[Paper Review] Phase Transition of Hard Disk Systems with Vicsek-type Interactions

Nobuaki Murase, Masaharu Isobe|arXiv (Cornell University)|Mar 18, 2026

Material Dynamics and Properties0 citations

TL;DR

The paper studies how self-propulsion with Vicsek-type alignment affects phase behavior in dense hard-disk systems using event-driven MD, revealing shifts in polar and orientational transitions and linking them to local structure and free-volume geometry.

ABSTRACT

The phase diagram of self-propelled hard disk systems with Vicsek-type alignment interactions was investigated by event-driven molecular dynamics simulations. The model incorporates two competing order parameters: the polar order-disorder transition associated with collective velocity alignment (Vicsek model) and the orientational order arising from solid-fluid transitions (Alder transition) induced by excluded volume effects. The incompressibility of hard disks suppresses motility-induced phase separation at high packing fractions. Distinctive fluctuations were observed near the transition point, accompanied by anomalous shifts in the transition point as functions of noise intensity and packing fraction. Analysis of local configurational parameters -- specifically, orientational order and circularity of free volume -- provides insight into the microscopic origins of these anomalous phase transition shifts.

Motivation & Objective

• Investigate how Vicsek-type alignment interacts with excluded-volume hard-disk physics in two dimensions.
• Characterize competing order parameters: polar order-disorder and six-fold orientational (Alder) transitions.
• Understand how incompressibility and self-propulsion influence solid-fluid transitions at high packing fractions.
• Analyze microscopic origins of anomalous transition shifts through local structure and free-volume geometry.

Proposed method

• Use an active-matter model with hard disks and Vicsek-type alignment updated at fixed intervals Δt* in an event-driven molecular dynamics (EDMD) framework.
• Incorporate excluded volume via hard-disk collisions; velocity directions are updated by the vector sum of neighboring velocities with noise (Eq. 1) and speed is updated accordingly (Eq. 2).
• Explore two order parameters: polar order ψ = |sum v_i| / sum|v_i| and hexatic orientational order Φ6^G and Φ6^L (Eq. 3–6).
• Analyze local structure through free volume v_f^i, free surface s_f^i, local circularity c_i^* and global circularity C^* (Eqs. 7–8).
• Perform simulations across packing fraction ν and noise η, with varying Δt* to study competition between collisions and Vicsek updates.
• Identify phase behavior and transition points by tracking ψ, Φ6^G, Φ6^L, and local structure metrics.

Figure 1: (Color online) The polar order parameter $\psi$ as a function of noise $\eta$ in densely packed hard disk systems with $\nu=0.72$ , for various values of the Vicsek interaction time interval $\Delta t^{*}$ . The inset shows the packing fraction $\nu$ dependence of $\psi$ at a fixed $\Delta

Experimental results

Research questions

• RQ1 How does Vicsek-type alignment interact with hard-disk excluded-volume effects to modify the phase diagram in 2D?
• RQ2 What is the impact of self-propulsion frequency (Δt*) on polar and orientational ordering in dense hard-disk systems?
• RQ3 How do local structural features, such as free-volume geometry and circularity, relate to anomalous shifts in transition points?
• RQ4 Does incompressibility of dense hard disks suppress motility-induced phase separation and how does this influence freezing/melting (Alder) transitions?
• RQ5 What microscopic mechanisms link local structure to global phase behavior in this active hard-disk system?

Key findings

• Polar order-disorder transitions occur around η ≈ 1.2π in dense hard-disk systems with Vicsek interactions.
• Increasing Vicsek interaction frequency (smaller Δt*) shifts the transition point due to competition with elastic collisions.
• Global hexatic orientational order Φ6^G shows a cusp near η ≈ 1.2π and shifts to higher ν as η decreases, indicating coupling between noise, Vicsek updates, and crystallization.
• Local orientational order Φ6^L remains relatively less affected than Φ6^G at certain η, with polycrystalline clustering forming grain boundaries.
• Free-volume geometry significantly influences dynamics: two distinct circularity ranges (c_i^* ≈ 1.4 and ≈ 2.3) correlate with rectangular free-volume shapes, promoting large-displacement hopping and fluidization.
• The shape, not just magnitude, of free volume controls mobility and fluidization, revealing a geometric mechanism for transition shifts.

Figure 2: (Color online) Contour map of the global orientational order parameter $\Phi_{6}^{\rm G}$ in the $\nu-\eta$ plane at $(N,\Delta t^{*})=(4096,10)$ . The snapshots color-coded by orientation direction $arg(\phi_{6}^{i})$ , at $\nu=0.72$ for $\eta=0.1,1.2\pi$ , and $1.9\pi$ (from left to righ