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

It is shown that the fraction $f_u$ of imaginary-frequency instantaneous normal 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 results $R\sim f_u$ and $S_c=a+b\ln {(f}_u).\mathrm{}$ The proportionality between R and $f_u$ is the basis of existing INM theories of diffusion, so the REM further confirms their validity. A link to $S_c$ opens new avenues for introducing INM into dynamical theories. Liquid states are usually defined by assigning a configuration to the minimum to which it will drain, but the REM naturally treats saddle barriers on the same footing as minima, which may be a better mapping of the continuum of configurations to discrete states. Requirements for a detailed REM description of liquids are discussed.