Fig. 2: Cooperative resetting exhibits a localization phase transition.
a Interpretation of the approximated resetting flux in Eq. (4) as the product of an effective resetting rate, the density of particles contributing to the flux, and the effective jump length of resetting events. Particles are shown as filled circles and dotted arrows in different shades of gray indicate different possible cooperative resetting interactions. A probability density function is sketched in red. Filled-out bold arrows indicate the strength of probability fluxes due to resetting events. The dotted vertical line indicates the position χ. b, c Agent-based simulations (dots) of N = 2 × 105 particles for cooperative and extrinsic resetting with ϱ = 21. b Histograms for cooperative and extrinsic resetting which both decay exponentially for α = 0. The histogram shows simulation data, the bold lines show the analytical result according to Eq. (4). Extrinsic resetting is depicted in gray; cooperative resetting is depicted in red. c The cumulative distribution is defined as \(\int_{x}^{\infty }{p}_{{{{\rm{s}}}}}(x){\mbox{d}}\,x\). Lines represent the predicted functional dependencies from Eq. (5). Simulation results are shown as dots. ϱ is calculated from the simulation parameters, and the offset of the theoretical prediction is fitted to the simulation data. Extrinsic resetting is shown in gray, and cooperative resetting is stratified in color for different values of α. d For α = 2 we find a phase transition dependent on the particle density. The top inset shows the increase in the variance over time. For particles performing Brownian motion, the variance increases as σ2 = 2Dt. The slopes of the increase of the variance in the long-time limit (thick lines in inlay) define an effective order parameter, which is shown in the main plot. If the rescaled density is below a critical density, ϱ < ϱc, the probability density function delocalizes. Crosses depict results from agent-based simulations of N = 2 × 105 particles. Circles depict values of ϱ that are represented in the top inlay with matched color. The dashed line is the critical density \(\sqrt{{\varrho }_{c}}=2.3\) estimated analytically; compare with the Supplementary Note 4. The bottom inlay shows a power-law dependence of the rate of dispersion on rescaled density in the vicinity of the phase transition. The solid line is a nonlinear least-squares fit with an exponent 1.223 ± 0.012 (95% confidence intervals). Magenta crosses show the numerically measured rate of dispersion and error bars indicate the 95% confidence intervals.
