Re: Re: heat diffusion calculation

From: Victor Ovchinnikov (ovchinnv_at_mit.edu)
Date: Mon Jun 18 2007 - 14:58:13 CDT

Jonathan,
The page numbers that you provide correspond to the calculations of the
temperature echos from an MD simulation (at least in the tutorial
version that I have)

Nevertheless, I can try to answer some of your questions:
1) Yes, it is a boundary condition to the diffusion equation on page 47.
2) Not sure where you are; step 11 on p.55 deals with temperature echos;
I'm certain that the script monitors the average temp of the system
including the outer shell; this is an approximation, since analytically,
the boundary has no volume, whereas in this case, it does; perhaps
that's why the agreement with the theoretical plot on p.49 is not
perfect.

3) The inner radius is 22, as defined in the VMD script that populates
the beta column of the PDB; the simulation domain is a sphere with
radius 26 Angstroms; so cutting off at 22 should apply to the outer
layer of the water molecules, since the diameter of a water moelcule is
roughly 3A. So the calculation _does_ take the radius into account; for
a larger radius, you would have to first solvate & equilibrate the
protein in a larger sphere; then when you ran VMD, increase to cutoff
accordingly.
4) Yes, the problem is that you need an analytical formula for the
solution of the heat equation in the rectangular domain; this should be
obtainable from a PDE book as a series solution (just like the spherical
formula) You would then need to take this solution and average over the
size of the box -- i.e. integrate over x,y,z & divide by the box size;
this would give you a an expression similar to the on p. 47

Regarding the physics of measuring diffusion, you are correct; it is
easiest to fix the temp. somewhere & put a probe at another location &
record your temperature values; This is the lab experiment, which would
give you a long-time averaged quantity (the duration of the experiment
is of the order of seconds). However, you can only do MD for a few
nanoseconds -- so your statistics for a quantity at one location would
be extremely poor. What the method outlined in the tutorial says, is
that you can still extract the diffusion coefficient from averaging over
multiple regions -- in this case the entire domain (which will give you
better statistics by orders of magnitude)

Best,
Victor
 

On Mon, 2007-06-18 at 12:16 -0500, Jonathan Lee wrote:
> Anybody? Thanks.
>
> Jonathan
>
>
> Jonathan Lee wrote:
> > Hello all,
> > I have some questions about the heat diffusion calculation in the
> > NAMD tutorial (page 53).
> >
> > 1) The shell is maintained at a temperature of 200, right? (As
> > opposed to just initialized to 200.)
> > 2) What is the temperature that is output (step 11, page 55)? Is that
> > the temperature of everything excluding the 200K shell?
> > 3) It seems to me that the calculation should take into account the
> > inner radius of the shell. If the radius is much larger, shouldn't it
> > take a longer time for the diffusion to occur?
> > 4) Can I do a similar calculation but with a rectangular prism domain
> > (i.e. fix the temperature at one end and calculate the heat diffusion
> > to the other end of the box)?
> >
> > Basically, my understanding is that the temperature should be
> > maintained in one region and measured in another region a finite
> > distance away. That distance (and the time of diffusion) should be
> > taken into consideration when finding the diffusivity. Am I
> > overlooking something? Thanks.
> >
> > Jonathan
> >
> >
> >
> >
>

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