Random energy model for dynamics in supercooled liquids:: N dependence -: art. no. 051110

The random energy model (REM) for the critical points (saddles and minima) of the potential energy landscape of liquids is further developed. While thermodynamic properties may be calculated from the unconditional distribution of states G(E), dynamics requires the distribution G(c)(E';E) of energies E' of neighbors connected to a state with energy E. Previously it was shown [T. Keyes, Phys. Rev. E 62, 7905 (2000)] that an uncorrelated REM, G(c)(E';E)=G(E'), is badly behaved in the thermodynamic limit N-->infinity. In the following, a simple expression is obtained for G(c)(E';E), which leads to reasonable N dependences. Results are obtained for the fraction f(u) of imaginary-frequency instantaneous normal modes, the configuration entropy S-c, the distributions of the different-order critical points, and the rate R of escape from a state. Simulation data on f(u)(T) and the density of minima rho(0)(E) in Lennard-Jones and CS2 are fit with the theory, allowing a determination of some model parameters. A universal scaling form for f(u), and a consequent scheme for calculating the mode-coupling temperature T-c consistently among different materials, is demonstrated. The dependence of the self-diffusion constant D upon R and f(u) is discussed, with the conclusion that Dproportional tof(u) in deeply supercooled states. The phenomenology of fragile supercooled liquids is interpreted. It is shown that the REM need not have a Kauzmann transition in the relevant temperature range, i.e., above the glass transition.