Linear response theory says that for small perturbations, the system's response is proportional to the applied force. Push gently, and the current — velocity, flux, displacement — scales linearly with the push. This is the foundation of transport theory: Ohm's law, Fick's law, Fourier's law all assume linearity at small driving forces.
Takeishi and Akimoto construct an exactly solvable model where linear response fails not approximately but exactly — the leading-order behavior is nonlinear. A driven particle experiences stochastic resetting whose rate depends on the particle's velocity: faster particles are reset more frequently, mimicking velocity-dependent scattering. The resetting couples the particle's state to the perturbation itself.
The steady-state mean velocity responds to the applied force F as ⟨v⟩ proportional to F^(1/(α+1)), where α parametrizes the velocity-dependence of the resetting rate. For any positive α, the exponent is less than 1 — the response is sublinear. Double the force, and the velocity increases by less than double. The nonlinearity is not a correction to a linear term; there is no linear regime. The first nonvanishing contribution is already nonlinear.
The mechanism: velocity-dependent resetting creates a feedback loop. A stronger force accelerates the particle, which increases the resetting rate, which returns the particle to zero velocity more frequently, which partially cancels the acceleration. The cancellation is exact enough to destroy the linear coefficient entirely.
Linear response fails when the perturbation changes the dissipation mechanism itself. If pushing harder also pushes back harder — through state-dependent coupling — the response is intrinsically nonlinear, and no amount of reducing the force will reveal a linear regime.