Breakdown of Linear Response Induced by Velocity-Dependent Stochastic Resetting

Abstract

Linear response theory lies at the foundation of transport phenomena, predicting that physical systems respond proportionally to weak external forces. Here we show that this principle can break down in a minimal nonequilibrium setting due to state-dependent stochastic resetting. We consider a driven Langevin particle subject to a resetting mechanism whose rate grows as a power of the particle velocity, motivated by transport processes where faster carriers experience more frequent scattering events. We derive the exact steady-state velocity distribution and establish a moment balance relation that links external driving, viscous dissipation, and resetting-induced dissipation. This relation reveals that the response is controlled by a nonlinear coupling between the velocity and the resetting rate, leading to nonlinear transport. In particular, the mean velocity obeys the exact power law $\langle v\rangle \propto F^{1/(α+1)}$, where $α$ characterizes the velocity dependence of the resetting rate. Our results provide a solvable example in which linear response fails at the level of the leading-order behavior and identify velocity-dependent resetting as a minimal dynamical mechanism for generating nonlinear transport in nonequilibrium steady states.

Figures (3)

• Figure 1: Statistical properties of the minimal deterministic model with velocity-dependent resetting for different values of the resetting exponent $\alpha=-0.5, 0,$ and 1. (a) Stationary velocity PDFs $P_{\mathrm{st}}(v)$ ($F=1$). Symbols denote numerical results, and dahsed lines show the exact analytical expression given by Eq. \ref{['eq: st dist det']}. (b) Mean velocity $\langle v \rangle$ as a function of the external force $F$. Symbols denote numerical results, while dashed lines represent the theoretical prediction from Eq. \ref{['eq: mean v case 1']}. The slopes are consistent with the exact scaling exponent $1/(\alpha+1)$, demonstrating nonlinear response for $\alpha \neq 0$ and recovery of linear response for $\alpha = 0$. (c) Inter-reset-time distribution $\psi(t)$ ($F=1$). Symbols denote numerical results and dashed lines correspond to the analytical expression in Eq. \ref{['eq: waiting-time dist det']}. The dependence on $\alpha$ illustrates how velocity-dependent resetting modifies the reset statistics.
• Figure 2: Stationary velocity distributions $P_{\mathrm{st}}(v)$ for the general dynamics with $\alpha = 1$. Symbols denote numerical results, while dashed lines represent the analytical expressions given by Eqs. \ref{['eq: st dist deterministic v>0']} and \ref{['eq: st dist deterministic v<0']}. The general dynamics ($D=1$, $\gamma=1$) is shown together with the limiting cases $D=0$ ($\gamma=1$) and $\gamma=0$ ($D=1$). (a) Distributions under a weak external force $F=0.1$. (b) Same data as in (a), shown for $v>0$ with a logarithmic scale on the vertical axis. (c) Distributions under a strong external force $F=10$.
• Figure 3: Mean velocity $\langle v \rangle$ as a function of the external force $F$ for the general dynamics ($D=1$, $\gamma=1$), together with the limiting cases $D = 0$ ($\gamma=1$) and $\gamma = 0$ ($D=1$). Symbols represent numerical simulations, while the solid lines indicate the asymptotic scaling: linear response $\langle v\rangle \propto F$ for small $F$ and nonlinear scaling $\langle v\rangle \propto F^{1/2}$ for large $F$, given by Eqs. \ref{['eq: mv small F case 2']} and \ref{['eq: mv large F case 2']}.