  In principle, all of the dynamical complexities of many-body systems are
encapsulated in the potential energy landscapes on which the atoms move - an
observation that suggests that the essentials of the dynamics ought to be
determined by the geometry of those landscapes. But what are the principal
geometric features that control the long-time dynamics? We suggest that the key
lies not in the local minima and saddles of the landscape, but in a more global
property of the surface: its accessible pathways. In order to make this notion
more precise we introduce two ideas: (1) a switch to a new ensemble that
removes the concept of potential barriers from the problem, and (2) a way of
finding optimum pathways within this new ensemble. The potential energy
landscape ensemble, which we describe in the current paper, regards the maximum
accessible potential energy, rather than the temperature, as a control
variable. We show here that while this approach is thermodynamically equivalent
to the canonical ensemble, it not only sidesteps the idea of barriers, it
allows us to be quantitative about the connectivity of a landscape. We
illustrate these ideas with calculations on a simple atomic liquid and on the
Kob-Andersen model of a glass-forming liquid, showing, in the process, that the
landscape of the Kob-Anderson model appears to have a connectivity transition
at the landscape energy associated with its mode-coupling transition. We turn
to the problem of finding the most efficient pathways through potential energy
landscapes in our companion paper.
