Equation-free dynamic renormalization in a glassy compaction model

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by  L. Chen

Equation-free dynamic renormalization in a glassy compaction model
L. Chen1, I.G. Kevrekidis1,2 and P.G. Kevrekidis3
1Department of Chemical Engineering and 2PACM,
Princeton University, 6 Olden Str. Princeton, NJ 08544 USA,
3Department of Mathematics and Statistics, University of Massachusetts, Amherst MA 01003-4515, USA
Combiningdynamicrenormalizationwithequation-freecomputationaltools,westudytheappar-
ently asymptotically self-similar evolution of void distribution dynamics in the diffusion-deposition
5
problemproposedbyStinchcombeandDepken[Phys. Rev. Lett. 88,125701(2002)]. Weillustrate
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fixedpoint and dynamicapproaches, forward as well as backward in time.
0
2
PACSnumbers: 05.10.-a,05.10.Cc,81.05.Kf,05.70.Fh
n
a
J
The study of glasses is an important topic under in- approach targets systems for which a fine scale, atom-

7 tense investigation for several decades now (see e.g., the istic/stochastic simulator is available, but for which no

review of [1]). Experimental studies in granular com- closed form coarse-grained, macroscopic evolution equa-
]
t paction (such as the ones in [2]) have offered consid- tion has been derived. The main idea is to substitute
f
o erable insights on the behavior and temporal dynamics function and derivative evaluations of the unavailable
s of glass forming systems. From the theoretical point of coarse-grainedevolutionequationwithshortburstsofap-
.
t view, a variety of modeling/computational approaches propriatelyinitializedfine scalecomputation. The quan-
a
m havebeen usedto understandbetter the keyobservables tities required for traditional continuum numerical anal-
and their evolution for this slow relaxationaltype of dy- ysis are thus estimated on demand from brief computa-
-
d namics. While direct (molecular dynamics) simulations tionalexperimentationwiththefinescalesolver. Wewill
n are widely used, perhaps more popular are kinetic ap- demonstrateequation-freefixedpointapproachestocom-
o
proachesincluding e.g.,the Fredrickson-Andersonmodel puting the self-similar shape and dynamics of the void
c
[ [3] and more recent variants thereof [4]. Another com- cumulative distribution function.
mon approach involves the use of mode coupling theory We start with a brief description of the model and
2
[5] examining the time evolution of the decay of density key (for our purposes) results of [8]. We then perform
v
fluctuations to separate liquid from glassy (non-ergodic equation-free dynamic renormalizationcomputations for
3
7 due to structural arrest) dynamics. More recently, the this problem. The coarse-graineddescription of the sys-
7 energy landscape of glass forming systems and the role tem evolution backwardin time is examined through re-
2 of its “topography” has become the focus of numerous verse coarse projective integration. We conclude with a
1
works [6]. brief discussion of potential extensions of the approach.
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The model consideredby Stinchcombe and Depken[8]
0 In a previous paper [7] we proposed a simple com-
/ paction “thought experiment” for hard spheres: the in- consists of unit-size grains interacting throughhard-core
t
a potentials and performing a Monte Carlo random walk.
sertionofahardsphereinagasofhardspheres(accepted
m While some of their results apply in any dimension, in
when the sphere does not overlap with previously exist-
- ing ones). Combining simple thermodynamic arguments this paper we will work in one spatial dimension; this is
d
forconvenience–ourapproachisnotlimitedto1d. While
n with equilibrium distribution results we argued that the
the grains diffuse on the line, as soon as a sufficiently
o evolution of the hard sphere density should be logarith-
c mic in time (as the maximal density is approached). large void is formed, it is instantaneously filled by an
: additional grain. We will work with a system of finite
v
In a recent paper [8] Stinchcombe and Depken pre-
i size andperiodic boundaryconditions; the systemsize L
X sented an interesting diffusion-deposition model exhibit- 5
(here of the order of 10 ) is large compared to the grain
r ing“glassy”compactionkinetics(seealso[9]forasimilar size, which is taken to be 1. If ρ is the system density,
a model example). The model provides a useful paradigm
δ = (1−ρ)/ρ, is the average void size. One of the main
for testing the hypothesis of self-similar evolution of the
findings in [8] is that the void density ǫ = 1−ρ goes to
system statistics –in particular, of the density and the
zero as an inverse logarithm
void distribution function– based on direct simulations.
