Invasion-wave-induced first-order phase transition in systems of active particles

An instability near the transition to collective motion of self-propelled particles is studied numerically by Enskog-like kinetic theory. While hydrodynamics breaks down, the kinetic approach leads to steep solitonlike waves. These supersonic waves show hysteresis and lead to an abrupt jump of the global order parameter if the noise level is changed. Thus they provide a mean-field mechanism to change the second-order character of the phase transition to first order. The shape of the wave is shown to follow a scaling law and to quantitatively agree with agent-based simulations.

Figure 1

(a) Steady state order parameter $\Omega$. (b) Speed of the invasion wave $v_W$ (circles) and average particle speed $u=\left|\mathbf{w}\right|/\rho$ (triangles) measured at the tip of the wave versus system size. The dash-dotted line shows the speed of sound in the disordered phase. All speeds are in units of $v_0$. Parameters: $R=1.5$, $v_0=0.97$, and $\eta =0.43$.

Figure 2

(a) The Fourier coefficients $g_k$ for a stationary invasion wave traveling into the positive $x$ direction. (b) The rescaled density ${\rho }_S=\rho L_x^{-2}{10}^8/2\pi$ versus rescaled position $X_S=x{L}_x^2{10}^{-7}$ for different system sizes. Results from agent-based simulations at $L=400$ are given by the fuzzy curve.

Figure 3

Order parameter versus noise $\eta$ for several system sizes $L$, and $R=0.1$, $v_0=1$. The blue diamonds are from agent-based simulations at $L=300$.