# Power-law Decay and the Ergodic–Nonergodic Transition in Simple Fluids

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## Abstract

It is well known that mode coupling theory (MCT) leads to a two-step power-law time decay in dense simple fluids. We show that much of the mathematical machinery used in the MCT analysis can be taken over to the analysis of the systematic theory developed in the Fundamental Theory of Statistical Particle Dynamics (Mazenko in Phys Rev E 81(6):061102, 2010). We show how the power-law exponents can be computed in the second-order approximation where we treat hard-sphere fluids with statics described by the Percus–Yevick solution.

## Keywords

Kinetic theory Ergodic-nonergodic transition Statistical mechanics Field theory Glass transition Mode-coupling theory Stochastic dynamics## References

- 1.Mazenko, G.F.: Fundamental theory of statistical particle dynamics. Phys. Rev. E
**81**(6), 061102 (2010)ADSCrossRefMathSciNetGoogle Scholar - 2.Andreanov, A., Biroli, G., Bouchaud, J.-P.: Mode coupling as a Landau theory of the glass transition. EPL (Europhysics Letters)
**88**(1), 16001 (2009)ADSCrossRefGoogle Scholar - 3.Kob, W.: Course 5: supercooled liquids, the glass transition, and computer simulations. In: Barrat, J.-L., Feigelman, M., Kurchan, J., Dalibard, J. (eds.) Slow Relaxations and Nonequilibrium Dynamics in Condensed Matter. Les Houches, vol. 77, pp. 199–212. Springer, Berlin (2003)Google Scholar
- 4.Mazenko, G.F.: Smoluchowski dynamics and the ergodic-nonergodic transition. Phys. Rev. E
**83**(4), 041125 (2011)ADSCrossRefGoogle Scholar - 5.Mazenko, G.F., McCowan, D.D., Spyridis, P.: Kinetic equations governing Smoluchowski dynamics in equilibrium. Phys. Rev. E
**85**, 051105 (2012)ADSCrossRefGoogle Scholar - 6.Berthier, L., Tarjus, G.: Critical test of the mode-coupling theory of the glass transition. Phys. Rev. E
**82**(3), 031502 (2010)ADSCrossRefGoogle Scholar - 7.Das, S.P., Mazenko, G.F.: Field theoretic formulation of kinetic theory: basic development. J. Stat. Phys.
**149**, 643–675 (2012)ADSCrossRefMATHMathSciNetGoogle Scholar - 8.Gotze, W.: Properties of the glass instability treated within a mode coupling theory. Zeitschrift fur Phys. B Condens. Matter
**60**, 195–203 (1985). doi: 10.1007/BF01304439 - 9.Barrat, J.L., Gotze, W., Latz, A.: The liquid-glass transition of the hard-sphere system. J. Phys.: Condens. Matter
**1**(39), 7163 (1989)ADSGoogle Scholar - 10.Brambilla, G., El Masri, D., Pierno, M., Berthier, L., Cipelletti, L., Petekidis, G., Schofield, A.B.: Probing the equilibrium dynamics of colloidal hard spheres above the mode-coupling glass transition. Phys. Rev. Lett.
**102**(8), 085703 (2009)ADSCrossRefGoogle Scholar - 11.van Megen, W., Underwood, S.M.: Dynamic-light-scattering study of glasses of hard colloidal spheres. Phys. Rev. E
**47**, 248–261 (1993)ADSCrossRefGoogle Scholar - 12.Bengtzelius, U., Gotze, W., Sjolander, A.: Dynamics of supercooled liquids and the glass transition. J. Phys. C: Solid State Phys.
**17**(33), 5915 (1984)ADSCrossRefGoogle Scholar - 13.Kim, B., Mazenko, G.F.: Mode coupling, universality, and the glass transition. Phys. Rev. A
**45**(4), 2393–2398 (1992)ADSCrossRefGoogle Scholar - 14.Gotze, W., Sjogren, L.: General properties of certain non-linear integro-differential equations. J. Math. Anal. Appl.
**195**(1), 230–250 (1995)CrossRefMathSciNetGoogle Scholar

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