6.1 Motivation and definition of the model
As outlined in section 1.4, it is a widespread view (see e. g. [Gol69, B¨as87, Sti95]) that glassy dynamics may be described by a thermally activated motion1of a “state point”
that jumps among the deep energy minima E of a configuration space, that we are referring to as “states”. This space is characterized by a given Density of States (DOS)
(E), a hopping ratew(Ei;Ef)between the states, and a connectivity. In the following we will define and motivate theQuenched Jump Model (QJM) that makes specific assumptions about these points.
A question of crucial importance is, which DOS (E) should be chosen to study aging. Since aging is a phenomenon that is observed for different materials as poly-carbonate, glycerol and spinglasses, and in different quantities as torque and dielectric permeability, it is legitimate to look for generic mechanisms. We will see in section 6.2 that for a finite glass transition temperature to exist in the QJM, the tail of(E)has to be exponential forE!-1. The occurrence of such a tail in real systems can be made plausible by the following argument: The aging processes extend to many orders of magnitude in time, where the system does not find an equilibrium state but explores deeper and deeper valleys in a rugged energy landscape. Thus the long–time dy-namics will be governed only by the deepest energy minima. This suggests a coarse–
grained description to be appropriate where the system performssuper–transitions between only these deepest minima of the energy landscape. In the illustrative fig. 6.1 these corresponds to the (few) minima in the shaded region of the energy axis. From this argument one expects not the microscopic DOSmik(E)to be relevant for the ag-ing dynamics, but the extreme values of this distribution density. As has been stated in chapter 5, the distribution of the minima ofNrandom variables that are distributed according to a given distribution function converges to one of three standard max–
limit distributions forN!1. We have seen in section 5.4.3 that for distributions, that are not too “heavy–tailed”, the extreme values in the limitN! 1will be distributed according to a GUMBEL distribution eq. (5.5c),
(E) = exp(-e-E=T0): Its density is
(E) = T -1
0 e
-E=T
0 exp(-e-E=T0); (6.1)
1The hopping motion can account only for “slow processes”, but is not appropriate for modeling “fast processes” as e. g. therelaxation. We are only concerned about slow dynamics.
E
Figure 6.1: Sketch of a rugged energy landscape of a glas with various metastable minima.
Within a coarse–grained description, the slow dynamics of glasses may be attributed to effec-tive “super–transitions” between the deepest minima belonging to the shaded area.
which asymptotically becomes
(E) T -1
0 e
-E=T
0 for E ! -1: (6.2)
From this general argument one may expect the deepest minima of the state energies to be exponentially distributed forE!-1, and indeed, mean–field theories of spin glasses [Bou97] and recent results from molecular dynamics simulations [Sch00b] sug-gest this to be the case. For simplicity we are using an exponentially decreasing DOS for all energies,
(E) = T -1
0 e
E=T
0 for -1<E0: (6.3)
One has to keep in mind that a real system will exhibit a finite ground–state energy
E
0. If however,E0cannot be reached on experimentally accessible time scales, we can formally perform the limitE0!-1in the theoretical modeling. For convenience we measure all temperatures and energies in units ofT0by introducing a scaled tempera-ture
T
T
0
: (6.4)
We now turn to the question of the transition rate between states. In the QJM we consider the state point to perform a POISSON process with the hopping rate intro-duced in section 3.6,
w(E
i
;E
f ) =
0 exp
-E
f
-(1-)E
i
; (6.5)
where the parameterwith0<1can be interpreted as a weighting factor between the energy of the initial stateEi and the energy of the final state Ef (cf. section 3.6).
6.2 Existence of a stationary distribution — the glassy phase This rate fulfills the detailed balance condition for any value of. Thetrap modelis obtained when setting=0, leading to a rate independent ofEf. For=1=2the rate depends onE=Ef-Eionly, and can be interpreted as a discrete version of a “random force model”. If the dynamics in the space of states is mapped onto a jump motion by means of some quantitative analysis, one would expect that the form of the jump rates is not unique but should vary with the position in configuration space. For the sake of simplicity we will treat the model first with one (fixed) value ofand extend the treatment later in section 8.6 to the case of a distribution ofvalues. The condition of eq. (3.32) for the raw barrier heightV0leads toV0>jE0j, whereE0is the ground–state energy. Thus for>0the inverse time scale 0=aexp(-V0=)is large compared to the attempt frequencya (cf. section 3.6). This cannot be surprising when taking into account that the model describes a coarse–grained dynamics between the deepest minima of the energy landscape. In the following we will choose-10 as unit of time,
0 1.
