Glassy dynamics in the asymmetrically constrained kinetic Ising chain
P. Sollich
1
and M. R. Evans
2
1
Department of Mathematics, Kings College London, Strand, London WC2R 2LS, United Kingdom
2
School of Physics, University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, United Kingdom
Received 18 March 2003; published 26 September 2003
We study the dynamics of the East model, comprising a chain of uncoupled spins in a downward-pointing
field. Glassy effects arise at low temperatures T from the kinetic constraint that spins can only flip if their left
neighbor is up. We give details of our previous solution of the nonequilibrium coarsening dynamics after a
quench to low T Phys. Rev. Lett. 83, 3238 1999, including the anomalous coarsening of down-spin domains
with typical size d
¯
t
T ln 2
, and the pronounced ‘‘fragile glass’’ divergence of equilibration times as t
*
=exp(1/T
2
ln 2). We also link the model to the paste-all coarsening model, defining a family of interpolating
models that all have the same scaling distribution of domain sizes. We then proceed to the problem of
equilibrium dynamics at low T. Based on a scaling hypothesis for the relation between time scales and length
scales, we propose a model for the dynamics of ‘‘superdomains’’which are bounded by up-spins that are frozen
on long time scales. From this we deduce that the equilibrium spin correlation and persistence functions should
exhibit identical scaling behavior for low T, decaying as g ( t
˜
). The scaling variable is t
˜
=( t / t
*
)
T ln 2
, giving
strongly stretched behavior for low T. The scaling function g ( • ) decays faster than exponential, however, and
in the limit T →0 at fixed t
˜
reaches zero at a finite value of t
˜
.
DOI: 10.1103/PhysRevE.68.031504 PACS numbers: 64.70.Pf, 05.70.Ln, 05.20.-y, 75.10.Hk
I. INTRODUCTION
The phenomenology of glassy systems—see, e.g., Refs.
1–4 for excellent reviews—has inspired many theoretical
descriptions and explanations. Experimentally, long relax-
ation times are observed; when these become much longer
than the observation time scale a glass transition is said to
occur. Other signatures of glassy dynamics are correlation
functions that can be fitted by a stretched exponential decay
law and aging phenomena 5 where, since the system is out
of thermal equilibrium, it keeps evolving as time goes by and
time-translation invariance is broken.
From a modeling perspective the same phenomenology
arises when one studies simple model systems by computer
simulation. Again, relaxation times can outstrip the time
available to run a simulation and one never explores the
equilibrium state.
The long relaxation times in glasses typically show a pro-
nounced divergence as the temperature T is lowered and are
often fitted experimentally by the Vogel-Tammann-Fulcher
VTF law
=
0
exp -A / T -T
0
. 1
The relaxation time may characterize, for example, the
time for a density fluctuation or an externally imposed stress
to relax. Although some heuristic justifications have been
offered 6, for practical purposes VTF is just a fit with three
parameters
0
, A , T
0
. For T
0
=0 it reduces to an Arrhenius
law. A system for which T
0
is small, so that one has some-
thing close to Arrhenius behavior, is referred to as a ‘‘strong
glass,’’ whereas a system exhibiting large deviations from
Arrhenius behavior is called a ‘‘fragile glass.’’ Generally, T
0
is much lower than the experimental temperatures so that the
mathematical singularity in the fit 1 is not physically rel-
evant in an experiment. From the theoretical point of view,
however, there has been a long debate over whether T
0
might
represent a true thermodynamic transition temperature which
would in principle be measurable in the limit of infinitely
slow cooling.
Although 1 is popular, it is not the only possibility for a
fit. For example, the exponential inverse temperature squared
EITS form
=
0
exp B / T
2
2
has been proposed as an alternative. This form does not ex-
hibit a singularity at any finite T. Experimentally, or in a
computer simulation, it is difficult to distinguish between
VTF and EITS behavior due to obvious limitations on the
longest accessible time scales; both can represent the experi-
mentally observed ( T ) in many materials 7. Theoretical
work is thus essential for clarifying whether VTF or EITS
might be more appropriate.
Stretched exponential decay of a relaxation function, let
us say an autocorrelation q ( t ), is expressed by the
Kohlrausch-Williams-Watt law
q t exp - t /
b
, 3
where the stretching exponent b 1. An heuristic explana-
tion for this law is that there is a broad distribution ( ) of
relaxation modes with decay constants ,
q t =
d
exp -t / . 4
For example, if one assumes ( ) exp(-a) then for large
t the dominant modes have =( t / a )
1/2
which leads to Eq. 3
with b =1/2. This however leaves the physical mechanisms
by which such a relaxation time distribution would arise un-
clear.
PHYSICAL REVIEW E 68, 031504 2003
1063-651X/2003/683/03150416/$20.00 ©2003 The American Physical Society 68 031504-1