Re: [Phys-l] resistors

["David Bowman" <David_Bowman@georgetowncollege.edu>]

Regarding John Denker's elegant hints about solving the
nearest-neighbor-linked resistor lattice problem:

> On 03/17/2008 08:40 AM, Dan MacIsaac wrote:
>
> > check out
> > http://xkcd.com/356/
> >
> > and hints for elegant solutions are welcome (no, I don't have one).
>
> Hint:  Fourier transform.

For sure.

> Lurid details:   http://www.geocities.com/frooha/grid/grid.html <http://www.geocities.com/frooha/grid/grid.html> 
>
> which cites
> D. Atkinson and F.J. van Steenwijk.
>  "Infinite resistive lattices."
>  Am. Jour. Phys., 67 486-492 (1999).
>
> BTW the Fourier idea is worth remembering, because it comes up
> fairly often.  Feynman uses it somewhere in volume II to calculate
> the evenness of illumination at desk-height from an array of lights
> at ceiling-height.
>
> The take-home message is that Fourier methods are not limited to
> waves.  Any sort of periodicity will do, even if not wave-like.
> Remember Fourier's book was _Théorie Analytique de la Chaleur_
> and heat conduction is famously too overdamped to be wave-like.

Yep. And the reason it works is because the translation
operator for any fixed-amount translation (in either time or
space or both) is an exponential of the derivative, (i.e. a complex
exponential of the momentum operator) which makes the momentum
operator commute with the translation operator, and they therefore
share eigenfunctions.  This makes a momentum basis (i.e.
Fourier series/transform) diagonalize just about any linear problem
involving spatial or temporal convolutions, nearest neighbor
couplings, discrete Laplacians, etc.

Also, this problem was discussed previously on this list in 2 threads
back in October, 1999.  The first thread where it appeared was one
whose subject was "Re: Let's outgrow puzzles".  A teaser for the
resistor problem at hand is given at:

https://carnot.physics.buffalo.edu/archives/1999/10_1999/msg00120.html <https://carnot.physics.buffalo.edu/archives/1999/10_1999/msg00120.html> 

The second thread had the more aptly named subject "Re: infinite
square lattice of resistors".  The solution to the resistor lattice
problem is outlined in the Phys-l archives at

https://carnot.physics.buffalo.edu/archives/1999/10_1999/msg00365.html <https://carnot.physics.buffalo.edu/archives/1999/10_1999/msg00365.html> 

David Bowman