Table Of ContentUsing the s-Ensemble to Probe Glasses Formed by Cooling and Aging
Aaron S. Keys1,2, David Chandler1 and Juan P. Garrahan3∗
1Department of Chemistry, University of California, Berkeley CA, 94720
2Lawrence Berkeley National Laboratory, Berkeley CA, 94720 and
3School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, United Kingdom
(Dated: April 1, 2014)
Fromlengthscaledistributionscharacterizingfrozenamorphousdomains,werelatethes-ensemble
method with standard cooling and aging protocols for forming glass. We show that in a general
class of models, where space-time scaling is in harmony with that of experiment, the domain size
4 distributionsobtainedwiththes-ensembleareidenticaltothoseobtainedthroughcoolingoraging,
1 but the computational effort for applying the s-ensemble is generally many orders of magnitude
0 smaller than that of straightforward numerical simulation of cooling or aging.
2
r
a Through biasing statistics of trajectory space, the so- Here, we take ∆t to be 103 integration steps, which is
M
called“s-ensemble”method,non-equilibriumphasetran- roughly the average time to complete an enduring dis-
0 sitions emerge between ergodic liquid-like states and dy- placement of one atomic diameter [8], and [···]iso indi-
namically inactive glass-like states. This class of transi- catesiso-configurationalaveraging,whichaveragesmany
3
tions are found in idealized lattice models [1, 2] and in trajectories of length ∆t, all starting from the same con-
] simulations of atomistic models [3–5]. In the latter case, figuration [14]. Particles colored red in Fig. 1a are those
h
it affords a systematic computational means of prepar- for which the iso-configuration averaged ¯r (∆t) is more
c i
e ing exceptionally stable glass-states states [6]. This Let- than 0.6σ from ¯ri(0), where σ is a particle diameter.
m ter draws the conclusion that the s-ensemble transition Smaller enduring displacements are colored by interpo-
- coincides with the physical glass transition [7], and the lating between red and blue, as noted in the color scale.
at ensemble of its inactive states are those of natural struc- The figure thus shows that enduring displacements oc-
t tural glass. Specifically, we derive correspondence be- curring over a short period of time, ∆t, take place in
s
. tween spatial correlations in the s-ensemble glass with sparse localized regions of space. These regions are the
t
a those in the glass produced with finite-rate cooling or excitations [8] in an otherwise rigid material.
m
aging. The correspondence provides a basis for an ex-
- traordinarilyefficientrouteforpreparingstructuralglass Similar pictures illustrating arrangements of enduring
d with molecular simulations. displacements are found without iso-configuration aver-
n
aging. See Fig. 1 of Ref. [8] and also Media 1 of Ref. [8].
o
c Space-time structure of glass-forming liquids. To Iso-configurationalaveragingservestoburnishthosepic-
[ begin, it is helpful to consider Figs. 1a and 1b, which tures [15]. Without that averaging, localized excitations
2 render trajectories of a two-dimensional 5 104-particle are already evident but more irregular. Importantly, the
v systeminafashionthatextendstheapproa×chofRef.[8]. excitations, often referred to as soft spots [16], change
6 The system is a liquid mixture at a temperature that is littleinsizeastheliquidiscooled,andfurther,thereare
0 80% below that of the onset temperature, T [8–12], and no inter-excitation correlations at equal times [8].
o
2
the trajectory runs for an observation time t 10τ.
7 obs ≈ Correlations develop over time through dynamics,
Here, τ stands for the equilibrium structural relaxation
.
