Kubo formula (linear response)

General expression for the linear response kernel/susceptibility in terms of equilibrium correlation functions; commutator form in quantum mechanics.

Kubo formula (linear response)

Setup

Perturb an equilibrium system by a weak field $h(t)$ coupled to an observable $B$ :

H \;\mapsto\; H - h(t)\,B.

For an observable $A$ , define the linear response kernel $R_{AB}(t)$ by

\delta\langle A(t)\rangle = \int_{-\infty}^{t} R_{AB}(t-s)\,h(s)\,ds.

The equilibrium expectation is taken in a canonical ensemble (classical) or a quantum Gibbs state (quantum).

Quantum Kubo formula (commutator form)

For a quantum system with Heisenberg time evolution, the Kubo formula states

R_{AB}(t) = \frac{i}{\hbar}\,\theta(t)\,\langle [A(t),B(0)]\rangle_{\beta},

where $[A,B]=AB-BA$ and $\langle\cdot\rangle_\beta$ is the expectation in the Gibbs state (equivalently built from the quantum partition function ).

The causal susceptibility is the Fourier transform $\chi_{AB}(\omega)=\int_0^\infty e^{i\omega t} R_{AB}(t)\,dt$ .

Classical analogue

In classical Hamiltonian systems, a parallel statement replaces commutators by Poisson brackets (or, equivalently, relates response to suitable time derivatives of equilibrium correlations). In many common conventions one may write

R_{AB}(t)=\beta\,\theta(t)\,\langle \dot{B}(0)\,A(t)\rangle_{\mathrm{eq}},

which is a standard route to the fluctuation–dissipation theorem .

Uses

Transport coefficients: Applying the Kubo formula with currents as observables yields the Green–Kubo relations .
Equilibrium structure: In quantum systems, analytic properties encoded by the KMS condition control the relationship between response and correlation functions.
Correlation input: The building blocks are equilibrium two-point correlations and ensemble averages .

Interpretation

Kubo’s formula is the precise statement that, to first order in the perturbing field, the deviation from equilibrium is determined entirely by equilibrium dynamical correlations of the unperturbed system.