Abstract
Kinetically constrained lattice models of glasses introduced by Kob and Andersen (KA) are analyzed. It is proved that only two behaviors are possible on hypercubic lattices: either ergodicity at all densities or trivial non-ergodicity, depending on the constraint parameter and the dimensionality. But in the ergodic cases, the dynamics is shown to be intrinsically cooperative at high densities giving rise to glassy dynamics as observed in simulations. The cooperativity is characterized by two length scales whose behavior controls finite-size effects: these are essential for interpreting simulations. In contrast to hypercubic lattices, on Bethe lattices KA models undergo a dynamical (jamming) phase transition at a critical density: this is characterized by diverging time and length scales and a discontinuous jump in the long-time limit of the density autocorrelation function. By analyzing generalized Bethe lattices (with loops) that interpolate between hypercubic lattices and standard Bethe lattices, the crossover between the dynamical transition that exists on these lattices and its absence in the hypercubic lattice limit is explored. Contact with earlier results are made via analysis of the related Fredrickson--Andersen models, followed by brief discussions of universality, of other approaches to glass transitions, and of some issues relevant for experiments.
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See e.g. P. G. Debenedetti and F. H. Stillinger, Nature 410:259 (2001).
P. G. De Benedetti, {\it Metastable Liquids} (Princeton University Press, 1997).
Ediger, Annu. Rev. Phys. Chem. 51:99 (2000).
E. R. Weeks J. C. Crocker A. C. Levitt A. Schofield D. A. Weitz (2000) Science 287 627 Occurrence Handle10.1126/science.287.5453.627 Occurrence Handle10649991
M. M. Hurley P. Harrowell (1995) Phys. Rev. E 52 1694 Occurrence Handle10.1103/PhysRevE.52.1694
R. Yamamoto A. Onuki (1998) Phys. Rev. Lett. 81 4915 Occurrence Handle10.1103/PhysRevLett.81.4915
C. Bennemann C. Donati J. Bashnagel S. C. Glotzer (1999) Nature 399 246 Occurrence Handle10.1038/20406
L. Berthier, Phys. Rev. E 69:020201(R) (2004).
F. Ritort P. Sollich (2003) Adv. Phys. 52 219 Occurrence Handle10.1080/0001873031000093582
J. Jackle (2002) J. Phys. Condens. Matter 14 1423 Occurrence Handle10.1088/0953-8984/14/7/303
W. Kob H. C. Andersen (1993) Phys. Rev. E 48 4364 Occurrence Handle10.1103/PhysRevE.48.4364
J. P. Garrahan D. Chandler (2003) PNAS 100 9710 Occurrence Handle10.1073/pnas.1233719100 Occurrence Handle12900500
G. Biroli M. Mézard (2004) Phys. Rev. Lett. 88:025501 (2002); O. Rivoire, G. Biroli, O. C. Martin, and M. Mézard, Eur. Phys. J. B 37 55
E. Marinari, and E. Pitard, A Topographic-Non-Topographic Paradigm for Glassy Dynamics, cond-mat/0404214.
D. R. Reichman, E. Rabani, and P. L. Geissler, Comparison of Dynamical Heterogeneity in Hard-Sphere and Attractive Glass Formers, cond-mat/0406136. P. L. Geissler, D. R. Reichman, Short-ranged attractions in jammed liquids: How cooling can melt a glass, cond-mat/0402673.
C. Chamon, P. Charbonneau, L. F. Cugliandolo, D. R. Reichman, and M. Sellitto, Out-of-equilibrium dynamical fluctuations in glassy systems, cond-mat/0401326.
