The problem I am interested in is central to the
treatment of ions (hard charged spheres, which
to begin with are approximated as points) in water
and in proteins. This is a central system in biology,
chemistry, physics, and stochastics, the latter
since it is the system that Brown originally studied
and is described by Einstein's treatment of Brownian
motion.

The problem is best introduced by considering
Na (positive hard spheres) and Cl (negative hard
spheres with similar but not equal diameters and
diffusion coefficients) in a dielectric background.

Many many chemists (including my friends Stuart
Rice, Doug Henderson, and Jean-Pierre Hansen) have
shown that a thermodynamic analysis of this system
is remarkably successful (but of course not totally
successful) in understanding the properties of ionic
solutions over a wide range of compositions.

We (see CV) have shown that a similar approach
works surprisingly well in biology, for ionic channels
(proteins with a hole down their middle, of enormous
importance in health and disease).

The problem starts by describing the motion of Na^+
as a Langevin equation (in the high friction limit if you wish)
in an electric field. The electric field MUST itself be calculated
from the charges present in the system (i.e., the other Na ions,
the Cl ions, and dielectric charges, and the boundary conditions).
The potential can NOT be independently prescribed because in
the physical world there are no sources to maintain that potential
fixed (in either space or time or both) as charges move in thermal
motion.

The Cl^- ion is described the same way.

The electric field is described by Poisson's equation ( i.e., for the
electric potential) and boundary conditions.

The standard problem is to find a description for the number density
of Na, the number density of Cl as a partial differential equation in
the spirit of a Fokker Planck equation.

The problem I pose is to find a similar description for the
probability of the

1) number density of charge

Q defined as the difference of the density of Na and Cl

and

2) for the total number density of both ions C

C defined as the sum of the density of Na and Cl.


The charge fluctuates, the 'concentration' C fluctuate.
A simulation can clearly keep track of the charge and the total
number density.

The question is can we write equations for these.


The reason this issue is so central is that it is an experimental
fact (verified by electrochemists since around 1890) that in many
domains the system of Na and Cl in water behaves according
to "Ohm's law" (i.e., one need be concerned only with charge,
to the first order, except perhaps one scaling constant) and
in other domains the system of Na and Cl in water behaves
according to "Fick's law" ( i.e., one need only be concerned
with the concentration, to first order).

Those domains are the domains that determine a large
fraction of biological behavior of nerve cells, muscle cells,
etc etc.

The mathematical question is can we derive these equations
and the domains in which they are valid.

There are then many things that could be done.

If one develops a mean field description, changing variables
from number density of each ion, to number density of
charge, and of all ions, gives large simplifications.

I imagine that the equivalent stochastic change of variable
(which is what I seek) would be a least as successful.

I suspect I have written a confusing description and fear that this may
discourage you, indeed if you have read this far.