Ikeda and Miyazaki Reply to Rolf Schilling and Bernhard Schmid [arXiv:1101.5577]

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by  Atsushi Ikeda

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Ikeda and Miyazaki Reply: In the preceding com- as the replica theory are “mean-field theories” a la spin-
ment [1], Schilling and Schmid (SS) criticized our recent glasses even in finite d’s. What we have discussed in
statementthatreconsiderationandrevisionofmodecou- ourpaperis the inconsistenciesbetweenthe twotheories
pling theory (MCT) from the groundup are in order [2]. in large but finite d’s (and not the effects of the finite-
Ourstatementisbasedonthefactthatlongtimelimitof sizenorfinite-dinconventionalcriticalphenomena). The
self-partofvanHovefunctionG(cs,∞) (r;d)exhibitsunphys- reason to consider large d’s was to avoid unwanted ap-
ical negative dips in as low dimension as d = 6 and the proximations for the static parameters.
depths ofthe dips increasewithdimension[2]. Although Finally, we agree with SS in that our conclusions
SS agreed with the existence of the dips, they suggested would not imply that MCT is invalid in a certain time
thattheseresultsarenotyetsufficienttoverifytheabove and temperature regime for two- and three-dimension
1 statement. Their suggestion is based upon two observa- liquids. One can not overstate the success of MCT
1 tions: (i) The negativedip ofG(cs,∞) (r;d) is demonstrated in predicting dynamical properties of the supercooled
0
to decrease as d increases. It is also shown that the dips liquids. The question is why MCT is so powerful and
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are very sensitive to the shape of the nonergodicity pa- robust, at least in low dimensions, despite of many
n rameter f(s)(k;d) and that an analytic function which uncontrolled and controversial approximations in its
a c
J fits f(s)(k;d) reasonably well can eliminate the negative derivation. Another question is in what sense MCT is
c
the mean-field theory, if there really exists a mean-field
1 dip. They also claim that such a glitch is not surprising
3 giventhenonlinearstructureofMCTandmaynotaffect theory of the glass transition. Demonstrating the
situations where MCT breaks down and fixing the
the quality of the theory. (ii) The long-time and large d
] breakdowns would help us to answer to these important
t limitshouldbetakenwithcaution. Numericalstudieson
of along-rangedφ4-model[3]showthatthe limits oft→∞ but unanswered questions. This is the reason why we
s andN →∞arenotcommutable,whereN isthenumber claim that “reconsideration and revision of MCT from
. the ground up are in order”.
t of interaction bonds which should be proportional to d
a
m for off-lattice liquids.
Regarding the point (i), we believe that our conclu- Atsushi Ikeda and Kunimasa Miyazaki
-
d sion is not affected by their objection. In Figure 1, we Institute of Physics, University of Tsukuba,
n show G(cs,∞) (r;d) multiplied by rd−1, or the probability Tennodai 1-1-1, Tsukuba 305-8571,Japan
o
density function, in much higher dimensions than those PACS numbers: 64.70.qj, 61.43.Fs,64.70.pm, 66.30.hh
c
[ shown in Ref. [2]. Similar dips have been observed for
(c)
1 the collective part of the van Hove function Gc,∞(r;d).
It clearly demonstrates that the amplitudes of the neg-
v
0 ative dips increase as d increases. The negative dips
[1] R. Schilling and B. Schmid, preceding comment.
4 remain noticeable even at d = 100 where the asymp- [2] A. Ikeda and K. Miyazaki, Phys. Rev. Lett. 104, 255704
8 totic scaling of MCT critical point ϕ (d) ∝ d2/2d is
mct (2010).
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observed [2, 4]). Note also that the probability density [3] W. Kob and R. Schilling, J. Phys.: Condens. Matter 3,
.
1 rd−1G(cs,∞) (r;d)ratherthanG(cs,∞) (r;d)isanaturalobserv- 9195 (1991).
0 [4] B. Schmid and R. Schilling, Phys. Rev. E 81, 041502
ables,becausethelatter,ifnegativeorpositive,becomes
1 (2010).
1 negligibly small for arbitrarynon-zero r’s as d increases.
: We emphasize that both the peculiar d-dependence of
v
i ϕmct(d) and the pathologicalnegative dips of G(cs,∞) (r;d)
X (s)
derive from the non-Gaussian shape of fc (k;d). One
r can not ignore this non-Gaussianity as a minor glitch ∞
a
hidden in MCT because the different scaling of ϕ (d)
mct
from the prediction of replica theory is a grave problem
whichmayundermine the mean-fieldpicture ofthe glass d=50 d=100
transition theory.
Regarding the second point (ii), we believe that our
argumentisirrelevanttotheconclusionsofRef.[3]where
the relaxation time of a long-ranged model was demon-
stratedtoincreasewiththesystemsizeN anddivergein FIG. 1: sdrd−1Gs,∞(r) for d = 50 (left) and 100 (right),
thermodynamiclimitN →∞intheorderedphase. This wheresd isthesurfaceareaofthed-dimensionalunitsphere.
thermodynamic limit should not be taken to be equiva-
lent to large-d limit in our MCT analysis; MCT as well