1
The techniques that we will use to test this hy- hǫ(t)i∼ , (1)
ln(t)
pothesis combine template-based dynamic renormaliza-
tion [10] with the so-called “equation-free” approach to asymptoticallyforlarget. Asimilarasymptoticbehavior
complex/multi-scale system modeling [11, 12]. Dynamic follows for δ.
renormalization has been developed in the context of Density provides a cumulative measure of the system
partial differential equations with self-similar solutions evolution(itisthezerothmomentoftheparticledistribu-
(e.g. blowups in finite time) [13]. The equation-free tionfunction). Wesetouttostudyindetailtheevolution
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20 to
1−ρ 1
18
δ ≡ ∼ . (2)
ρ Aln(t+t0)+B
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Replotting the data in view of the above relation col-
14 lapses the Fig. 1 curves onto a single line; the role of t0
ρ1−) 1 willbe discussedbelow. This implies thatthe dynamical
ρ/(12 0.8ρ, t evolutionofδ eventuallyfollowsthedifferentialequation:
1
F0.6
10 CD0.4 0.5
dδ 1
8 0.2 00 0.5 1 ∼−δ2exp − . (3)
0 dt (cid:18) Aδ(cid:19)
0 0.2 0.4 0.6
x
6
(and hence a similar equation can easily be obtained for
100 102 104 106 108
t ǫ). The dynamics of (3) share the asymptotic behavior
of Eq. (1).
Wenowtesttheself-similarityhypothesisfortheCDF
FIG. 1: Dynamic evolution of the model for different initial-
izations. Dotted (resp. dash-dotted) lines: starting with the evolution. If the dynamics of the CDF are self-similar
self-similar (resp. uniform) void CDF. Solid (resp. dashed) and stable, all initial distributions will asymptotically
lines: evolution replotted in terms of an appropriate “initial approach the same self-similar shape (modulo rescaling)
time” t0; notice the collapse of the curves on a straight line, while they densify and slow down. Our goal is to find
seetext. TheinsetshowsvoidCDFsatdifferenttime/density
the self-similar shape at a convenient scale – a relatively
instances(color-codedasinthemainFigure);theyallcollapse
low density, when the evolution dynamics are still fast.
upon rescaling to a single CDF (smaller inset).
Our main assumption is that the void CDF is a good
observable for the system dynamics – i.e., that a macro-
scopic evolution equation (possibly averaged over sev-
Consistent Microscopic ICs Microscopic
Timestepper eral experiments) conceptually exists for its dynamics,
M RESTRICT even though it is not available in closed form. Starting
µ LIFT void CDF @ t=τ with a given void CDF0 with δ = δ0 we lift to an en-
Macroscopic Description RESCALE semble of microscopic realizations of it – that is, grain
(void CDF @ t=0) configurations possessing the given void CDF. This
void CDF @ t=τ
“lifting” is accomplished here by randomly selecting the
FIG. 2: Schematic of coarse dynamic renormalization. voids from CDF0 as grains are “put down” on the line;
δ0 indirectly selects the scale at which we will perform
our computation. We then evolve each of these real-
izations based on true system stochastic dynamics for
of the void cumulative distribution function, CDF(x,t). a time horizon τ. Finally, we obtain the ensemble av-
This function is defined between 0 and 1 in the void size eraged void CDF(x,τ) and its δ(τ) = δ1. This is the
x,sinceatnotimedowegetvoidslargerthan1(themo- restriction stepofequation-freecomputation: evaluating
menttheyappear,theygetfilled). Thedottedlines(and the macroscopic observables of detailed, fine scale com-
the dash-dot line) in Fig. 1 show the evolution of δ(t) putations. We now rescale our macroscopic observable
for various initial void distributions (the choice of initial (the CDF(x,τ)), using the ensemble average δ1, to the
void distributions for these transients will be discussed original δ0: C\DF(δ0x/δ1,τ)=CDF(x,τ). (Clearly this
below). Thestochasticsimulationuses100,000particles; \
requires the largest void in the rescaled CDF to be less
sequential update is used, with a maximum spatial step
than 1 – a condition that one may expect to prevail at
sizeof0.5. Thesemilogplotclearlyindicatestheasymp-
high enough densities). We then discard the simulations
totic logarithmic regime in time. The inset shows the
weperformed,andstartanewensembleofsimulationsat
shape of the void CDF for various simulations and vari- \
the original density, but with the more “mature” CDF.
