The velocity autocorrelation function tells you how a particle's speed at one time relates to its speed at another time. The force autocorrelation function tells you how the forces on a particle relate across time. These measure different things — one is kinematic, the other dynamic. There is no obvious reason they should be proportional.
Lüders, Mandal, and Franosch proved they are strictly proportional in equilibrium. For Brownian systems at thermal equilibrium, the velocity autocorrelation is exactly the negative of the force autocorrelation, scaled by a known constant. The proof is not perturbative or approximate. It is exact: a consequence of the fluctuation-dissipation theorem applied to the full many-body dynamics.
The proportionality holds because equilibrium enforces a specific relationship between fluctuations and dissipation. The forces a particle experiences from its neighbors and the velocities it exhibits in response are locked together by detailed balance — the same constraint that makes equilibrium stationary. The cross-correlations between displacements and interaction forces vanish identically. Nothing is left over.
In nonequilibrium systems — driven, active, or aging — this proportionality breaks. The cross-correlations that vanish in equilibrium become nonzero. The mirror between velocity and force develops distortions. The distortions are not noise. They are the signature of whatever is driving the system out of equilibrium.
The through-claim: equilibrium is a constraint, and the constraint is the diagnostic. You don't detect nonequilibrium by measuring something new. You detect it by measuring the same two things — velocity autocorrelation and force autocorrelation — and checking whether they still mirror each other. If they do, the system is in equilibrium. If they don't, the mismatch tells you about the driving. The deviation from the mirror is the signal.
The equilibrium relation gives you something for free: a baseline that requires no model of the nonequilibrium physics. You don't need to know what's driving the system. You just need to check whether the mirror still holds.