Kubo response and fluctuation–dissipation

Let a weak force f couple through an energy perturbation $-f(t)x$. The change in mean position is a convolution of f with a retarded kernel R. Retarded means that the response vanishes before the perturbation: future forcing cannot affect an earlier observation. Since integrating R times force over time gives a length, R has units of length divided by force and time. Its time integral is the static susceptibility, with units of displacement per force. Confusing the kernel with this integrated quantity loses a timescale.

Linear response retains only the part proportional to the perturbation amplitude. It does not mean the unperturbed system has no fluctuations, nor that every driven problem has an exactly linear solution. A nonlinear system may have a valid weak-field response around one equilibrium while behaving differently under large forcing. In the harmonic example below, the mean equation happens to be exactly linear for any applied force within the model. That special solvability should not be promoted into a general claim about strongly driven matter.

For $U=kx^2/2-f(t)x$ and constant mobility mu, the Itô equation has drift $-\mu kx+\mu f$ and noise amplitude $\sqrt{2\mu k_BT}$. Averaging gives a deterministic linear equation for the mean. An impulse force produces a response $R(t)=\mu e^{-\mu kt}$ for positive t. Separately, equilibrium Ornstein–Uhlenbeck fluctuations have covariance $C(t)=(k_BT/k)e^{-\mu k|t|}$. Differentiating this covariance for positive time reproduces the response after multiplication by minus inverse thermal energy.

The equality ties a dissipative mobility measured by forcing to a correlation measured while no force is applied. Its minus sign is essential because the covariance decays. The equilibrium variance fixes the static susceptibility through $\chi_{\mathrm{stat}}=C(0)/(k_BT)=1/k$. These expressions concern a time-even classical coordinate with the stated coupling convention. Quantum Kubo relations involve operator commutators and frequency-dependent thermal factors, while velocity observables and magnetic fields require careful time-reversal conventions. One must match the formula to the observable and regime.

If an experiment violates the equilibrium relation, possibilities include active forcing, a nonstationary preparation, unresolved slow variables, measurement filtering, or an inappropriate response protocol. The mismatch alone does not identify a unique mechanism. One must first compare correlations and responses at matched timescales, units, and baseline conditions. Replacing temperature by a fitted number can sometimes match one observable or frequency range, but does not prove that the entire driven system has a thermodynamic temperature or obeys all equilibrium identities.

A system can also be far from equilibrium while admitting useful linear response about its driven steady state. That response is linear in an additional small perturbation, but need not obey the equilibrium correlation formula written above. Generalized response theories can involve extra dynamical correlations and information about steady currents. Conversely, fluctuation theorems in later lessons can remain valid for large driving when their own microscopic-reversibility assumptions hold. Linear-response restrictions and far-from-equilibrium identities are different statements with different domains of validity.