Re: infinite square lattice of resistors

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Ok, since it looks like everyone may have given up on this problem by
now I think it may be time to spill the beans on its solution.

The problem of an infinite nearest-neighbor-bond-connected Bravais
lattice of resistors can be formally solved by Fourier transforming
Kirchoff's current law applied to each node to the corresponding equation
in the Fourier reciprocal space of the 1st Brillouin zone which turns the
linear Kirchoff current law operator equation into an algebraic equation.
Then transforming back to the original lattice completes the solution.
The reason this works is that the lattice has translational invariance
under lattice translations and the Kirchoff law merely corresponds to a
discrete version of Laplace's equation at all points without a nodal
connection to the outside current source.  At the nodes connected to the
external current source Kirchoff's current law is a discrete version of
the Poisson equation whose point source is effectively the amount of net
currrent injected into the node from the outside. Fourier transformation
solves this problem since a Fourier transformation will diagonalize any
tranlationally invariant linear operator.  The reason that such a
diagonalization occurs is that such a linear translation operator is
merely an algebraic function (actually it happens to be a complex
exponential) of the crystal momentum operator and the crystal momentum is
diagonal in Fourier reciprocal space.  In the case of the square lattice
the formal solution for the effective resistance R_eff as a function of
the pair (m,n) of m units along one n.n. basis direction and n units
along the other n.n. direction between the two external electrodes
connections is a double integral over integration variables (x,y) whose
domain of integration is the square: 0 <= x <= [pi], 0 <= y <= [pi]
where:

R_eff = r*dble_int{dx*dy*(1 - cos(m*x + n*y))/(2 - cos(x) - cos(y))}/[pi]^2

Here r is the resistance of each bond-connected resistor.  Using symmetry
it is possible to show that for (m,n) = (1,0), (0,1), (-1,0), and (0,-1)
the effective nearest neighbor resistance is: R_eff(1,0) = r/2.  The
integral can be integrated in closed form for (m,n) along the diagonal
directions (i.e. m = +/- n).  Using these known diagonal values, using
the square symmetry, and using the fact that the solution for R_eff obeys
the discrete Laplace equation throughout the lattice (but with a unit
source at the origin (m,n) = (0,0) and a boundary condition that R_eff
vanishes at the origin) we can recursively solve for all other (m,n)
pairs throughout the rest of the lattice by systematically working out
from the origin.  For the case for which Leigh asked Herb (i.e.
(m,n) = (1,1) ) the solution ends up being R_eff = (2/[pi])*r.

> Okay, somebody let me in on the trick for the resistance across a single
> diagonal of the infinite square lattice of say one-ohm resistors. I'd like
> both the elegant solution from symmetry which Leigh implied exists, and I'd
> like to know the formal Fourier method to go between any pair of junctions --
> or at least a sketch of the basic idea behind the latter if it's very
> complicated.

Is the above sketch sufficient?

> ...
> For that matter, what's the resistance along a single edge rather than a
> diagonal?

It's r/2.  (This is derivable from symmetry).

> Or what are the resistances for a simple cubic lattice along an edge,

It's r/3. (Also derivable by symmetry)

> face diagonal, and body diagonal?

I can't finish evaluating the necessary 3-d triple integral to find the
answer in closed form for these other instances.  A numerical evaluation
of the integral is needed.

It is possible in the case of the simple hypercubic d-dimensional family
of lattices to turn the d-dimensional integral into a 1-dimensional
integral over a d-fold product of modified Bessel functions I_m(x) --
each of whose order m is the integer displacement along each direction of
the lattice.  Unfortunately, I can't seem to be able to evaluate the
integral in closed form for cases other than the nearest neighbor distance
(and only the infinite distance case in 3-d).

Also, in the case of the 2-d triangular lattice case it is also possible
to evaluate the integral along a symmetry direction and recursively work
one's way out from the origin to find all the other values at other
coordinates.

Note that symmetry considerations result in the interesting (to me at
least) outcome that for *any* nearest-neighbor-bond-connected type of
Bravais lattice the effective resistance across any nearest neighbor bond
is R_eff = 2*r/z  where z is the coordination number of the lattice (i.e.
each lattice point has z nearest neighbors).

David Bowman
David_Bowman@georgetowncollege.edu