VOLUME 54, NUMBER 2 PHYSICAL REVIEW LETTERS Hydrodynamic Theory of the Glass Transition 14 JANUARY 1985 Shankar P. Das and Gene F. Mazenko The James Franck Institute and Department of Physics, The University of Chicago, Chicago, Illinois 60637 and Sriram Ramaswamy Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania I 9104 and John J. Toner IBM T. J. Watson Research Center, Yorkto~n Heights, New York 10598 (Received 23 July 1984) We demonstrate that the equations of hydrodynamics for a simple compressible fluid, including nonlinearities and thermal fluctuations, display a transition, at sufficiently high densities and low temperatures, to a glass. The theory leads to predictions for the behavior of viscosities, sound speeds, and correlation functions near the transition. PACS numbers: 64.70.Ew The equations of hydrodynamics for a simple com- pressible fluid display a transition at sufficiently high densities and low temperatures to a "phase" with the properties' of real glass. A remarkable feature of this transition is that it is purely dynamical: All static prop- erties, like thermodynamic derivatives, are regular as one goes through the transition. The shear and bulk viscosities diverge at the glass transition temperature TG. There is a discontinuous jump in the longitudinal sound speed as the temperature T is lowered past TG. For T ( TG, transverse sound waves propagate at arbi- trarily low frequencies co, with a speed whose zero- frequency limit vanishes discontinuously at TG. The density-density correlation function acquires a visco- elastic peak at ~=0 as T TG, with a width which drops continuously to zero at TG and stays there for T~ TG. This behavior of the transverse modes can be inter- preted in terms of a shear modulus which vanishes discontinuously at TG. The viscoelastic response of the system in the large co~ limit, where ~ is a charac- teristic time, shows a nonuniversal power-law behavior in keeping with a vast number of observations in glassy systems. Finally, our theory leads naturally in the glass phase to a yield stress of the order o; the shear modulus. The dynamical mechanism (nonlinear density fluc- tuations) which leads to this glass transition was re- cently discovered by Leutheusser and Yip (LY)" s in the kinetic theory of hard spheres. As will become clear below, the origins of their model differ consider- ably from ours, and their results emerge as a low-order approximation to our more general results. The equations governing the low-frequency and long-wavelength behavior of an isotropic compressible fluid are those of fluctuating nonlinear hydrodynamics including dissipation and thermal fluctuations. These equations describe the dynamics of the conserved or slow variables in the problem, namely, the mass, momentum, and energy densities p, g, and e. Since e plays a secondary role in the development, we shall ig- nore it in what follows. Results which follow in its in- clusion will be given elsewhere. The hydrodynamic equations can be derived by use of standard methods6 with the usual results "dp/"dt='7 g, (2) where qo and (0 are the bare shear and bulk viscosi- ties and V— = g/p. f, is a Gaussian noise source whose statistics are related to qo and (o via a fluctuation- dissipation theorem in the conventional fashion. The other term, pV;5FU/5p, involves the "potential" part of the effective free energy F [p, g] which governs the equal-time correlations of the fields p and g. Quite generally one can write8 F[p, g] = J d'x(g'/2p)+ FU[p], (4a) F [p]= Jii d3~[f(p)+ —, ' C(~p)~+ ] (4b) where f is analytic in p with a quadratic minimum at Equations (1) and (2) are just the continuity equation expressing conservation of mass and momentum. The second term on the left of (2) is the usual convective term. o-;J is the dissipative part of the stress tensor given by tr, ID= — 7io('7, VJ + 'vrj V, — —' , 5;~O' V) — (OB, J V V,