Research Article
Received 4 July 2011 Published online in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/mma.2672
MOS subject classification: 35K59; 70F40; 70F45; 74B15; 74B20
Friction dominated dynamics of interacting
particles locally close to a
crystallographic lattice
M. Bodnar
a
*
†
and J. J. L. Velázquez
a
Communicated by M. Lachowicz
A system of particles, in general d-dimensional space, that interact by means of pair potentials and adjust their positions
according to the gradient flow dynamics induced by the total energy of the system is studied. The case when the range
of the interaction is of the same order as the mean interparticle distance is considered. It is also assumed that particles,
locally, are located close to some crystallographic lattice. An appropriate system of equations that describes the evolution
of macroscopic deformation of the crystallographic lattice, as well as the system that describes the evolution of the main
crystallographic directions is derived. Well-posedness of the derived system is studied as well as the stability of the
particle system. Same examples of potentials that yield stable and unstable systems are given. Copyright © 2012 John
Wiley & Sons, Ltd.
Keywords: hydrodynamic limit; interacting particles; deformation; almost crystallographic lattices
1. Introduction
The goal of this paper is to study the evolution of a set of particles that remain close to a crystallographic lattice and whose positions
fX
h
.˛, t/g evolve by means of the following system of equations
d
dt
X
h
.˛, t/ D
X
ˇ ¤˛
rV
h
.X
h
.˛, t/ X
h
.ˇ, t// , X
h
.˛, t/ 2 R
d
. (1.1)
Equations with the form (1.1) typically describe systems dominated by friction, in contrast with systems where inertia plays a role.
Because of this assumption, there exists no second derivative of the particle position with respect to time on the left-hand side of (1.1).
The term crystallographic is used in this paper in a mathematical sense and it refers to the existence of specific groups of symmetry
(crystallographic groups) that approximately describe the arrangements of particles. This type of crystallographic arrangement is not
uncommon in biological systems (for instance the structure of Elodea cells, plagiomnium affine or cells from iris petal, see e.g. [1, p. 89,
Fig. 5.18(b)]).
Models with the form (1.1) are used often in mathematical biology, specifically in the so-called individual cell-based models
(cf. [2–10]). These models assume that cells that can be considered as point particles interact by means of pair potentials and adjust
their positions according to the gradient flow dynamics induced by the total energy of the system. Models that take into account
stochasticity in a cell motion can be found, for instance in [8–16] and references therein.
We will assume in the following that V
h
rescales as
V
h
.x/ D
h
H
d
V
1
x
H
, (1.2)
where H D h
#
for some # 0. Here, h is a typical distance between particle, and H is the range of the interaction of the potential.
a
Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland
b
Institut für Angewandte Mathematik, Endenicher Allee 60, 53115 Bonn, Germany
*Correspondence to: M. Bodnar, Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw,
ul. Banacha 2, 02-097 Warsaw, Poland.
†
E-mail: mbodnar@mimuw.edu.pl
Copyright © 2012 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2012