One-Dimensional Coulomb Multiparticle Systems

Abstract

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.

Phase transitions in the one-dimensional Coulomb gas ensembles

“…The results are listed in the next theorem in order of increase external force. Recall that the model when F = 0 was treated in [7] (it is a particular case of part (a) of the following theorem). We shall use notation g(N) = Θ(h(N)) if for some constants 0 < c < C ch(N) < g(N) < Ch(N).…”

“…To solve equation (4.4) first we consider EX k,λ . Notice, that in [7] one can find the principal term of the asymptotic of this value. Here using the arguments of [7] we get more details.…”

“…i.e., the right-hand side does not depend on λ. Notice, that Diaconis and Freedman [4] used this property for the identically distributed random variables, their result was then used in [7] to treat the case without external force. Here we show that a similar argument works even without condition on the identity of the distributions.…”

“…Here we focus on the model which describes the charges with nearest neighbour Coulomb interaction on an interval in a presence of external force. This model was introduced and studied recently by Malyshev [6], and then by Malyshev and Zamyatin [7]. Following the previous authors we call this model 1-dimensional.…”

“…The methods we use here develop the probabilistic approach of [7] (and [4]) but now in an inhomogeneous setting.…”