Competition in a system of Brownian particles: Encouraging achievers

TL;DR

This work introduces a Brownian-particle system with inter-agent competition in which, at a constant rate, two randomly chosen particles compete and the one closest to the origin is reset to the origin, effectively discouraging underachievement. In the large-N limit, the bulk dynamics obey a nonlocal hydrodynamic equation that yields a stationary density with a power-law tail $|x|^{-3}$ and a self-similarly expanding halo, while finite-N effects cap the swarm radius at scales $ar{ℓ} o ext{const} imes oot 2 elax N$ and induce finite fluctuations; the current leader, however, continues a Brownian exploration with $ar{X}(t) o ext{const}+rac{2}{ oot 2 elax \pi} oot t$ scaling. The analysis is extended to an arbitrary small number $n$ of competing particles, giving heavier tails in the steady state and a general scaling form for the leader dynamics of the $n-1$ persistent leaders. These results are supported by comprehensive Monte-Carlo simulations and illuminate mechanisms that can generate broad wealth distributions and inequality-like features in interacting-agent systems.

Abstract

We introduce and study analytically and numerically a simple model of inter-agent competition, where underachievement is strongly discouraged. We consider $N\gg 1$ particles performing independent Brownian motions on the line. Two particles are selected at random and at random times, and the particle closest to the origin is reset to it. We show that, in the limit of $N\to \infty$, the dynamics of the coarse-grained particle density field can be described by a nonlocal hydrodynamic theory which was encountered in a study of the spatial extent of epidemics in a critical regime. The hydrodynamic theory predicts relaxation of the system toward a stationary density profile of the "swarm" of particles, which exhibits a power-law decay at large distances. An interesting feature of this relaxation is a non-stationary "halo" around the stationary solution, which continues to expand in a self-similar manner. The expansion is ultimately arrested by finite-$N$ effects at a distance of order $\sqrt{N}$ from the origin, which gives an estimate of the average radius of the swarm. The hydrodynamic theory does not capture the behavior of the particle farthest from the origin -- the current leader. We suggest a simple scenario for typical fluctuations of the leader's distance from the origin and show that the mean distance continues to grow indefinitely as $\sqrt{t}$. Finally, we extend the inter-agent competition from $n=2$ to an arbitrary number $n$ of competing Brownian particles ($n\ll N$). Our analytical predictions are supported by Monte-Carlo simulations.

Figures (10)

• Figure 1: The steady state density profile for pair competition with $N=10^2$ particles, as observed in simulations (points) and predicted by Eq. \ref{['rho-2:sol']} (solid line).
• Figure 2: The shape function $R(\xi)$, see Eq. \ref{['rxt']}, determined by solving Eqs. (\ref{['R:eq']}) and (\ref{['R:BC']}) numerically. The dashed line shows the small-$\xi$ asymptotic (\ref{['R:small1']}).
• Figure 3: The time-dependent numerical solution for $r(x,t)$ approaches the steady state solution $r_0(x)$. Solid lines: $r(x,t)$ at times $0$, $5$, $200$ and $400$ (from left to right). The last two lines coincide with each other and with the analytical prediction (\ref{['r2:sol']}) for the steady state (dashed line).
• Figure 4: The time-dependent numerical solution for $r(x,t)$ exhibits an expanding halo as predicted by Eq. (\ref{['rxt']}). Shown is the ratio $r(x,t)/r_0(x)$ at times $50$, $100$, $200$ and $400$ (from left to right).
• Figure 5: (a) The ratio $r(x,t=100)/r_0(x)$ (solid line) is compared with the shape function $R(\xi)$ from Eq. (\ref{['R:eq']}) (dashed line). The horizontal axis is rescaled to the distance $x_{1/2}(t)$ where the depicted functions are both equal to $1/2$. (b) $\ln x_{1/2}(t)$ versus $\ln t$ at $t=50$, $100$, $200$ and $300$ (symbols). A fit (straight line) gives the dynamic exponent $0.51$ in agreement with Eq. (\ref{['rxt']}).