ISSN 0032-9460, Problems of Information Transmission, 2011, Vol. 47, No. 2, pp. 190–200.
Pleiades Publishing, Inc., 2011.
Original Russian Text
V.A. Malyshev, 2011, published in Problemy Peredachi Informatsii, 2011, Vol. 47, No. 2, pp. 117–127.
Critical States of Strongly Interacting
Many-Particle Systems on a Circle
V. A. Malyshev
Laboratory of Large Random Systems, Faculty of Mathematics and Mechanics,
Lomonosov Moscow State University
Received December 27, 2010; in ﬁnal form, February 22, 2011
Abstract—Normally, two scales are considered in multicomponent systems, namely, macro
and micro scales. Moreover, it is accepted that macro variables are completely deﬁned by
micro variables. We show that in the considered particle system with strong local (Coulomb)
interaction this is not always the case. Namely, information concerning some macro variables
can be encoded on a scale much ﬁner than the micro scale.
In multicomponent systems with strong local interaction, one encounters phenomena that are
absent in standard systems of statistical physics and other multicomponent systems. Namely, a sys-
tem with N components in a bounded volume of order 1 (macroscale) has a natural microscale of
. Applying a macroscopic force (of order 1) to the system, and thus to any of its compo-
nents, one normally gets changes in the macroscale itself and, simultaneously, small (of order
changes in the microcomponents; see, for example, . In systems considered below, with strong
Coulomb repulsion between particles, however, one can observe an inﬂuence of such force on the
equilibrium state only on a scale much smaller that the standard microscale. In other words, infor-
mation about the macroforce is not available on both the macroscale and the standard microscale
but is available only on a ﬁner scale. If this phenomenon does not depend on continuity properties
of the applied force, then the mere existence of the equilibrium depends essentially on continuity
properties of the external force.
Model. Consider a system
0 ≤ x
(t) < ... <x
(t) <L (1)
of identical classical point particles on the interval [0,L] with periodic boundary conditions (i.e., on
acircleS of length L). Dynamics of this system of points is deﬁned by a system of N equations
+ F (x
) − A
where A ≥ 0, F is an external force, and the interaction is given by
)+...+ V (x