ous instances in time, within the logarithmic regime; all
The map from current void CDF to future void CDF
shapeswhenappropriatelyrescaledwiththeaveragevoid
is the “coarse timestepper” of the unavailable equation
size collapse ona single curve (smaller inset). These two
for the macroscopic observable evolution; the composi-
observations (logarithmic time dependence, and collapse
tion of this map with the rescaling step constitutes the
of the distributions upon rescaling) clearly suggest the
“renormalized coarse timestepper”, the main tool of our
possibility of self-similar evolution dynamics.
dynamic renormalization procedure, schematically sum-
In particular, the graph of the time evolution of the marized in Fig. 2. Given this map, several approaches
densitiessuggeststhatδbecomeseventuallyproportional to the computation of the long-term coarse self-similar
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dynamics exist. The simplest is successive substitution fixedpointproblem. Fig. 3bshowsthe iteratesofsucha
–werepeatcoarserenormalizedtime-steppingagainand matrix-free fixed point computation for Eq. (4) through
again, and observe the approach of the void CDF to its a Newton-Krylov GMRES algorithm applied to a 100-
self-similar shape at the scale we chose (parametrizedby point uniform finite difference discretization of the void
δ0). The fixed point of this procedure lies on the group CDF in x; our initial guess was a uniform void CDF
orbitofscaleinvariancefortheCDF dynamics;itwasse- (δ0 = 0.2), and we converge to the self-similar shape
lected through a “pinning” or “template” condition (our within a few iterations; the inset shows the evolution
choice of δ0). This dynamic renormalization iteration, of the norm of the residual vector with iteration num-
with τ = 4 is shown in Fig. 3a, starting from a uni- ber. Lifting from this coarse description (100 numbers)
form voiddistribution with δ0 =0.2; the inset shows the to realizations of the CDF (200,000 particles) involved
third iteration before (dotted line) and after (solid line) linear interpolation of the CDF. Relatively short runs
rescaling. (τ = 1) were used to construct the coarse renormalized
timestepper in Fig. 3b; its fixed point is independent of
1 1 the reporting horizon τ.
Wrapping traditional numerical algorithms around
n n
0.8 0.8 coarsetimesteppers enablesseveralcomputationaltasks,
1 of which the fixed point computation above is only one
100
0.6 0.8 0.6 example. Another interesting example is the so-called
F F
CD 0.6 CD 10−1 coarse projective integration; here, short runs of the
0.4 CDF 0.4 R stochastic simulation are used to estimate the local time
0.4
10−2 derivatives of the macroscopic observables (e.g. of the
0.2
0.2 0.2 void CDF); these estimates are used to “project” the
00 0.5 1 10−30 2 4 6 voidCDF “far”intothe future. The simplestsuchalgo-
x n
0 0 rithm is projective forward Euler; see [11] for multi-step
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
x x and implicit coarse projective integration. One does not
evolve the microscopic problem; one evolves the coarse-
FIG. 3: a. Successive evolution of an initially uniform CDF grainedclosureofthemicroscopicproblem,whichispos-
profile with renormalization: the inset shows the profile at
tulated to exist, but is not explicitly available. Here we
the n=3iteration before (dashed) and after (solid) rescaling.