L. Berthier J. P. Garrahan (2003) Phys. Rev. E 68 041201 Occurrence Handle10.1103/PhysRevE.68.041201
A. Lawlor D. Reagan G. D. McCullagh P. DeGregorio P. Tartaglia K. A. Dawson (2002) Phys. Rev. Lett. 89 245503 Occurrence Handle10.1103/PhysRevLett.89.245503 Occurrence Handle12484955
S. Whitelam L. Berthier J. P. Garrahan (2004) Phys. Rev. Lett. 92 185705 Occurrence Handle10.1103/PhysRevLett.92.185705 Occurrence Handle15169509
S. Franz R. Mulet G. Parisi (2002) Phys. Rev. E 65 021506 Occurrence Handle10.1103/PhysRevE.65.021506
J. Kurchan L. Peliti (1997) ArticleTitleand M Sellitto: Europhys. Lett. 39 365–370 Occurrence Handle10.1209/epl/i1997-00363-0
A. Barrat J. Kurchan V. Loreto (2000) ArticleTitleand M Sellitto,. Phys. Rev. Lett. 85 5034–5038
J. Kurchan, Proceedings in ref. 28.
L. Berthier (2003) Phys. Rev. Lett 91 055701 Occurrence Handle10.1103/PhysRevLett.91.055701 Occurrence Handle12906606
Y. J. Jung J. P. Garrahan D. Chandler (2004) Phys. Rev. E 69 061205 Occurrence Handle10.1103/PhysRevE.69.061205
M. Sellitto J-J. Arenzon (2000) Phys. Rev. E 62 7793 Occurrence Handle10.1103/PhysRevE.62.7793
H. M. Jaeger J. B. Knight R. P. Behringer (1996) ArticleTitleRev Mod. Phys. 68 1259–1273 Occurrence Handle10.1103/RevModPhys.68.1259
A.-J. Liu (2001) and S.-R. Nagel ed.) Jamming and Rheology: Constrained Dynamics on Microscopic and Macroscopic Scales (Taylor and Francis London
G. H. Fredrickson H. C. Andersen (1986) Phys. Rev. Lett. 53:1244 (1984); J. Chem. Phys. 83:5822 (1985); G. H. Fredrickson and S. A. Brawer, J. Chem. Phys. 84 3351
T. M. Liggett (1985) Interacting Particle Systems Springer-Verlag New York
W. Gotze in Liquids Freezing and the Glass Transition, Hansen J.P., Levesque D., and J. Zinn-Justin ed. (North-Holland Amsterdam, 1991), pp. 287–503.
L. J. P. Bouchaud J. F. Cugliandolo M. Kurchan Spin Mézard Random Glasses Singapore Fields (1998) World Scientific L. F. Cugliandolo Les Houches Session 77 July 2002 J-L. Barrat, J. Dalibard, J. Kurchan, M. V. Feigel’man eds
C. Kipnis S. R. S. Varadhan (1986) ArticleTitleComm Math. Phys. 104 1–19 Occurrence Handle10.1007/BF01210789
H. Spohn (1991) Large Scale Dynamics of Interacting Particles Berlin Springer
H. Spohn (1990) ArticleTitleJ Stat. Phys. 59 1227–1239 Occurrence Handle10.1007/BF01334748
P. Harrowell (1993) Phys. Rev. E 48 4359 Occurrence Handle10.1103/PhysRevE.48.4359
R. Kohlrausch (1847) Ann. Phys 12 393
G. Williams D. C. Watts (1980) Trans. Faraday Soc 66 80
D. A. Huse D. S. Fisher (1987) Phys. Rev. B 35 6841
C. Toninelli G. Biroli D. S. Fisher (2004) Phys. Rev. Lett. 92 185504 Occurrence Handle10.1103/PhysRevLett.92.185504 Occurrence Handle15169499
C. Toninelli G. Biroli (2004) J. Stat. Phys. 117 27 Occurrence Handle10.1023/B:JOSS.0000044063.86539.19 Occurrence HandleMR2098557
M. D. Ediger, How molecules move near the glass transition temperature, Invited talk at the APS March meeting.
R. Cerf F. Manzo (2002) Stoch. Proc. Appl. 101 69 Occurrence Handle10.1016/S0304-4149(02)00124-2
J. Reiter J. Chem (1991) Phys 95 544
A. C. D. van Enter J. Adler J. A. M. S. Duarte (1990) J. Stat. Phys. 60 323 Occurrence Handle10.1007/BF01314923
L. Bertini, and C. Toninelli, in Exclusion processes with degenerate rates: convergence to equilibrium and tagged particle, cond-mat/0304694, accepted by J. Stat. Phys.