1 time. It is about 105 integration steps at this particular which is illustrated with Fig. 1b. That picture shows
0 constant-value surfaces of [a(r,t)] . The surfaces form
temperature, and t , being 10 times longer, provides iso
4 obs connected tubes or lines in space time – excitation lines
1 ampleopportunitytoobservethenatureofdynamichet-
[17] – indicating that excitations facilitate birth (and
: erogeneity in the system.
v death) of adjacent excitations. Larger amplitude fluc-
Mostmotionsinglass-formingliquidsareirrelevantvi-
Xi brations, and the amplitudes of most of those vibrations tuations of the surface occur less frequently than smaller
amplitudefluctuations,indicatingthatthefacilitateddy-
r are similar in size to typical enduring displacements [8].
a namics is hierarchical [18]. Lowering temperature re-
Irrelevant vibrations can be filtered out by focusing on
duces the number of excitations or soft spots, which re-
inherent structures [13]. The set of particle positions at
duces the probability that soft spots can connect, which
time t, r (t) , evolves by molecular dynamics; the in-
i
{ } reduces the rate at which the system can relax.
herent structure, ¯r (t) , is the position of the potential-
i
{ }
energy minimum closest to r (t) . The renderings in
{ i } Thisbehaviorisfoundconsistentlyinglass-formingliq-
Fig. 1 refer to [a(r,t)] , where
iso uidsforalltemperaturesbelowtheonset,i.e.,T <T [8].
o
Throughout this regime, it is characterized by simple
N
a(r,t)=(cid:88) ¯r (t+∆t) ¯r (t) δ(r ¯r (t)). (1) equations for space-time scaling and for the equilibrium
i i i
| − | − distribution of distances between neighboring soft spots,
i=1
2
a b c
- d=2, N=5x104 d=3, N=106
∆ ri / σ 1.3
1.1
≥ 0.6 1.2 F(r)1.0
0.5 1.1
0.4 Z(r) 5 r /1 0σ 15 20
1.0
0.3
0.2 0.9 average A
e low 0.1%
0.1 c A
a 0.8
0.0 ps time 10τ 5 10 15 20
r / σ
FIG.1. Softspotsandspace-timebubbles. (a)Burnishedexcitationsforthed=2supercooledWCAliquidmixture[8],with
∆r¯ =[|¯r (∆t)−¯r (0)|] . (b) Excitation lines for a subsection of the system shown in (a). (c) The functions Z(r) and F(r)
i i i iso
demonstrating correlation holes for inactive subsystems of a supercooled d=3 WCA liquid mixture [21].
P ((cid:96)): Z(r) = a(r) /a. Here, a is the equilibrium aver-
eq ∆v
(cid:104) (cid:105)
age of a(r,t), and is that average conditioned
∆v
(cid:104)···(cid:105)
(cid:96)/σ =(τ /τ )1/β˜γ. (2) on a low activity in the sub-volume at the origin. Sim-
(cid:96) o
ilarly, we have computed the radial distribution of mo-
and bility, F(r)= a(r) 0/ag(r), whereg(r)N/V isthemean
(cid:104) (cid:105)
particle density at r given a particle is at the origin and
Peq((cid:96))=(cid:96)−eq1exp(−(cid:96)/(cid:96)eq), (cid:96)eq =σexp(β˜/df), (3) (cid:104)···(cid:105)0 is the equilibrium average given a particle at the
originhasjustthencompletedanenduringdisplacement
Here,β˜=J /T J /T ,whereJ istheenergyofanex- of at least 0.3σ. The red lines of Fig. 1c refer to the
σ σ o σ
citationwithend−uringdisplacementsofthecharacteristic 0.1% least active sub-volumes. For F(r), that means the
structurallength, σ, 1/τ istherelaxationrateonlength central displacing particle is within such a low-activity
(cid:96)
scale(cid:96),d isthefractaldimensionalityofdynamichetero- sub-volume. The equilibrium Z(r) exhibits no structure,
f
geneity,andγistheproportionalityconstantforlogarith- and the equilibrium F(r) decays over the length scale of
mic growth of excitation energy with respect to length a single excitation. In contrast, the atypical low-activity
scale [19]. The parabolic law [8–10], τ =τ exp(β˜2γ/d ), functionsshowsignificantanti-correlationbetweenneigh-
o f
follows from space-time scaling, Eq. (2), evaluated at boring excitations.
