Visualizing free-energy landscapes for four hard disks

We present a simple model system with four hard disks moving in a circular region for which free-energy landscapes can be directly calculated and visualized in two and three dimensions. We construct several energy landscapes for our system, and we explore the strengths and limitations of each in terms of understanding system dynamics, in particular the relationship between state transitions and free-energy barriers. We also demonstrate the importance of distinguishing between system dynamics in real space and those in landscape coordinates, and we show that care must be taken to appropriately combine dynamics with barrier properties to understand the transition rates. This simple model provides an intuitive way to understand free-energy landscapes, and it illustrates the benefits that free-energy landscapes can have over potential energy landscapes.