ISSN 0032-9460, Problems of Information Transmission, 2015, Vol. 51, No. 1, pp. 31–36.
Pleiades Publishing, Inc., 2015.
Original Russian Text
V.A. Malyshev, 2015, published in Problemy Peredachi Informatsii, 2015, Vol. 51, No. 1, pp. 36–41.
Phase Transitions in the One-Dimensional
V. A. Malyshev
Laboratory of Large Random Systems, Faculty of Mathematics and Mechanics,
Lomonosov Moscow State University, Moscow, Russia
Received September 26, 2014; in ﬁnal form, December 13, 2014
Abstract—A Coulomb medium is a system of N charged particles of equal charge on an
interval with nearest-neighbor Coulomb interaction and constant external electric ﬁeld. We
show that, asymptotically as N →∞, stable conﬁgurations have four possible phases of the
particle density depending on the external ﬁeld, which is assumed to be a function of N.
Moreover, we ﬁnd these phases explicitly.
The problem of ﬁnding N-point particle conﬁgurations on a manifold having minimal energy
(or even ﬁxed conﬁgurations) was regarded to be important long ago . Therefore, we should
say some words about the history of this question. First of all, we consider systems of particles
with equal charges and with the Coulomb interaction. Immediately, the problem is separated into
thecaseofsmallN, where the problem is ﬁnding such conﬁgurations explicitly, and the case of
large N, where asymptotics is of the main interest. Already J.J. Thomson (who discovered the
electron) suggested the problem of ﬁnding such conﬁgurations on a sphere, and an answer for
N =2, 3, 4 has been known for more than 100 years, but for N = 5 the solution was obtained quite
recently . In the one-dimensional case, already T.J. Stieltjes studied the problem on an interval
with logarithmic interaction and found its connection with zeros of orthogonal polynomials on the
corresponding interval; see [3,4]. However, the problem of ﬁnding minimal energy conﬁgurations
on the two-dimensional sphere for any N and for power interaction (sometimes, this is called
the seventh S. Smale’s problem; it is also connected with the names of F. Risz and M. Fekete)
was completely solved for quadratic interaction only (see [5–7] and review ). For more general
compact manifolds, see a survey .
Here we follow an alternative direction: namely, we study how a conﬁguration could change in the
presence of a weak or strong external force. It appears that even in a simpliﬁed one-dimensional
model with nearest-neighbor interaction there is an interesting structure of ﬁxed points (more
exactly, ﬁxed conﬁgurations), rich both in the number and in the charge distribution. For the
constant force case, we ﬁnd four phases of the charge density, with respect to a parameter which is
the ratio of the interaction strength constants and the value of the external force.
By a Coulomb medium we call the space of conﬁgurations
−L ≤ x
< ... < x
of N + 1 point particles with equal charges on the segment [−L, 0]. Here N is assumed to be
suﬃciently large; however, some results are valid for any N ≥ 2. We assume a repulsive Coulomb