(cid:96)=(cid:96) .
eq
Preparing glassy states. The statistical weight for
Contributions to P ((cid:96)) with short inter-excitation
eq
these low-activity regions are enhanced by shifting to
lengths, (cid:96)<(cid:96) , come from regions with excitation lines
eq
a non-equilibrium s-ensemble distribution, P [x(t)]
thatconnectandreorganize. Contributionswith(cid:96) (cid:96)eq s ∝
(cid:29) P [x(t)]exp( s [x(t)]) [2–4, 22]. Here, P [x(t)] is the
come from regions of rigidity – the empty regions of 0 − A 0
equilibrium distribution functional for trajectories x(t)
Fig. 1b, so-called “bubbles” in space-time [17]. When
of length t , and [x(t)] is the net dynamical activity
the liquid transforms into glass, the temporal extents of obs A
[23],
those bubbles grow to very long times, and excitation
lines rarely or never touch, yielding a striped structure (cid:90) (cid:90) tobs
of trajectory space [20]. In that case, excitations are no [x(t)]= dr dt a(r,t). (4)
A
longer uncorrelated, and a non-equilibrium correlation V 0
length, (cid:96)ne, gives the average or most probable separa- For an ergodic equilibrium system, = tobsVa. De-
A
tion of excitation lines. viations from this equilibrium value are measures of
Figure1cshowsthattheequilibriumglass-formingliq- non-ergodic non-equilibrium behavior. Time integrals of
uidalreadycontainstheseedsofthisnon-equilibriumcor- other quantities, not just the activity as in (4), can also
relation length. Specifically, for a 106-particle WCA liq- serve as suitable order parameters to distinguish ergodic
uid mixture [21] in d=3 at T =0.7T , Fig. 1c contrasts and non-ergodic behavior. Time integration is the cru-
o
the equilibrium concentration of excitations with that cial feature. Fluctuations are then intimately related to
surroundingadynamicallyinactivesub-volume. Thetra- the behavior of time correlators.
jectory length is t = 50τ 103∆t. The net dynami- Remarkably,forsystemsatT <T ,themarginalequi-
obs o
cal activity in a sub-volume≈∆v is (cid:82)tobsdt(cid:82) dra(r,t). librium distribution for exhibits fat tails at low activ-
0 ∆v A
For Fig. 1c, we have partitioned the total volume V ity [1], so that the non-equilibrium mean, , changes
s
(cid:104)A(cid:105)
into cubes, each of size ∆v = 125V/N, and computed abruptly around a transition value of s. For s < s∗,
3
the material is a normal melt, and for s>s∗, the mate- t , the system can relax domains that are smaller than
age
rialisaninactiveamorphousphase–aglass. Theabrupt a characteristic length [20, 27] (cid:96) = (t /τ )1/β˜γ, with
ne age o
changetendstoadiscontinuityasNtobs →∞. Theglass β˜ 3. We use tage 106 so as to produce an average
transition in the s-ensemble is thus a first-order transi- ≈ ≈
non-equilibriumspacingbetweenexcitationsofabout10.
tion [2–4].
In the second, cooling, the model is equilibrated at a
Even more remarkably, the transition can be obtained
temperature T =1 and then cooled to zero temperature
with tobs much shorter than time scales required to pro- at a rate of ν = 10−5. A glass transition occurs at the
duce glass from standard cooling protocols. By cooling
stage where 1/ν dτ/dT , which gives T 0.48 and
g
at a rate ν, a glass transition occurs at the temperature ≈ | | ≈
thus (cid:96) 10 and τ 106. In other words, excitations
T , where ν−1 dτ/dT . The time scale for that ne ≈ g ≈
g ≈ | |T=Tg in the glass are frozen in with a typical spacing of about
process is τ = τ(T ). The transition freezes excitations
g g 10,andthetimescaletocreatethematerialisabout106.
separated by the non-equilibrium length, (cid:96) = (cid:96) (T ).
