Subject: Re: CCL:100% parallel algorithm in MD

From: [email-masked] (Song Ling)
Date: Thu, 27 Oct 1994 11:07:49 -0700

You proposed that by reversing all momenta (velocities when they are 
proportional to momenta) you can connect two pieces of integrated 
trajectories together to form a longer one, unfortunately this is 
incorrect in general.  Here is a simple example, the trajectory of 
a harmonic oscillator in phase space is a closed topological circle,
and for an initial condition (x,p) in the first quadron (spell), 
when you reverse its momentum (velocity), you will end up in the 
4th quadron, integrating the trajectory for a time dt, your first 
trajectory is still in the first quadron, while your another trajectory 
is still in the 4th quadron, the two pieces simply don't connect.

The problem originated from that you didn't think about the trajectories
as phase space objects (flows), you had a coordinate space picture.

It is indeed possible to connect two pieces of trajectories to form a 
longer one in phase space, the way to do it is to integrate backward in
time for -dt, and then integrate forward in time for dt, endding up a
trajectory of "length" 2dt.  Do not reverse the signs of the momenta 
(velocities) of your initial point.  (You stay at the same point in 
phase space.)  But this approach may be error prone depending on how 
many time steps you integrate your trajectory, if your system is very
sensitive to initial conditions, the chances are after you integrate a
trajectory from point A to point B, your integrate backward from B
(with -dt) you may not end up in A.

You seemed to be a little confused with the concept of "time reversal
symmetry", to be safe let's say you have a conservative system (non-
dissipative), the total energy is a constant of the motion, time 
reversal symmetry says that looking at the trajectories in phase space
you don't find the arrow of time.  In the harmonic oscillator example
above, you have a topological circel in the phase space, you can 
parameterize it such that a point on it runs counterclockwise or 
clockwise, corresponding to running time forward or backward.  If you
have a damped harmonic oscillator, your trajectory spirals into the 
origin, the fixed point, now you have an arrow of time, past is mapped 
into the future.

In short, you do not apply time reversal to join two pieces of trajectories
together; you integrate forward and backward in time to do that.  I don't
know if that will save you computer time.