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* Matthew Moss, 09.03.2007 21:38:
> Strictly speaking, as this is a torus, you could put it anywhere
> and be at "the center".

Actually it only looks as if we are acting on a torus. The reason is
as follows:

We start with the physical problem that you have a center at which
freezing starts and an area around this center that can be dealt with
as if it is infinite in extension (meaning that we need not take into
account the surroundings).

For the simulation being feasible it is unavoidable to restrict the
simulation not only to a certain granularity but also to a certain
size of the simulated area[1] which means introducing a boundary into
the simulation that is not present in the simulated situation.

Now that we have an artificial boundary we need to find some clever
boundary conditions that limit artefacts to an unavoidable minimum.
Unless freezing takes place next to the boundary we can assume that
the leftmost region of the simulated area is not too different from
what is immediately to the right of the rightmost region of the
simulated area so that we can use it as an approximation of the true
state for this outside region (similar holding for the other sides).
This approximation obviously is quite good as long as the freezing
has not reached the boundary. But what if that happens? In the case
of the simulation we are dealing with it still is not too bad. The
reason for this is that the growth of the crystal starts at the
center of the simulated are so that the asymmetry present is a
reasult of the growth but not of the system itself being asymmetric.

In other words: While it at first sight may seems that the periodic
boundary conditions allow the center to be anywhere the truth is that
(quite the other way round) the fact that the center is in the middle
of the simulated area allows to use periodic boundary conditions (as
they are called) even after the frozen area has reached the
boundaries of the simulated area which otherwise would mark the end
of the validity of the simulation.

[1]  Many of you may have heard that this is the second major issues
of the simulation of weather and climate simulation - the other
being the impossibility to simulate all the complex phenomena
involved in these processes (not even the most imporant ones).

Josef 'Jupp' Schugt
--=20
Blog available at http://www.mynetcologne.de/~nc-schugtjo/blog/
PGP key with id 6CC6574F available at http://wwwkeys.de.pgp.net/

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