Non-negative differential mobility of one-dimensional underdamped Langevin dynamics: Proof via mapping to a marginally stable oscillator

For an underdamped particle driven by a constant non-conservative force across a periodic potential in one spatial dimension at finite temperature, numerical studies have shown that the differential mobility is non-negative. Here, an analytical proof will be given that is based on expressing the differential mobility by the long-time mean position of a marginally stable harmonic oscillator subject to noise in its stiffness.