In [Sch73] SCHER and LAXhave introduced the Continuous–Time Random Walk (CTRW) model, where the diffusion of a particle on a lattice is described by trapping times: A particle sitting on a lattice site since timetperforms a transition to a nearest–
neighbor site at timet+, whereis drawn from a distribution density(). Since in the trap model (=0) each state energyEdetermines exactly one hopping ratew(E)= exp(-E=), it can be mapped to a model that is similar to the CTRW, by defining trapping times
1
w(E)
= e E=
: (6.6)
The model differs from the CTRW in one important point: The trapping timesare assigned to the states once and forever, while in the CTRW a new trapping time is drawn after each jump. For an exponentially decreasing DOS (6.3) the distribution density of trapping times reads
() =
-1- for 1 < 1; (6.7)
which is a PARETOdensity. For>0it is convenient to operate with thedefined in eq. (6.6) as well, even when these no longer have the meaning of a trapping time.
The hopping rate eq. (6.5) can then be written as
w
i;j
=
j
1-i
: (6.8)
Concerning the connectivity, we will make the simplest possible assumption for the QJM, i. e. that the states are arranged on ad–dimensional hypercubic lattice.
6.2 Existence of a stationary distribution — the glassy phase
In order to reach an equilibrium distributionpeq(E)for long times, the normalization integral of the BOLTZMANNdistribution
Z
0
-1 dEe
-E=
(E) = Z
0
-1 dEe
(1-1=)E
(6.9)
10
-210
-110
010
110
210
310
-210
010
210
410
610
810
10t
t 1=3 hr
2
(t)i
Figure 6.2:Mean square displacement of the QJM ford=10,=1=3and=1=4.
has to exist, which is obviously true only for>1. For<1 (orT<T0) the parti-cle is exploring deeper and deeper traps as time proceeds, without ever reaching an equilibrium state. We will call this low–temperature phase the “glassy phase” of the QJM. We note that for the existence of a glassy phase it is not necessary for the DOS to be strictly exponential, but that it is sufficient that the DOS has anexponential tailfor
E!-1and that the glass transition temperature
T
0
lim
E!-1
ln(E)
E
-1
(6.10) is finite.
BelowT0the dynamics of the model differs strongly from the dynamics aboveT0, which can be seen both in the mean square displacement and in the mean trapping time: In the trap model the absence of an equilibrium state for<1 is reflected in the fact that the mean trapping time becomes infinite. For the case>0the same holds true for an “effective trapping time”eff
P
n w(E
0
;E
n )
-1, wherenlabels the nearest–neighbors of the initial state. The non–equilibrium nature of the dynamics is reflected also in the fact that the mean square displacement never becomes diffusive but stays subdiffusive for all times, see fig. 6.2. The resulthr2(t)it can be easily understood for the trap model, since the diffusion is normal in the number of hops,
hr 2
(N)iN, and from eq. (5.21) witht=Pi
iit followsN(t)t. From Monte–Carlo simulations (for a description see appendix D) we know that hr2(t)i t remains valid also for>0. Arguments why this is expected are given in appendix F.3.
6.3 Initial condition and the aging function
6.3 Initial condition and the aging function
Slow dynamics in glassy systems starts when the system is quenched with a high cool-ing rate from a temperature above the glass transition temperatureT0to a temperature belowT0. To model this quench in the jump model we choose a temperatureT=1 fort< 0and change the temperature at timet=0instantaneously toT<T0which is kept constant fort>0. An infinite high temperature fort<0implies that all states are occupied with the same probability att=0.
Since the physical observables are functions of the coordinates of the configuration space, they will essentially decorrelate after the system has undergone a single transi-tion from one deep minimum to another one. Hence, a “generic correlatransi-tion functransi-tion”
in the model is to consider the probability that the state point has not jumped between
t
w and tw+t. We denote this two–time probability, after performing the disorder average, asaging function
C(t
w +t;t
w )
Y
i=0 Z
0
-1 dE
i
!
p(E
0
;:::;E
;t
w )e
-t P
j=1 w(E
0
;E
j )
: (6.11)
For>1an equilibrium state is reached in the limittw!1, and the aging function becomes time–translational invariant,
C t+t
w
;t
w
Ceq(t) for tw !1: (6.12)
If, on the other side, there is no equilibrium state to be reached fortw!1as we have shown it is the case for<1, the aging function will stay dependent ontw even for
t
w
!1. This situation is shown in fig. 6.3.
