Re: funny capacitor

["John S. Denker" <jsd@MONMOUTH.COM>]

At 12:08 PM 3/18/01 -0600, brian whatcott wrote:

> It was a week or two ago that Leigh pointed out the unphysicality
> of conceptualizing the iterations of a spread sheet as the time
> evolution of a physical system.

That's true as far as the physics goes, although I don't recall any such
message from Leigh.

> I think this misconception originated with a post of John Denker's.

Really?  What's the evidence of that?

According to my records, the only statement I've previously made on this
topic is on my web page,
  http://www.monmouth.com/~jsd/physics/laplace.html
and includes the statement "The sequence defies simple description, and it
has nothing to do with the physics."

> It may well be that a spread sheet in fact proceeds monotonically
> in a direction that ends in a minimal energy state.

The basic relaxation algorithm *will* proceed monotonically.

My fancy version, if you fool around with the over-relaxation parameter,
may become non-monotonic.  Indeed it may not converge at all if you get too
carried away.

>  It would be no criticism of an algorithm for the procedure if it
> had intermediate states portraying high energy conditions.
>
> The spreadsheet is merely a computing convenience. It is not a
> real time simulation.

It's obviously absurd to use "real time" and "static" (as in
"electrostatic") in the same sentence.

The real issue here is whether the electrostatic energy can serve as a
Lyapunov function which is monotonically decreased by the update
algorithm.  Ludwik checked and found empirically that it seemed to be.  I
checked the equations and found that the *local* energy is minimized when
any given cell is equal to the average of its four neighbors.  Therefore it
is true -- step by step and cell by cell (not just sweep by sweep) -- that
every step of the basic relaxation algorithm decreases the energy (or
leaves it unchanged if it is already minimal).

It's an easy calculation -- check it yourself.

Definition:
  http://www.math.pku.edu.cn/stu/wsxy/sxrjjc/wk/Encyclopedia/contents/Lyapu
http://www.math.pku.edu.cn/stu/wsxy/sxrjjc/wk/Encyclopedia/contents/Lyapunov
Function.html

16,000 examples:
  http://www.google.com/search?q=lyapunov+function