Global density equations for interacting particle systems with stochastic resetting: from overdamped Brownian motion to phase synchronization

Abstract

A wide range of phenomena in the natural and social sciences involve large systems of interacting particles, including plasmas, collections of galaxies, coupled oscillators, cell aggregations, and economic ``agents'. Kinetic methods for reducing the complexity of such systems typically involve the derivation of nonlinear partial differential equations for the corresponding global densities. In recent years there has been considerable interest in the mean field limit of interacting particle systems with long range interactions. Two major examples are interacting Brownian particles in the overdamped regime and the Kuramoto model of coupled phase oscillators. In this paper we analyze these systems in the presence of local or global stochastic resetting, where the position or phase of each particle independently or simultaneously resets to its original value at a random sequence of times generated by a Poisson process. In each case we derive the Dean-Kawasaki (DK) equation describing hydrodynamic fluctuations of the global density, and then use a mean field ansatz to obtain the corresponding nonlinear McKean-Vlasov (MV) equation in the thermodynamic limit. In particular, we show how the MV equation for global resetting is driven by a Poisson shot noise process, reflecting the fact that resetting is common to all of the particles and thus induces correlations that cannot be eliminated by taking a mean field limit. We then investigate the effects of local and global resetting on nonequilibrium stationary solutions of the macroscopic dynamics and, in the case of the Kuramoto model, the reduced dynamics on the Ott-Antonsen manifold.

Figures (12)

• Figure 1: Stationary solution of the 1D McKean-Vlasov Eq. (\ref{['MV1']}) for $V(x)=x^4/4-x^2/2$ and no resetting ($r=0$). Plot of the first moment $m_0(\ell)$ as a function of $\ell$ and various inverse temperatures $\beta$. The nonzero intercepts with the diagonal determine the positive definite solution $\ell_0$. We also take $\lambda=1$.
• Figure 2: Plot of Green's function $G(x,y)$ satisfying Eq. (\ref{['G']}) for fixed $y$ and various resetting rates $r$. Other parameters are $\beta \mu =1$, $y=1$ and $D=1$.
• Figure 3: Plot of Green's function $G(x,y)$ satisfying Eq. (\ref{['G']}) for fixed $y$ and various inverse temperatures $\beta$. Other parameters are $\mu =1$, $y=1$, $r=0.25$ and $D=1$.
• Figure 4: Solution of the1D McKean-Vlasov Eq. (\ref{['MV1']}) for $V(x)=\nu x^2/2$. Plot of the effective reset location $x_0(r)$, Eq. (\ref{['x0r']}), as a function of $r$ for various values of $\lambda$. Other parameters are $\nu =0.5$, $\beta =1$, $D=1$, and $x_0=4$.
• Figure 5: Solution of the1D McKean-Vlasov Eq. (\ref{['MV1']}) for $V(x)=\nu x^2/2$ and $r>0$. Plot of the NESS $\phi(x)$ as a function of $x$ for various resetting rates $r$ and weak coupling $\lambda=0.5$. Other parameters are $\nu =0.5$, $\beta =1$, $D=1$, and $x_0=4$. Filled circles indicate $x_0$.