6.4 Precursors of aging for
> 1In the high–temperature regime>1we will compute two time scales that are mea-suring the decay of the aging functionCeq(t)in equilibrium. One might assume that the equilibrium correlation time also gives an estimate for the equilibration time when quenching the system from a higher temperature to a temperatureT>T0.
The probability distributionp(E;tw)of finding the system in a state with energyE at timetw fortw !1approaches the BOLTZMANNequilibrium distributionpeq(E)/
10
-3Figure 6.3: Aging function C(t + tw;tw) of the QJM from Monte–Carlo simulations for
(d;;)=(10;1=4;3=8)and different waiting timestw. The symbols refer to waiting times
110
8(),5108 (),2109 (),5109 ( ),11010 (N),41010(4),11011(H),31011 (O), and11012(), respectively.
(E)exp(-E=). Thus the aging functionCeq(t)can be calculated asymptotically,
Ceq(t) =
The equilibrium correlation timeeq may be estimated from the decay ofCeq(t). By identifyingeq with a time, where the correlations have decayed to a certain fraction
f, we find a VOGEL–TAMMANN–FULCHER law
6.5 Failure of the EMA to account for aging whereA=(1-)jln(f=c)jdepends only weakly on, but strongly on. This yields a divergence of the correlation time when!1+is approached. Eq. (6.14) gives a hint that T0 may have to be interpreted as a VOGEL–TAMMANN–FULCHER temperature instead of a calorimetric glass transition temperature.
Another way to define a correlation time in equilibrium is the “terminal time” (see also [Mon96]),
Eq. (6.15a) yields an algebraic divergence at a higher temperature
T
?
= (2-)T
0
: (6.16)
The existence of a second characteristic temperature for a similar model was first pointed out by Odagaki [Oda95].
6.5 Failure of the EMA to account for aging
In chapter 3 we have developed an Effective Medium Approximation scheme to ac-count for dispersive transport in stationary systems with quenched site energy disor-der. Thus it seems obvious to ask whether this EMA can be used to describe physical aging dynamics as well. However, we will see that an EMA scheme as presented in chapter 3 is not capable of describing this class of non–equilibrium phenomena, but that other concepts are needed. The argument is presented for the trap model (=0in eq. (6.5) ) to keep the derivations simple, but can be straightforwardly extended also to the case>0, since state energies of neighboring states are uncorrelated. For the notation of this section, cf. chapter 3.
The probability densityp(E;tw)can be expressed by the lattice GREENmatrix,
p(E;t
where the sum is taken over all states. This leads to the simple form
C(t
for the aging function. When performing a LAPLACEtransform with respect totwand using the definition eq. (2.7b) ofHn;k(t)this becomes
^
C(t;z) = 1
N X
n;k H
n;k (z)e
-E
n -w(E
n )t
: (6.19)
In the EMA as presented in chapter 3 one substitutesHn;k(z) by Fn;k(z;E) given by eqs. (3.2a,3.21), yielding the intermediate result
^
C
EMA
(t;z;E) = 1
N
a
2
d
e
-E-w(E)t Z
q 21. BZ
d d
q
P
n;k e
iq (xn-x
k )
z
E
+2(z;E) P
m
(1-cosqma) : (6.20) When the double–sum is performed, with
X
n;k e
iq (xn-x
k )
= N
2
a
d
Æ(q)
theq integral gives 1=zE. This can be identified as the LAPLACE transform of eE which does no longer depend upontw, yieldingCEMA(t;E)=exp(-w(E)t). As for the diffusion coefficient, the quantity has to be averaged with the BOLTZMANN equi-librium density to obtain
C
EMA
(t) = 0
Z
-1
dEpeq(E)e-w(E)t; (6.21) which corresponds toCeq(t). Thus the aging function can never show aging behavior with our EMA method, whatever transition ratew(E)and DOS(E)are chosen.