Exact results and instabilities in the harmonic approximation of active crystals

TL;DR

The paper develops an exact harmonic-theory framework for a two-dimensional triangular lattice of active particles with distance-dependent nearest-neighbor interactions, retaining full lattice anisotropy via a non-diagonal dynamical matrix and incorporating both thermal and persistent active noise. By solving the linear Langevin equations in Fourier space, it derives exact displacement correlators, analyzes crystalline order via Mermin’s criterion, and computes the mean-squared particle separation, internal energy, and entropy production, highlighting how activity alters energy distribution across modes. A key finding is that 2D active crystals lack long-range translational order in the harmonic limit, yet maintain local crystalline integrity up to a buckling threshold set by the pair-potential derivatives; beyond this threshold, the harmonic description breaks down and higher-order terms become essential. The work provides closed-form expressions for observables and clarifies the role of mode-by-mode entropy production, offering a rigorous link between microscopic interactions and macroscopic non-equilibrium behavior in dense active matter, while outlining the limitations of fixed adjacency and the need to extend the framework to evolving particle contacts.

Abstract

Condensates of active particles such as cells form almost-crystalline lattices which play a central role in many biological systems. Typically, their properties have been determined merely by analogy to the rather trivial one-dimensional case, leaving a gap between experimentally accessible observables and suitable theoretical models. Within a harmonic approximation, we characterise analytically a two-dimensional triangular lattice of active particles that interact with their nearest neighbours through a general pair potential, obtaining exact expressions for the correlators. We study this "active crystal" as a means of characterising active matter in the dense phase. Our treatment correctly approximates arbitrary pair potentials, rather than demanding an unphysical non-singular bilinear form. We retain "off-diagonal" terms that are routinely neglected despite quantifying the anisotropy of the particles' local potential. From the exact expressions for the correlation matrices, we derive exact results that shed light on the presence (or absence) of crystalline order. We further calculate the mean-squared particle separation, energy, entropy production rate and the onset of a pressure-induced instability resulting in the breakdown of the harmonic approximation. The entropy production rate is found to have a general form that is valid for generic active particles and lattice geometries, while resembling that of non-interacting "active modes".

Figures (10)

• Figure 1: Schematic representation of how the bulk of a condensate of densely packed active particles, interacting through repulsive pair potentials, can be approximated by a harmonic active crystal. (a) Illustration of the repulsive pair potential felt by nearest neighbours of the central particle. (b) In the bulk of the condensate, particles form an ordered lattice. Particle interactions are illustrated by springs, which are typically under pressure. Arrows indicate the direction and magnitude of each particle's self-propulsion $\mathbf{w}_{\mathbf{n}'}(t)$.
• Figure 2: (a) Three particles from a triangular lattice and the response of the lower one to external pressure (thick arrows). The grey springs are initially all equally compressed. As a result of the displacement, the blue spring relaxes and the red spring gets more compressed. (b) At the same time, the projection $\mathbf{f}_\pm$ of the forces $\mathbf{F}_\pm$ along the axis of motion changes.
• Figure 3: Contribution $(2\pi)^{-2}\int_{-\pi}^\pi \!\mathrm{d}^{2}p\, [(1+\mu^{-1}\tilde{\lambda}^+(\mathbf{k}))^{-1}+(1+\mu^{-1}\tilde{\lambda}^-(\mathbf{k}))^{-1}]$ as a function of $U"(\ell)/\mu\in[0.1,4]$ and $U'(\ell)/(\ell U"(\ell)\in[-0.333,2]$ to the energy per particle, Eq. (\ref{['eq:specific_energy']}).
• Figure 4: Illustration of the distance vector $\mathbf{u}^{}_{\mathbf{n}}\@ifempty{t}{}{(t)}-\mathbf{u}^{}_{\mathbf{n}+\mathbf{m}_q}\@ifempty{t}{}{(t)}-\ell\mathbf{a}_q$ between two particles that have been displaced away from their equilibrium lattice positions.
• Figure 5: Potential landscape $\sum_{q=1}^6U(|\mathbf{u}^{}_{\mathbf{n}}\@ifempty{}{}{()}-\ell\mathbf{a}_q|)$ of a single particle $\mathbf{n}$ as a function of its displacement, $\mathbf{u}^{}_{\mathbf{n}}\@ifempty{}{}{()}=(x,y)^\mathsf{T}$, assuming all surrounding particles are motionless, $\mathbf{u}^{}_{\mathbf{n}+\mathbf{m}_q}\@ifempty{}{}{()}=\mathbf{0}$ for $q=1,\ldots,6$. Parameters: $\kappa=1$, $z_0=1.3$ and $\ell=1$, Eq. (\ref{['eq:harmonic_pot']}). (a) Harmonic pair potentials, Eq. (\ref{['eq:harmonic_pot']}), as a function of the distance $|\mathbf{u}^{}_{\mathbf{n}}\@ifempty{}{}{()} - \mathbf{u}^{}_{\mathbf{n} + \mathbf{m}_q}\@ifempty{}{}{()} - \ell \mathbf{a}_q|$. Since the potential is a function of the mutual distances, some noticeable peaks appear at the position of neighbouring particles. (b) Harmonic approximation, Eq. (\ref{['eq:harmonicApproximation_explicit']}), using the same parameters as in subfigure (a).