ne eq g The third case, the s-ensemble protocol, produces a
FromEq.(2), (cid:96)ne/σ =(τg/τo)1/β˜gγ. Thislengthmustbe similar glass in a much shorter time. A similar inter-
largeiftheglasspersistsforlongtimes. Thus, inviewof excitation distance is targeted with trajectories run at
Eq. (3), β˜g >1. Typically, τg >1010τo and (cid:96)ne (cid:38)10σ. T = 0.72 for which τ 300 and β˜ 0.3. The glass
Ontheotherhand,withthes-ensemble,thesamelarge transition from the co≈oling protocol≈occurs at β˜
g
non-equilibrium length can be obtained with any pos- 1/(1/2) 1 = 1. Accordingly, from Eq. (5), the s≈-
itive value of β˜. In that case, from Eq. (2), (cid:96)ne/σ = ensemble−transitionfortrajectoriesatT =0.72produces
(tobs/τo)1/β˜γ. As such, the glass with (cid:96)ne when tobs 100. To apply the s-
≈
ensemble,weusethetotalnumberofenduringkinksasa
tobs/τo =(τg/τo)β˜/β˜g. (5) measure of dynamical activity. An enduring kink at site
i is a change in n that persists for at least a mean ex-
The ratio β˜/β˜ can be much smaller than 1. In practice, i
g changetime[26]. AtT =0.72andt 100,forsystem
obs
β˜/β˜g 1/10. Thus, the simulation time required to size N chosen, the s-ensemble glass tra≈nsition occurs at
≈
prepare a glassy state in the s-ensemble, tobs, is many s∗ 10−2 =O(1/N) [33] (see Supplemental Material).
orders of magnitude shorter than the time to prepare F≈igure 3 compares the non-equilibrium correlation
glass by straightforward cooling, τg. lengths and distribution functions for the three differ-
ent preparation protocols. It also shows Z((cid:96)), which is
Illustration with the East model. Equation 5 fol- the relative concentration of enduring kinks a distance (cid:96)
lows from well-tested scaling relationships, and there is from the 0.1% least-active domains of the equilibrium
some empirical evidence that glasses produced with the East model. It exhibits a correlation hole in a fash-
s-ensemble do indeed coincide with natural structural ion similar to the analogous Z(r) in the WCA mixture,
glass [6, 24]. Nevertheless, this relationship is not yet Fig. 1. The East model thus illustrates how preparation
tested explicitly. Here, we do so for the East model [25], of glass, which necessarily requires long physical times,
thesimplestofmodelsconsistentwithphenomenologyof canbeaccomplishedinsimulationinmuchshortertimes
structural glasses and glass formers [10, 20, 26]. throughapplicationofthes-ensemble. Equation(5)pro-
In brief, the East model consists of a d = 1 lattice videsthekeyforunderstandingpriorsuccessesinprepar-
with N sites, each with variables ni = 0,1. The equi- ing glassy states through applications of the s-ensemble
librium concentration of excitations is ni = c. At low inatomisticmodels. However, ifthesimulationboxsize,
(cid:104) (cid:105)
temperatures, c exp( 1/T). (We take 1 as the energy L, is smaller than target non-equilibrium length, the s-
∼ −
scale and length scale for the model.) Sites with ni = 1 ensemblemethodpreparesadistributionofglassystates,
can facilitate a spin flip at the adjacent site ni+1. The all of which correspond to inactive domains in glasses
corresponding transition rates are given for a site i by with (cid:96) >L.
ne
ki,0→1 = ni−1c/(1 c) and ki,1→0 = ni−1. The dynam- Natural dynamics changes (cid:96) continuously, and at the
−
ics of this model is hierarchical [25, 27]. Its structural pointwherethesystemfallsoutofequilibrium(cid:96) =(cid:96) .
