fully-penetrable-particle-fluid

Metropolis-Hastings sampling of the the canonical ensemble of a simple fluid interacting with a single step potential. Particles are fully penetrable in the sense that it takes finite energy to get any given particle pair to zero distance.

What is happening here?

In the code we perform a metropolis-hastings sampling of the canonical ensemble for a system of penetrable spheres (PS) and penetrable disks (PD). The "penetrable" is meant in the sense that they interact via the pair potential $u_E(r)=E,\Theta(\sigma-r)$. This treatment of penetrable spheres/disks can be understood as a variation on the problem of predicting the free energy of a system of rigid spheres (RS) or rigid disks (RD), also referrred to as hard spheres or hard disks: when $E\gg k_\text{B}T$, a pair of particles would need an unrealistically high energy to overcome the barrier $E$ to closing in to any distance shorter than $\sigma$, effectively making them rigid. The family ${u_E}_{E\in[0,\infty)}$ in this sense connects the ideal gas, interacting with $u_0(r)\equiv 0$, to the rigid particle fluid, doing so with potentials that have a clear mathematical resemblance with rigid particle interaction.

The overlap energy $E$ (or penetration energy barrier) is the only energy scale in the system besides temperature, such that the phase diagram only depends on the ratio $$\varepsilon=\frac{E}{k_\text{B}T}$$ Besides the reduced overlap energy $\varepsilon$, the system behaviour is still shaped by the packing density $$\Phi=B_d\left(\frac{\sigma}{2}\right)^d\rho$$ where $\rho$ is number density and $B_d$ the volume of a unit sphere in $d$ dimensions (which reads, forexample, $B_2=\pi$ and $B_3=\frac{4}{3}\pi$ in $d=3$).

For now ...

Code development is on low flame. As soon as it is reliable I seek to find and explain the various behaviour regimes observed in simulation.

Copyright notice

© 2026 Miriam Derla - CC BY-SA 4.0