So in this new chapter, we will discuss the Poisson-Boltzmann equation, which actually

is quite different from the Boltzmann equation and the so-called Poisson and Planck equations.

So the main difference to what we did so far is that now we are dealing with particles

that carry a charge.

These could be either electrons or semiconductors, so all your computer and cell phone chips

are based on electrons moving inside a semiconductor, or it could be like in the human body, ions

that go through cell membranes, so-called ion channels, or nowadays a lot in batteries

like lithium ions or in the human body, you for example have sodium and chloride.

So the difference to what we did so far is that at the microscopic level, each particle

carries a charge, so they are still moving, but they can have like a plus charge or a

minus charge or say a multiple of an elementary charge if they are ions.

Macroscopic, this means we have two different densities.

One is the usual particle density rho, and the second is the charge density.

This is now a science density, C times rho, where C is a positive or negative number depending

on the charge of an elementary particle.

So it's a multiple of the elementary charge, it can be positive or negative.

So C can either be positive or negative, so rho is not necessarily positive.

Now we have several ways to derive a macroscopic equation for the particle density and velocity.

The first is to assume we have just a flow, Euler, with additional coulomb forces.

So this means we start with the Euler equations.

We assume there is some friction that we just use as a linear friction.

So for simplicity we set the friction coefficient to 1.

Then we have the usual Dijkstra force, and now this comes from a coulomb force.

And we also assume there is a little bit of pressure, and we assume the pressure to be

linear, so this will be, as usual, gradient P divided by rho.

P of rho is linear, and the term in the equation is gradient P divided by rho, as usual.

And the coulomb force between two particles can be written as follows, k is actually proportional

to the, as we have seen also in the microscopic thing, is a constant that is not really important,

times C squared times the gradient of G.

Or, if you want to write it, let's keep it like this.

And G was the coulomb kernel, is 1 over the norm of x in 3D.

So this is how we could start writing a model.

What is interesting is we can rewrite this in terms of what is the electric field.

And the electric field is actually minus this constant times C times

gradient of G, convoluted with rho, or in other words, it's minus this constant times

gradient of G, convoluted with the charge density C times rho.

That's how we can write it at the full space, and then the equation that we have

is minus U plus C times the electric field plus D times gradient of rho divided by rho.

We could also look at the electric potential.

That's the variable we want to formulate everything in the end.

Phi is equal to G, that is constant, and G convoluted with the charge density.

And so, as usual, we have the relation the electric field as a vector is minus gradient

of the electrical potential.

This is the microscopic derivation.

Actually, it turns out if you compute the divergence of the electrical field, so minus

laplacian of phi, then we get back the charge density.

So if you compute for this Coulomb force, formally, the laplacian becomes a delta distribution

and the convolution of this concentrated delta with C times rho just gives C times rho.

There's sometimes a different normalization made here with epsilon, the dielectric constant,

in order to account for changing materials, and then you write it typically like this.