Section 4. Dynamics, relaxation and glass transition Dynamics of complex systems above the glass temperature R.M.C. de Almeida a , N. Lemke b , P. Jund c , R. Jullien c , I.A. Campbell c,d, * , D. Bertrand e a Instituto de Fisica, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, 91501-970 Porto Alegre, RS, Brazil b Centro de Ci^ encias Exatas e da Terra, Unisinos Av. Unisinos, 950 93022-000 S~ ao Leopoldo, RS, Brazil c Laboratoire des Verres, Universit e de Montpellier II, 34095 Montpellier cedex 5, France d Laboratoire de Physique des Solides, Centre Universitaire, B^ atiment 510, 91405 Orsay cedex, France e LPMCT, INSA, 135 avenue de Rangueil, 31077 Toulouse cedex 4, France Abstract We propose a phase space approach to understanding non-exponential relaxation above the glass temperature in complexphysicalsystems.Relaxationofasystemconsistingofmanyinteractingelementsatoms,molecules,spins,etc.) at a given temperature can be considered as the random walk of the point representing the whole system among those con®gurations which are thermodynamically permitted at that temperature. The phase space is a closed sphere-like) high dimensional space. It can be demonstrated numerically that random walks on a fractal inscribed in a sphere-like space lead to stretched exponential relaxation of the memory function. As stretched exponential decay is observed experimentally and numerically in very many glassy systems as the freezing temperature is approached, we suggest that this is the signature of a ubiquitous fractal phase space structure necessarily preceding a glass transition. Relaxation that resembles stretched exponential decay but which is not strictly stretched exponential could indicate a subtly dif- ferent physics. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 75.10.Nr; 75.40.Mg; 61.20.Lc 1. Introduction It is known that many glassy systems, particu- larly spin glasses and `fragile' structural glasses, have non-exponential long time relaxation for a wide range of temperature above the glass transi- tion, T g , as well as creep and aging eects for temperatures below the transition. The latter ef- fects have been extensively studied and have been widely discussed in terms of a `rough landscape' phase space, meaning that the system is only free tograduallymodifyitsstatebygoingthroughvery speci®c routes in con®guration space. In terms of spins, the con®guration at a given instant may well be such that it is permitted to turn spin i but not spin j, but once spin i has turned spin j becomes free to turn, and so on. Much less theoretical at- tention has been paid to the region above T g , but to be coherent the stretched exponential relaxation whichappearstobeubiquitousinthistemperature regime calls for an explanation in the same sort of language. In 1854 Kohlrausch [1] used a phenomenologi- cal expression Journal of Non-Crystalline Solids 287 2001) 201±209 www.elsevier.com/locate/jnoncrysol * Correspondingauthor.Tel.:+33-467143410;fax:+33-467 14 34 98. E-mail address: campbell@ldv.univ-montp2.fr I.A. Camp- bell). 0022-3093/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII:S0022-309301)00628-7