Age-structured hydrodynamics of ensembles of anomalously diffusing particles with renewal resetting

TL;DR

This work develops an age-structured hydrodynamic framework to describe the collective dynamics of many anomalously diffusing particles under renewal resetting, focusing on scaled Brownian motion with diffusion D(t) ~ t^{2H-1}. It analyzes three resetting protocols—Model A with independent resets to the origin, Model B with resets of the farthest particle, and a scaled Brownian bees variant where the farthest reset targets a random particle—revealing non-equilibrium steady states and distinct stationary density profiles. In the N -> ∞ limit, it derives explicit steady-state densities; Model A recovers the single-particle reset distribution, while Models B and scaled Brownian bees exhibit compact-support densities for all H > 0 due to global inter-particle correlations. The results demonstrate the versatility of age-structured HD for renewal processes and lay groundwork for incorporating fluctuations via fluctuating hydrodynamics in globally coupled anomalous-diffusion systems.

Abstract

We develop an age-structured hydrodynamic (HD) theory which describes the collective behavior of $N\gg 1$ anomalously diffusing particles under stochastic renewal resetting. The theory treats the age of a particle -- the time since its last reset -- as an explicit dynamical variable and allows for resetting rules which introduce global inter-particle correlations. The anomalous diffusion is modeled by the scaled Brownian motion (sBm): a Gaussian process with independent increments, characterized by a power-law time dependence of the diffusion coefficient, $D(t)\sim t^{2H-1}$, where $H>0$. We apply this theory to three different resetting protocols: independent resetting to the origin (model~A), resetting to the origin of the particle farthest from it (model~B), and a scaled-diffusion extension of the ``Brownian bees" model of Berestycki et al, Ann. Probab. \textbf{50}, 2133 (2022). In all these models non-equilibrium steady states are reached at long times, and we determine the steady-state densities. For model A the (normalized to unity) steady-state density coincides with the steady-state probability density of a single particle undergoing sBM with resetting to the origin. For model B, and for the scaled Brownian bees, the HD steady-state densities are markedly different: in particular, they have compact supports for all $H>0$. The age-structured HD formalism can be extended to other anomalous diffusion processes with renewal resetting protocols which introduce global inter-particle correlations.

Figures (6)

• Figure 1: Steady-state age-structured density $n_s(x, \tau)$ at $x=1$ versus $\tau$ for model $A$ with $r=D=1$ and four different values of $H$ (see legend). Symbols: simulations with $N = 10^5$ particles. The simulated density histograms are computed by averaging over 100 configurations observed at different times at intervals of $\Delta t = 10$ between them. Black dashed lines: Eq. \ref{['nsA']} for each of the $H$ values.
• Figure 2: Steady-state total density $u_{\text{s}}(x)$ for model A with $r=D=1$. (a) $u_{\text{s}}(x)$ for $H=1/5$ (blue), $1/2$ (black), $4/5$ (magenta), and $2$ (brown), see legend. These results were obtained by numerically evaluating the integral in Eq. \ref{['usA']}. (b) The $H$-dependence of the maximum density reached at $x=0$ for $0<H<1$. For $H>1$ the maximum density diverges, but this singularity is integrable. (c) and (d): $u_{\text{s}}(x)$, see Eq. \ref{['usA']}, alongside with its large- and small-$x$ asymptotics (\ref{['largexA']}) and (\ref{['smallxA']}), respectively, for $H=1/5$ (c) and $H=4/5$ (d).
• Figure 3: Steady-state total density $u_s(x)$ for model A with $r=D=1$ and four values of $H$ (see legend). Symbols: simulations with $N = 10^5$ particles. The simulated density histograms are computed by averaging over 100 configurations observed at different times at intervals of $\Delta t = 10$ between them. Black dashed lines: Eq. \ref{['usA']} for each of the $H$ values.
• Figure 4: Steady-state age-structured density $n_s(x, \tau)$ at $x=1$ for model $B$ with $r=D=1$ and four different values of $H$ (see legend). Symbols: simulations with $N = 10^5$, where the simulated density histograms are computed by averaging over 100 configurations observed at different times at intervals of $\Delta t = 10$ between them. Black dashed lines: Eq. \ref{['nsB']} for each of the $H$ values.
• Figure 5: Steady-state density $u_{\text{s}}(x)$ for model B with $r=D=1$. (a) $u_{\text{s}}(x)$, see Eq. \ref{['usB']}, for $H=1/5$ (blue), $1/2$ (black), $4/5$ (magenta), and $2$ (brown). (b) The support radius $L=L(H)$ as described by Eq. \ref{['LvsH']} (blue solid line) and its limit for $H\to \infty$, Eq. \ref{['limitL']} (red dashed line). (c) The $H$-dependence of the maximum density reached at $x=0$ for $0<H<1$. For $H>1$ the maximum density diverges, but this singularity is integrable. (d) Comparison of the theoretically predicted $u_{\text{s}}(x)$, Eq. \ref{['usB']} (black dashed lines), with simulations (symbols) with $N = 10^5$ particles and four different values of $H$ (see legend). The density histograms are computed via averaging over 100 configurations of the system, observed at intervals of $\Delta t = 10$.