We consider the system of particles with equal charges and nearest neighbour Coulomb interaction on the interval. We study local properties of this system, in particular the distribution of distances between neighbouring charges. For zero temperature case there is sufficiently complete picture and we give a short review. For Gibbs distribution the situation is more difficult and we present two related results.

Many electric phenomena are not well understood and even might seem mysterious. More exactly, most are not still deduced from the microscale version of Maxwell equations on rigorous mathematical level. For example, even in the standard direct or alternative current, the electrons move along hundred kilometers of power lines, but the external accelerating force acts only on some meters of the wire. Here what one can read about this in “The Feynman Lectures on Physics” (see [

“…The force pushes the electrons along the wire. But why does this move the galvanometer, which is so far from the force? Because when the electrons which feel the magnetic force try to move, they push - by electric repulsion - the electrons a little farther down the wire; they, in turn, repel the electrons a little farther on, and so on for a long distance. An amazing thing. It was so amazing to Gauss and Weber - who first built a galvanometer - that they tried to see how far the forces in the wire would go. They strung the wire all the way across the city….”

This was written by the famous physicist Richard Feynman. However, after that, this “amazing thing” was vastly ignored in the literature. For example, the Drude model (that can be found in any textbook on solid state physics, see, e.g., [

Many more questions arise. For example, why the electrons in DC move slowly, but such stationary regime is being established almost immediately. In [

In fact, Ohm’s law is formulated on the macroscale (of order one); one-dimensional movement of

Besides other nonsolved dynamic problems like discharge, lightning, global current, bioelectricity, and so forth, also local and global properties of equilibrium configurations of charged particles in external electric fields are not at all studied (note that the mathematical part of equilibrium statistical physics has been developed mostly on the lattice). The equilibrium configurations can be either ground state (zero temperature) or Gibbs states. Ground states are easier to describe and we give short review of results in Section

Consider systems of

More interesting is the case of large

In this section we review recent results concerning nonzero external force. Moreover, we consider not only global minima but even more interesting case of local energy minima. It appears that even in the simpler one-dimensional model with nearest neighbour interaction there is an interesting structure of fixed points (more exactly, fixed configurations), rich both in the number and in the charge distribution.

We consider the set of configurations of

It is evident that if

Assume that

However, the monotonicity assumption in this theorem is very essential. An example of strong nonuniqueness (where the number of fixed points is of the order

One-dimensional case shows what can be expected in multidimensional case, which is more complicated but has great interest in connection to the static charge distribution in the atmosphere or in the living organism.

To discover phase transitions one should consider asymptotics

The necessity to consider cases when ^{−1},

Below this section we assume for simplicity that

Both contraction cases are related to the discharge possibility; as after disappearance of the external force, discharge can be produced, the strength of which depends on the initial concentration of charged particles.

We consider the set

We will consider the probability density:

Below we put

We want to note here that there exist many papers, related to the famous Kac mean field model, where conditional independence (chaos) of

We consider the densities having the following asymptotic behaviour as

Under this condition and if

If

It is of interest to know the behaviour of the covariance for densities with hyperexponential decrease at zero.

We will prove here Theorem

(1)

(1) By abelian theorem, see [

(2) The function

Let

The family of densities (

The normalized moment

For any

For some

(1) It is similar to the proof of (1) in Lemma

(2) Put

(3) Note that always

Assume conditions (1)–(3) of Lemma

We change a bit the standard proof of local limit theorem. Let

The estimate for

Assume conditions (1)–(3) of Lemma

By Lemma

By Lemma

As

By definition (

Again we introduce the exponential family of densities

We will use modified Bessel function of the second kind defined as follows:

Let again

(1) As

(2) As

(3) There exists a unique

(1) We can write

Put

(2) Mathematical expectation is

Covariance is equal to

(3) It follows from continuity of

The exponential family (

The normalized moment

For any

For some

It is similar to Lemma

Using Lemmas

The authors declare that there is no conflict of interests regarding the publication of this paper.