apply coarse projective integration backward in time (as
b. Newton-Krylov-GMRES fixed point renormalization cal-
discussedin[15]thisisappropriateforsystemswithlarge
culation,startingfromuniformvoidCDF;theinsetshowsthe
decrease of the norm of the residual vector at each Newton time scale separation between fast stable dynamics and
step. slow dynamics). Starting with the self-similar CDF at
some (relatively large)density (see Fig. 4 and its insets)
Successive substitution will converge to stable self- we lift to microscopic realizations, evolve briefly forward
similar solutions. It is also, however, possible, to use in time, and estimate the rate of void CDF evolution.
fixed point algorithms to convergeto the self-similarvoid This rate (estimated from the short pink run) is used
CDF at δ0. Ineffect,wewishtosolveafunctionalequa- toprojectthe voidCDF fora(longer)intervalbackward
tion for the fixed point of the renormalized timestepper: in time (in blue). The forward simulations are then dis-
carded, and new simulations are initialized at the “ear-
\ lier” void CDF. Repeating the procedure clearly shows
CDF(x,τ)−CDF(x,0)=0. (4)
anaccelerationinbackwardtime asthe density becomes
Fixed point algorithms (like Newton’s method) require lower;theprobabilityoflargevoidsbecomesincreasingly
the repeated solution of systems of linear equations in- larger, the backward dynamics evolves faster and faster
volving the Jacobian matrix of some discretization of and their fluctuations intensify, making rate estimation
Eq. (4). Since we have no closed form equations for fromtheshortforwardrunsdifficult. Fig. 4seemstosug-
the void CDF evolution, this Jacobian is unavailable; gesta“reversefinitetime”ratesingularity. Thisisonlya
yetequation-freemethodsofiterativelinearalgebra(like visualsuggestion,however;asthedensitybecomeslower,
GMRES [14]) allow the solution of the problem through wedonotexpectanevolutionequationforthevoidCDF
a sequence of matrix-vector product estimations. In our to provide a good model. More detailed physical mod-
casetheseestimationsareobtainedthroughthe“lift-run- eling of the void-filling process will be required for the
restrict-rescale” protocol performed at appropriately se- study of this backward dynamics at lower densities to
lected nearby initial distributions (for details, see [11]). become meaningful.
Short bursts of stochastic simulation from nearby initial It is this “visual suggestion” of a backward explosion
voiddistributions allowsus to estimate the action of the of the density evolution rate that suggested fitting the
Jacobian on selected perturbations, and hence the solu- data as a function of t+t0; we find that, if we initialize
tion of linear equations and ultimately, of the nonlinear with the self-similar CDF (at whatever density), evolu-
4
20 evolution in time) or in renormalized space (renormal-
1
ized void CDF evolution in logarithmic time). We be-
18 lievethatthebridgingofcontinuumnumericaltechniques
DF0.5
C with microscopic simulations we illustrated here may be
16 helpful in the coarse grained study of glassy dynamics
0
0 0.2 0.4 throughatomistic/stochasticmodels,especiallywhenthe
x
ρ)14 dynamicsdependonparametersofthemicroscopicrules.
−
ρ/(112 15.4
15.2
Acknowledgments IGK acknowledges the support of a
10 ρρ/(1−)141.58 NSF-ITRgrant. PGKacknowledgesthesupportofNSF-
DMS-0204585, NSF-CAREER and the Eppley Founda-
14.6
8 tion for Research.
14.4
−0.5 0 0.5 −1.5 −1
t × 104
6
−5 −4 −3 −2 −1 0 1
t x 104
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