Global perspectives on the energy landscapes of liquids, supercooled liquids, and glassy systems: The potential energy landscape ensemble

Abstract

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.3

Computer simulations of glasses: the potential energy landscape

“…This finding supports the validity of the "potential energy landscape ensemble" paradigm [127,128] (see section 5.5). Since the Hessians are known at all saddle points, it is possible to compute the partition function at the saddles, which can be used in transition state theory [126].…”

“…In a related approach, Wang et al [127] propose the "potential energy landscape ensemble", which is defined as the set of configurations for which the potential energy is less than or equal to some constant "landscape energy", E L . This has the effect of deemphasising the concept of dynamics between basins as a process of surmounting activation barriers.…”

NVU dynamics. II. Comparing to four other dynamics

“…5 In the thermodynamic limit these points are almost all of potential energy very close to U 0 . We refer to Paper I for further discussion of how NVU dynamics relates to these earlier works.…”

“…The NVU "heat bath" is provided by the multitude of configurational degrees of freedom. [3][4][5] The present paper compares NVU dynamics to other molecular dynamics, including stochastic ones. We first compare to NVE dynamics, which is also deterministic, and conclude that for large systems the two dynamics are basically equivalent.…”

“…6 In contrast, all points on have the same probability in NVU dynamics and there are no energy barriers -all barriers are of entropic nature defining unlikely parts of that must be passed. [3][4][5] Despite the absence of energy barriers in the ordinary sense of this term, NVU dynamics is fully able to describe locally activated events (hopping processes between local potential-energy minima). The NVU "heat bath" is provided by the multitude of configurational degrees of freedom.…”