----> t2 using  t+1 as the previous
step. Do you think the connectiong is smooth and neat ?
        As stated by many people that the past of the trajectory is lost
due to the numerical error in integration scheme, I would believe so.
Anyway, if both trajectories are still in the equilibrium state, they
should represent the phase space correctly. 
        Some ideas ?
                                                      my best regards,
                                                        Teerakiat
----------------------------------------------------------------------------
Teerakiat Kerdcharoen (NOY)

Institute of General and Inorganic and Theoretical Chemistry
Innrain 52a, A-6020 Innsbruck AUSTRIA
e-mail:  [email-masked], [email-masked], [email-masked] 
      :  [email-masked], [email-masked] ( Bangkok )
Research :  Molecular Dynamics simulations
         :  Computer Aided Molecular/Material Designs
-----------------------------------------------------------------------------
***  I have no past and no future. I just have today.

---

From: J Scott Shaffer <[email-masked]>
Date: Mon, 31 Oct 1994 09:50:52 -0800

Dear Dr. Kerdcharoen:

I have entered this discussion a little late, but here goes.  Song
Ling has presented these arguments earlier; here they are again in a
slightly different form.

Let's talk only about integrating Newton's equations, yielding MD
simulations in the NVE ensemble.  There are two points to be made:

(1) Disregarding numerical error in the time integration, to generate
    continuous trajectories, one must reverse the sign of the time
    step, dt ----> -dt, keeping the same values of the positions *and*
    velocities.

    Correct time correlation functions cannot be obtained by reversing
    the signs of velocities and connecting the two resulting
    trajectories.  By reversing the velocities, you change the
    location of the system in phase space, as illustrated below, by
    reflecting the momenta through the P=0 axis.  Thus, any time
    correlation functions that are calculated by joining the two
    trajectories would be incorrect, since the trajectories are not
    continuous in phase space.

The phase space (Q,P): The points ta, t0, t1, and t2 are those plotted
by Dr. Kerdcharoen in his first summary.  The point t0' is the phase
point obtained by reversing the velocities of the system at point t0;
the point t2' is the final location of the system in phase space after
integrating Newton's equations from point t0'.

                             P
                              |          
                              |          
                              |     ta \
                              |         \
                              |           \
                              |             \                t1
             t0'              |               \ t0  ......./
  t2'         .               |              ....+ 
    . .       .               |              /
       . .     .              |       ...../
         . . . .              |     *'
                              |    t2
                              |
______________________________|___________________________________ Q

(2) With numerical error in the integration of the equations of
    motion, no trajectory is reversible in practice.  This is
    illustrated nicely in Figure 3.1 of Allen and Tildesley, "Computer
    Simulation of Liquids", where they show that trajectories that
    should in principle be identical actually diverge from each other
    rapidly because of numerical error in the time integration.  

    This is not necessarily a problem in terms of calculating ensemble
    averages from MD.  In fact, if one were to use the dt ----> -dt
    reversal described in (1), this numerical error would be
    *essential*, otherwise you would end up back in the unequilibrated
    starting configuration!

Scott

=====================================================================
 J. Scott Shaffer                       [email-masked]
 Department of Chemical Engineering     
 University of Southern California      (213) 740-2067
 Los Angeles, CA  90089-1211  USA       (213) 740-8053 (FAX)
=====================================================================