One-dimensional Coulomb multi-particle systems

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.


Introduction
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 "Feynman lectures on physics" ( [22]. volume 2, 16-2): "...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. 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 for example [1]), considers free (non-interacting) electrons and constant external accelerating force acting along all the wire, without mention where this force (or field) comes from.
Many more questions arise. For example, why the electrons in DC moves slowly but such stationary regime is being established almost immediately. In [4,5] it was demonstrated rigorously that even on the classical (not quantum) level that the stationary and space homogeneous flow of charged particles may exist as a result of self-organization of strongly interacting (via Coulomb repulsion) system of electrons. This means that the field accelerating the electrons is created by the neighboring electrons via some multiscale selforganization.
In fact, Ohm's law is formulated on the macroscale (of order one), one-dimensional movement of N electrons is described on the microscale (of order N −1 ), but the accelerating force is the corollary of the processes on the so called submicroscale (of the order N −2 ). To show this we used only classical nonrelativistic physics -Newtonian dynamics and Coulomb's law, but also the simplest friction mechanism, ignoring where this friction mechanism comes from.
Besides other not solved dynamic problems like discharge, lightning, global current, bioelectricity etc, 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 Part 1 of the paper. Study of local structure of Gibbs configurations is at very beginning and we present in Part 2 two new results with complete proofs.

Ground state configurations
Consider systems of N particles with equal charges, Coulomb interaction and external force F on an manifold. Even when there is no external force, the problem appears to be sufficiently difficult, and was claimed important already long ago [8]. For example, J. J. Thomson (who discovered electron) suggested the problem of finding such configurations on the sphere, and the answer has been known for N = 2, 3, 4 for more than 100 years, but for N = 5 the solution was obtained only quite recently [11].
More interesting is the case of large N, where the asymptotics N → ∞ is of main interest. In one-dimensional case T. J. Stieltjes studied the problem with logarithmic interaction and found its connection with zeros of orthogonal polynomials on the corresponding interval, see [9], [10]. However, the problem of finding minimal energy configurations on two-dimensional sphere for any N and power interaction (sometimes it is called the seventh problem of S. Smale, it is also connected with the names of F. Risz and M. Fekete) was completely solved only for quadratic interaction (see [12], [14], [15] and review [13]). For more general compact manifolds see review [16].
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.

The model
We consider the set of configurations of N + 1 point particles with equal charges on the segment [−L, 0]. Here N is assumed to be sufficiently large. We assume repulsive Coulomb interaction of nearest neighbours, and external force α ext F 0 (x), that is the potential energy is given by where α ext , α int are positive constants. This defines the dynamics of the system of charges, if one defines exactly what occurs with particles 0 and N in the points 0 and −L correspondingly. Namely, we assume completely inelastic boundary conditions. More exactly, when particle x 0 (t) at time t reaches point 0, having some velocity v 0 (t − 0) ≥ 0, then its velocity v 0 (t) immediately becomes zero, and the particle itself stays at point 0 until the force acting on it (which varies accordingly to the motion of other particles) becomes negative. Similarly for the particle x N (t) at point −L.

Problem of many local minima
It is evident that if F 0 ≡ 0, then there is only one fixed point with Thus it is the global energy minimum. More general result is the following Theorem 1 Assume that F 0 (x) is continuous, nonnegative and monotonic. Then for any N, L, α ren the fixed point exists and is unique.
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 of N) is very simplefor a function F 0 (x) with the only maximum inside the interval. Namely, on the interval Then there exists C cr > 0 such that for all sufficiently large N and α ren = cN, c > C cr , one can show using similar techniques that for any odd N 1 < N there exists fixed point such that Moreover, any such point will give local minimum of the energy. One-dimensional case shows what can be expected in multi-dimensional case, which is more complicated but has great interest in connection to the static charge distribution in the atmosphere or in the live organism.

Phase transitions
To discover phase transitions one should consider asymptotics N → ∞, with the parameters L, F 0 (x) being fixed. Then the fixed points will depend only on the "renormalized force" F = αext α int F o , and we assume that the renormalized constant α ren = αext α int can tend to infinity together with N, namely as α ren = cN γ , where c, γ > 0.
The necessity to consider cases when α ren depends on N, issues from concrete examples where α ren ≫ N. E.g. the linear density of electrons in some conductors, see [1], is of the order N ≈ 10 9 m −1 , α int = e 2 ǫ 0 ≈ 10 −28 and α ext = 220 volt meter e = 220 × 10 −19 (in SI system). Thus α ren has the order 10 11 . This is close to the critical point of our model, which, as it will be shown, is asymptotically c cr N, that is close to 4 × 10 9 in our case.
Below this section we assume for simplicity that F 0 > 0 is constant. We formulate now the assertions proven in [3,2,7].
Critical force For any N, L there exists F cr = F cr (N, L) such that for the fixed point the following holds: Multiscale phase The case when α ren does not depend on N was discussed in details in [3,2], there are no phase transitions but it is discovered that the structure of the fixed configuration differs from (2) only on the sub-micro-scale of the order N −2 . More exactly, consider more general case when V (x) = |x| −b , b > 0. Then the following holds: if F does not depend on N then for any k = 1, ..., N Uniform density We define the density ρ(x) so that for any subintervals I ⊂ [−L, 0] there exist the limits , then the density exists and is strictly uniform, that is for all k = 1, ..., N as Non-uniform density If F = cN and 0 < c ≤ c cr , then x N = −L and the density of particles exists, is nowhere zero, but is not uniform (not constant in x).
Weak contraction If F = cN and c > c cr , then as N → ∞ and the density on the interval (− 2 √ c , 0) is not uniform.
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 shall consider the probability density