ne eq
relaxation slows by twelve orders of magnitude as T de- The equilibrium length, (cid:96) = (cid:96) , is equivalent to a or
eq
creases from 1 to 0.2 [28], it obeys space-time scaling c. Because the length scale ch(cid:104)an(cid:105)ges continuously as a
of Eq. (2) and the parabolic law with γ 1/2ln2 [28– glass former falls out of equilibrium, the glass transition
≈
30]. (A different value of γ applies for aging regimes has the appearance of a second-order transition. How-
[31].) Anequilibriumtrajectoryofthemodelisshownin ever, the time-integrated order parameters, the distribu-
Fig. 2a. Three protocols for preparing non-equilibrium tion of (cid:96), and the connection between (cid:96) and c, all change
glass states are illustrated in Figs. 2b, 2c and 2d. abruptly. The change becomes singular in the limit of
In the first, aging, the model is initially equilibrated infinite time, manifesting the first-order non-equilibrium
at T =1 and then instantly quenched to T =0.25, after transition that underlies the glass transition.
which it runs at that the low temperature for times t ,
age
where τ(1) t τ(0.25) 3 109. During the time Acknowledgements. We thank D.T. Limmer, R.L.
age
(cid:28) (cid:28) ≈ ×
4
a equilibrium b aging c cooling d s-ensemble
100 100 100 100
50 50 50 50
e
c 0 0 0 0
a
p 0 5x105 0 1x106 0 1x105 0 75 150
s
time
FIG.2. TrajectoriesofexcitationsintheEastmodel. (a)EquilibriumdynamicsforT =0.45overatimescalespanningabout
50 structural relaxation times at that temperature. (b) Aging dynamics after a quench from T =1 to T =0.25. (c) Cooling
at a rate ν =10−5. (d) Trajectory from the s-ensemble at T =0.72 and s>s∗ ≈10−2, trajectories running for about 1/2 a
structural relaxation time at that temperature.
Jack, P. Sollich, T. Speck and Y.S. Elmatad for help- provided computational resources.
ful discussions. Salaries were supported by the Director,
OfficeofScience,OfficeofBasicEnergySciences,andby
theDivisionofChemicalSciences, Geosciences, andBio-
sciences of the U.S. Department of Energy at LBNL, by
∗ Correspondingauthor,[email protected]
theLaboratoryDirectedResearchandDevelopmentPro-
[1] M. Merolle, J. Garrahan, and D. Chandler, Proc. Natl.
gram at Lawrence Berkeley National Laboratory under
Acad. Sci. USA 102, 10837 (2005).
Contract No. DE-AC02-05CH11231, and by Leverhulme
a b c [2] J.P.Garrahan,R.L.Jack,V.Lecomte,E.Pitard,K.van
- d=2, N=5x104 Trust grant no.dF=/30, N01=11406/BG. NSF award CHE-1048789 Duijvendijk, and F. van Wijland, Phys. Rev. Lett. 98,
∆ ri / σ 1.3 195702 (2007).
≥ 0.6 1.2 F(r)11..01 [3] Ldl.eOr,.SHcieedngcees,3R23.,L1.3J0a9ck(,2J0.09P)..Garrahan, andD.Chan-
0.5 1.1a b [4] T. Speck and D. Chandler, J. Chem. Phys. 136, 184509
0.4 Z(r)1 5 r /1 0σ 15 20 cooling (2012).
1.0 s-ensemble
00..32 Z(�) 0.9 eavqeurilaibgreiu Am P(�) aging [5] T10.9S,p1e9ck57,0A3.(M20a1li2n)s., andC.P.Royall,Phys.Rev.Lett.
e 0.9 cloowo l0in.1g% [6] R.L.Jack,L.O.Hedges,J.P.Garrahan, andD.Chan-
0.1 c A
a 0.8 dler, Phys. Rev. Lett. 107, 275702 (2011).
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on different theoretical approaches see, e.g., M. Ediger,
25 25 25
c aging d cooling e s-ensemble C. Angell, and S. Nagel, J. Phys. Chem. 100, 13200
20 20 20 (1996); V. Lubchenko and P. G. Wolynes, Annu. Rev.
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` 10 T=0.2 10 10 T=0.5 476, 51 (2009); D. Chandler and J. P. Garrahan, Annu.
