Several systems, such as proteins and glasses, are characterized by a complex potential energy surface, i.e., there are many minima that are separated by barriers of differing heights spanning the entire gamut of energy scales. 1-3 For these systems, it is meaningful to characterize the distribution of energy barriers, g(E), which is notoriously very difficult to compute. In several papers, we have proposed and developed related methods for calculation of g(E) from the temperature-dependent

more »

... tion f u (T) of unstable "instantaneous normal modes." We have obtained g(E) in peptides 4 and proteins 5 and the unit density Lennard-Jones liquid. 6 Our methods are necessarily approximate and rely on physically motivated simplifications. Probably the most important simplification is that the potential energy landscape is assumed to look the same from all the minima. This excludes the existence of "correlation" where the height of a barrier depends upon the depth of the connected minima. The assumption is clearly stated in our work. For example, 7 "We now invoke a major simplifying assumption, the equivalent minima model-the topology of the potential surface is identical when viewed from each minimum." Again, 5 "We assume ... the potential energy as seen from the minimum of any basin will be identical to any other." In the accompanying Comment, Zwanzig 8 considers a one-dimensional potential energy function, U(x), generated by successive placement of randomly chosen parabolas of alternating downward ͑barrier͒ and upward ͑minimum͒ curvature along the Uϭ0 line; the minima have negative energy and the barriers positive energy. This model has a strong correlation. A minimum with negative energy E ␣ will have relative barrier energies E such that Eу͉E ␣ ͉. There is no possibility that a deep minimum will adjoin a low barrier. Thus, the model is antithetical to the assumptions in our papers. Zwanzig demonstrates that the Straub and Thirumalai ͑ST͒ integral equation theory, 4,5 starting from the exact g(E), yields an incorrect T dependence of f u (T) at low T. For that example, the ST equation results in a linear T dependence while the exact result varies as (1/T)exp(Ϫ1/T). We consider this neither surprising nor a damaging criticism of our work, since Zwanzig's model has strong correlation, explicitly excluded in our theories. It is therefore interesting and informative to repeat Zwanzig's calculation for a onedimensional potential with zero correlation. We choose a one-dimensional rough potential of a form originally studied by Zwanzig: 9