Entropy, dynamics, and instantaneous normal modes in a random energy model

It is shown that the fraction f(u) of imaginary-frequency instantaneous nor mal modes (INM) may be defined and calculated in a random energy model (REM ) of liquids. The configurational entropy S-c and the averaged hopping rate among the states, R, are also obtained and related to f(u), with the resul ts R similar to f(u) and S-c = a + b ln(f(u)). The proportionality between R and f(u) is the basis of existing INM theories of diffusion, so the REM f urther confirms their validity. A link to S-c opens new avenues for introdu cing INM into dynamical theories. Liquid states are usually defined by assi gning a configuration to the minimum to which it will drain, but the REM na turally treats saddle barriers on the same footing as minima, which may be a better mapping of the continuum of configurations to discrete states. Req uirements for a detailed REM description of liquids are discussed.