Rev.Phys.Chem.61,191(2010);K.BinderandW.Kob,
5 5 5
Glassy materials and disordered solids (World Scientific,
0 0 0 2011); L. Berthier and G. Biroli, Rev. Mod. Phys. 83,
11010 0 1 1001020 1 0 010100e04 +10 e01+150e60 +16 1e010+70800180 0 1 00100104e 0 + 1 0 e150+160 e 6 + 10 1e70+180811001 110020 1 10003 0 1 1000400
t ν -1 t 587(2011);G.BiroliandJ.P.Garrahan,J.Chem.Phys.
age obs
138, 12A301 (2013).
FIG.3. DistributionsofEast-modelglasses. (a)Therelative [8] A.S.Keys,L.O.Hedges,J.P.Garrahan,S.C.Glotzer,
concentrationofexcitationsadistance(cid:96)fromoneoftheleast and D. Chandler, Phys. Rev. X 1 (2011).
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T =0.5(b)DistributionsofdomainlengthsP((cid:96))forsystems 100, 9710 (2003).
aged, cooled and driven with s, all designed to yield (cid:96) ≈ [10] Y. S. Elmatad, D. Chandler, and J. P. Garrahan, J.
ne
10. Here,(cid:96)representsthedistancebetweenfrozenexcitations Phys. Chem. B 113, 5563 (2009).
(called “super-spins”[27]). Aging was done at T = 0.25 for [11] Y. S. Elmatad, D. Chandler, and J. P. Garrahan, J.
t = 2.5×105 after quenching from T = 0.4. Cooling to Phys. Chem. B 114, 17113 (2010).
age
T = 0 was done with a cooling rate of ν = 10−6. s-enemble [12] To is the crossover temperature below which dynamics
trajectories were carried out at T =0.72 for a time duration is spatially heterogeneous. Below that temperature, the
of t = 320. (c-e) Growth of (cid:96) as a function of relevant structural relaxation time is super-Arrhenius, obeying
obs ne
timevariables. Dashedlinesin(c)referto(cid:96) =(t /τ )1/β˜γ; theparaboliclaw[9,10],andabovethattemperature,the
ne age o structuralrelaxationtimeisArrheniusorsub-Arrhenius.
dashed line in (d) refers to (cid:96) (T ); dashed lines in (e) refer
eq g Reversibletransportpropertiesatverylowtemperatures
to (cid:96) =(t /τ )1/β˜γ.
ne obs o canbepredictedquantitativelybyextrapolationfrombe-
5
havior near but below T [8, 11]. Thus, we see is no ev- θ(···)istheunitHeavisidefunction,thesumoniisover
o
idence for another dynamical crossover in reversible su- all particles, and the sum on t is over time intervals of
percooled glass forming liquids. width ∆t extending from 0 to t .
obs
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obscure structure, especially for short trajectories. Bur- [28] D.J.Ashton,L.O.Hedges, andJ.P.Garrahan,J.Stat.
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eq
Anderson, Phys. Rev. Lett. 53, 958 (1984). relevant for aging protocols) γ = 1/ln2 [32]. For longer
[19] For d = 1, d = 1. For dimensions d = 2 and 3, nu- (cid:96), γ is reduced by entropic effects to a limiting value
f
merics[8]findsd ≈1.8and2.4,respectively,andγ isof 1/2ln2 [30].
f
orderone,withaspecificvaluethatissystemdependent. [32] P. Sollich and M. R. Evans, Phys. Rev. Lett. 83, 3238
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Stat. Phys. 127, 51 (2007). [35] G. M. Torrie and J. P. Valleau, J. Comp. Phys. 23, 187
[23] There is significant freedom in the specific choice of dy- (1977).
