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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. 

We perform computational studies of jammed particle packings in two dimensions undergoing isotropic compression using the well-characterized soft particle (SP) model and deformable particle (DP) model that we developed for bubbles and emulsions. In the SP model, circular particles are allowed to overlap, generating purely repulsive forces. In the DP model, particles minimize their perimeter, while deforming at the fixed area to avoid overlap during compression. We compare the structural and mechanical properties of jammed packings generated using the SP and DP models as a function of the packing fraction ρ, instead of the reduced number density φ. We show that near jamming onset the excess contact number Δz=z-z J and shear modulus G scale as Δρ 0.5 in the large system limit for both models, where Δρ=ρ-ρ J and z J ≈4 and ρ J ≈0.842 are the values at jamming onset. Δz and G for the SP and DP models begin to differ for ρ≥0.88. In this regime, Δz∼G can be described by a sum of two power-laws in Δρ, i.e. Δz∼G∼C 0 Δρ 0.5 +C 1 Δρ 1.0 to lowest order. We show that the ratio C 1 /C 0 is much larger for the DP model compared to that for the SP model. We also characterize the void space in jammed packings as a function of ρ. We find that the DP model can describe the formation of Plateau borders as ρ→1.0. We further show that the results for z and the shape factor A versus ρ for the DP model agree with recent experimental studies of foams and emulsions.