Gibbs distribution
on S(N, L) with respect to the Lebesgue measure ν on S(N, L), where is the function on the set of configurations called the energy, and V is the function on the segment [0, ∞], called potential. Most interesting case for us is the Coulomb repulsive potential Equivalently one could say that we consider the sum of N independent identically distributed positive random variables ξ k , each having density g(u), u > 0, further assumed to be smooth, for simplicity. Then the conditional density of the vector {ξ 1 , ..., ξ N }, under the condition S N = L, that coincides with (7), if we put It is clear that conditional distributions P (ξ k < x|S N = L) are the same for all k. In particular < ξ k |S N = L >= L N Note that in the limiting case β = ∞ the distribution is concentrated in the unique fixed point be the n times convolution of f (x). Then the conditional variance is We want to note here that there exist many papers, related to the famous Kac mean field model, where conditional independence (chaos) of ξ k is proved under various conditions, see for example [17] and references therein. We follow here another goal -to reveal possible multiscale local structure in the Gibbs situation, which could resemble zero temperature case structure, discussed in Part I.

Results
We consider the densities having the following asymptotic behaviour as x → 0 It is of interest to know the behaviour of the covariance for densities with hyperexponential decrease at zero.

General power asymptotics
We will prove here Theorem 2. Instead of one density g(x) it is useful to consider the family of densities (such trick has been used in some large deviations problems, see [18,19]) where λ ≥ 0 and Let ξ λ,k be random variables with density h λ (x). Put m λ = Eξ λ,k , σ 2 λ = Dξ λ,k and denote the conditional densities of ξ λ,k under the condition that S N = 1 It is easy to check that f λ (x) in fact does not depend on λ.
Proof. 1) By abelian theorem, see [21] p. 445 Theorem 3, and the condition g(x) ∼ c 0 x α−1 as It follows that 2) The function m λ is monotonically decreasing in λ. Thus for any N there exists λ N such that m λ N = 1 N From 1) it follows that λ N ∼ αN and σ 2 λ N ∼ α −1 N −2 as N → ∞. Let φ λ (t) be the characteristic function of ξ λ .

Lemma 2
The family of densities (11) has the following properties:

for some
Proof.
1) It is similar to the proof of 1) in lemma 1.
2) Put Let us show that for some δ > 0 as λ → ∞. For this can write f (t, λ) as Taking into account that Putting y = λx in the last integral we get for some δ > 0, then For J 2 (λ) we get the estimate From these estimates (13) follows.
3) Note that alwaysˆ1 for some p > 1. Without loss of generality one can assume that 1 < p ≤ 2. By Hausdorff-Young inequality Lemma 3 Assume conditions 1-3 of lemma 2 and that λ N are defined by the condition where o(1) tends to 0 uniformly in x ≥ 0.
Proof. We change a bit the standard proof of local limit theorem. Let p N (z) be the density of the standard deviation where S N,λ = ξ 1,λ + . . . + ξ N,λ is the sum independent random variables having density (11). Let q(z) be the standard Gaussian density. The inverse Fourier transform gives where ψ λ (t) = φ λ (t)e −itm λ . Denote is bounded uniformly in λ (lemma 2, part 1). Then For I 1 we have the estimate (see [20] p. 109, lemma 1) For I 2 we have by parts 2 and 3 of lemma 2, for N sufficiently large, where γ < 1 and q > 1.
The estimate for I 3 is trivial. Thus, where O(N −1/2 ) does not depend on x ∈ R. Lemma follows as The lemma is proved.

Lemma 4 Assume conditions 1-3 of lemma 2. Then the conditional variance
Proof. By lemma 3 we have where the term o(1) tends to 0 uniformly in x ∈ [0, 1]. By (12) the conditional variance is Substituting (15) to this expression we get The lemma is proved.
Proof. By definition (11) and lemma 1 as N → ∞ Changing variable y = Nx we have which gives the lemma and the theorem.

Coulomb case
Again we introduce the exponential family of densities where the function g(x) satisfies condition (10) with β > 0 and We will use modified Bessel function of the second kind defined as follows: where α ∈ R and z > 0. For K α (z) we know the asymptotic expansion as z → ∞ Let again ξ λ be a random variable with the density h λ (x) and put m λ = Eξ λ and σ 2 λ = Dξ λ .
2)Mathematical expectation is By part 1 of this lemma we havê Covariance is equal to It follows from part 1 that as λ → ∞ Using asymptotic expansion (18) we get as λ → ∞ 3) It follows from continuity of m λ as function of λ and (21) that there exists λ N such that m λ N = 1 N and then λ N ∼ βN 2 as N → ∞. By (22) Lemma is proved.

for some
Proof. Similar to lemma 2. Using lemmas 3, 4, 6 we find By lemma 6 λ N ∼ βN 2 and σ 2 λ N ∼ 2 −1 β −2 N −3 as N → ∞, then We split the integral in (23) into two integrals By condition (10) To find the asymptotics of D The function s(x) can be expanded at the neighborhood of N −1 using Taylor's formula: After cancellations as N → ∞. For the second integral D