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isA[x(t)]=(cid:80) (cid:80) θ(|¯r (t+∆t)−¯r (t)|−σ),whereσ is (2008).
i t i i
a particle diameter or a fraction of a particle diameter,
Supplemental Material
Aspects of the s-ensemble results are detailed in Fig. 4. The s-ensemble is sampled according to the methods
outlined in Refs. [4, 34]. We use standard transition path sampling with both shooting and shifting moves to sample
trajectory space. The s-ensemble at each state point is sampled within 20 simulation windows, w, each with a
differenttargetvalueofactivity, . Trajectoriesareacceptedorrejectedaccordingtoastandardumbrellasampling
w
criterion[35], withaharmonicbiAasingpotentialactingontheactivityforeachwindow, W = k( [X] )2, where
w
A −A
[X] is the total number of enduring kinks for the trajectory X. At the state points considered, optimal sampling
A
is obtained for k 105. Replica exchange between windows is implemented to facilitate the sampling of glassy
≈
states with low activity, which are inherently slowly-evolving. Un-biased statistical averages are obtained using the
multi-state Bennet acceptance ratio method [36].
Whereas aging tends to eliminate short domains because larger domains are kinetically frozen, the s-ensemble
eliminates short domains because of the statistical penalty imposed by the field s. In both cases, domains that are
shorter than (cid:96) relax on average while larger domains remain intact. For aging, (cid:96) t1/β˜γ, where β˜ coincides
ne ne age
∼
with the temperature of the quench. For the s-ensemble, (cid:96) t1/β˜γ, where β˜ coincides with the temperature of the
ne ∼ obs
s-ensemble trajectories.
The probability density of intensive activity, A [X]/Nt , is plotted in Fig. 4(a) as a function of t for s=0.
obs obs
≡A
For all t , the activity distribution is non-Gaussian and exhibits a fat tail for low values of activity. This is the
obs
signature of a low-activity phase that can be accessed by driving the system with s. Fig. 4(b) shows that, for s
exceeding a critical value s∗ (i.e., the value of s that maximizes dA/ds), the system undergoes a phase transition into
this inactive state. The value of s∗ tends to zero as t , as [X] grows extensively with time. The sharpness
obs
of the transition is quantified by a susceptibility χ(s) →dA∞/ds =A(cid:10)A2(cid:11) A 2, plotted in Fig. 4(c). The length (cid:96)
ne
≡ −(cid:104) (cid:105)
exceeds the system size N in the limit t , and the system forms a single domain of length N. (The final spin
obs
→∞
cannot be eliminated due to the boundary conditions and facilitation rules.) Activity fluctuations at s=s∗ therefore
scale proportionally with Nt , as illustrated in the inset of Fig. 4(c). This scaling is the hallmark of a first-order
obs
dynamical phase transition at s = s∗. If (cid:96) exceeds N, the system undergoes a first order transition to an ideal
ne
6
a b c
00 0.0055 0.5 0.55
0.0045 200.088 N =64 0.4 05.45 00..55
ln[P(A)]ln[P(K)]11--11-505505 12625486t000obs = K(s) 0003A x 10 ...000000...000000123000432555234 s* x 1000...00040246000 5 05120 250 86 1 10K(s)000K(s)0 000 (cid:1)...000(s) x 10123 ...000555234 ...000123...432555234 (cid:1)0(s*).205000 5 050 0 1 100000
-2-200 5120 0.0011 tobs x 10-3 0.1 0.11 tobs x 10-3
10240 0.0005 0.0 05.05
2-255 0 0 0 00
0.010 10 .200 20 .300 30 4.00 40 .500 50 .600 6 07.007 -0.005 .00 0 .0 05 0..11 0. 105 0..22 0 . 205 0..33 0 .3 50 0..44-0.0-05.050 .00 0 0 0.0.0 055.0 50 0..11 0 .10 .01.1 505.15
A Kx 103 ss s ss
FIG. 4. First-order dynamical phase transition in the s-ensemble with activity measured in terms of enduring kinks. (a)
Probability of observing a trajectory with intensive activity A as a function of t for the d=1 East model at T =0.72 and
obs
N =64. (b) A as a function of s for the same systems sampled in (a). The value of s∗ is plotted as a function of t in the
obs
inset for different system sizes N. (c) Susceptibility χ(s) as a function of s. The inset shows the peak of the susceptibility,
χ(s∗) as a function of t for different N.
obs
inactive phase; otherwise, the transition is smooth and the system falls into a striped phase.
