Active elastic theory of self-aligning solids
Active disordered solids including dense human crowds and epithelial cells under confinement exhibit striking system-scale oscillatory spatiotemporal patterns. These are linked to a local feedback mechanism, self-alignment, that aligns the direction of a particle's motility vector to the total force. Simulations and experiments of solids made of such agents show these spontaneous oscillation patterns. Theoretically, they have been linked to both a nonlinear bifurcation and to selection of long-wavelength normal modes. Here we derive a closed nonlinear equation for the displacement field of active self-aligning 2d solids subject to angular noise. Along the normal modes of the solid, the dynamics of the mode amplitudes correspond to nonlinearly damped and stochastically driven harmonic oscillators. To linear order, we show that the system transitions from Active Brownian type correlated motion to oscillatory motion that increasingly condenses onto the lowest modes of the solid. We compare the analytical predictions for the mode spectra with simulations, finding excellent agreement approaching the transition from the disordered side. Strong nonlinearities manifest deep in the oscillating phase at strong alignment and small noise, consistent with the previous observations. At the continuum level, we derive a closed-form nonlinear wave equation for self-aligning solids. The transition to undamped oscillations is a second order dynamical phase transition driven by the competition between noise and alignment. At the linear level, we predict travelling acto-elastic waves, together with the emergence of system-scale oscillations for confined systems, consistent with observations in tissues and crowds. Our framework extends the understanding of self-aligning solids, which are pervasive among artificial and biological systems across multiple scales.