B. DeFinetti (1931) Atti R. Acc. Naz. Lincei serie 6 4 251
H.-O. Georgii (1979) Lecture Notes in Mathematics Vol. 760 Springer Berlin
J. Adler (1991) Physica A 171 435
J. Chalupa P. L. Leath G. R. Reich (1979) J. Phys C 12 L31
M. Aizenmann J. L. Lebowitz (1988) J. Phys 21 3801
A. E. Holroyd (2003) Probab. Theory Rel 125 195
C. Toninelli, G. Biroli, and D. S. Fisher, in preparation.
P. De Gregorio A. Lawlor P. Bradley K. A. Dawson (2004) Phys. Rev. Lett. 93 025501 Occurrence Handle10.1103/PhysRevLett.93.025501 Occurrence Handle15323924
L. Chayes V. J. Emery S. A. Kivelson Z. Nussinov J. Tarjus (1996) Physica A 225 129
N. C. Wormald (1999) Models of Random Regular Graphs in Surveys in Combinatorics J. D Lamb D. A Preece (Eds) London Mathematical Society Lecture Note Series Vol. 216 Cambridge University Press UK 239
M. E. Fisher J. W. Essam (1961) J. Math. Phys. 2 609 Occurrence Handle10.1063/1.1703745
M. Aizenman J. T. Chayes L. Chayes C. M. Newman (1988) J. Stat. Phys. 50 1 Occurrence Handle10.1007/BF01022985
S. Franz G. Parisi (2000) J. Phys.: Condens. Matter 12 6335 Occurrence Handle10.1088/0953-8984/12/29/305
M. Pretti (2003) J. Stat.Phys 111 993 Occurrence Handle10.1023/A:1022862618478
R. H. Schonmann (1992) Ann. Probab 20 174
R. Cerf (1999) ArticleTitleE. Cirillo Ann. Probab. 27(4) 1837
S. J. Pitts T. Young H. C. Andersen (1992) J. Chem. Phys. 113:8671 (2000); J. Reiter, F. Mauch, and J. Jackle, Physica A 184 458
M. Sellitto, G. Biroli, and C. Toninelli, Facilitated spin models on Bethe lattice: bootstrap percolation, mode-coupling transition and glassy dynamics, cond-mat/0409393
R. Renner H. Lowen J.-L. Barrat (1995) Phys. Rev. 52 5091
R. Du A. Grosberg T. Tanaka M. Rubinstein (2000) Phys. Rev. Lett. 84 2417 Occurrence Handle10.1103/PhysRevLett.84.2417 Occurrence Handle11018899
L.-F. Cugliandolo J. Kurchan (1993) ArticleTitlePhys Rev. Lett. 71 173–176 Occurrence Handle10.1103/PhysRevLett.71.173
T. R. Kirkpatrick D. Thirumalai (1988) Phys. Rev. A 37 4439 Occurrence Handle10.1103/PhysRevA.37.4439 Occurrence Handle9899574
G. Biroli J.-P. Bouchaud (2004) Europhys. Lett. 67 21
G. Tarjus, D. Kivelson, S. Mossa, and C. Alba-Simionesco, Disentangling density and temperature effects in the viscous slowing down of glassforming liquids, cond-mat/0309579.
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Toninelli, C., Biroli, G. & Fisher , D.S. Cooperative Behavior of Kinetically Constrained Lattice Gas Models of Glassy Dynamics. J Stat Phys 120, 167–238 (2005). https://doi.org/10.1007/s10955-005-5250-z
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DOI: https://doi.org/10.1007/s10955-005-5250-z