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Directed percolation transition to active turbulence driven by non-reciprocal forces

Juliane U. Klamser, Ludovic Berthier

Abstract

We numerically study the collective dynamics of dense particle assemblies driven by non-reciprocal pairwise forces of amplitude $κ$. At a critical value $κ_{\rm c}$, the system undergoes a dynamical phase transition from an absorbing state ($κ< κ_{\rm c}$) to a chaotic steady state ($κ> κ_{\rm c}$). The chaotic phase is marked by nontrivial spatiotemporal velocity correlations and mixing, reminiscent of active turbulence in self-propelled systems. The sharp onset of chaos shows critical scaling consistent with the universality class of directed percolation. We argue that this transition is generic to a broad class of locally-driven, dense disordered materials.

Directed percolation transition to active turbulence driven by non-reciprocal forces

Abstract

We numerically study the collective dynamics of dense particle assemblies driven by non-reciprocal pairwise forces of amplitude . At a critical value , the system undergoes a dynamical phase transition from an absorbing state () to a chaotic steady state (). The chaotic phase is marked by nontrivial spatiotemporal velocity correlations and mixing, reminiscent of active turbulence in self-propelled systems. The sharp onset of chaos shows critical scaling consistent with the universality class of directed percolation. We argue that this transition is generic to a broad class of locally-driven, dense disordered materials.

Paper Structure

This paper contains 5 equations, 4 figures.

Figures (4)

  • Figure 1: (a) The repulsive forces $\mathbf{F}_{12}$ and $\mathbf{F}_{21}$ between particles 1 and 2 with orientations ${\bf u}_1$ and ${\bf u}_2$ are modulated by the angles $\alpha_{12} \neq \alpha_{21}$, resulting in non-reciprocal pairwise interactions. The orientations act as an 'eye' for the particles. (b) Typical velocity field in chaotic steady state at large enough $\kappa=0.08 > \kappa_{\rm c}$ (colors code for amplitudes, arrows for orientations). Large scale correlated regions of fast and slow velocities are organized along streams and vortices. (c) Corresponding vorticity field, with stream lines.
  • Figure 2: (a) Time evolution of the mean-squared velocity for different values of $\kappa$, starting from random initial conditions. The time $t_{\rm f}$ to enter an absorbing state with vanishing velocity increases with $\kappa$. Above $\kappa_{\rm c}$, the system remains active within the numerical observation time. (b) Normalized velocity autocorrelation for different $\kappa$, showing a two-step decay to a constant, defining a microscopic collision timescale $\tau_{\rm c}$, and a much longer persistent time $\tau_{\rm v}$.(c) The mean freezing time $\langle t_{\rm f} \rangle$ and the rescaled persistent time $\tau_{\rm v} / \tau_{\rm c}$ diverge on both sides of the absorbing phase transition at $\kappa_{\rm c}$.
  • Figure 3: (a) Longitudinal and (b) transverse velocity correlations, Eq. \ref{['Eq:C(r)']}, for different values of $\kappa>\kappa_{\rm c}$. (c) Kinetic energy spectrum $E(k)$, Eq. \ref{['Eq:E(k)']}, for the same parameters. All three functions demonstrate the growth of a correlation lengthscale diverging towards $\kappa_{\rm c}$, opening a regime of scale-free correlations.
  • Figure 4: Directed Percolation scaling laws (solid black line) for timescales (a), lengthscales (b), and the activity order parameter (c) describe the numerical data very well (symbols) using known DP values for $(\nu_\parallel, \nu_\perp, \beta)$ in two dimensions.