Green–Kubo relations

Transport coefficients expressed as time integrals of equilibrium current autocorrelation functions (linear response).

Green–Kubo relations

Statement (generic form)

In equilibrium (typically with respect to a canonical ensemble ), many linear transport coefficients can be written as time integrals of equilibrium time-correlation functions (a special case of Kubo linear response ).

A common template is

\mathcal{L}_{\alpha\beta} = \int_0^\infty \langle J_\alpha(0)\,J_\beta(t)\rangle_{\mathrm{eq}}\,dt,

up to conventional prefactors (often $1/(k_B T)$ and/or volume factors) depending on the definition of the currents and forces.

Here $\langle \cdot \rangle_{\mathrm{eq}}$ is an ensemble average in equilibrium, and $\langle J_\alpha(0)J_\beta(t)\rangle$ is a two-point time correlation .

Standard examples (common conventions)

Electrical conductivity

Let $J(t)$ be the total electric current in volume $V$ . Then

\sigma = \frac{1}{k_B T\,V}\int_0^\infty \langle J(0)\cdot J(t)\rangle_{\mathrm{eq}}\,dt.

Shear viscosity

Let $P_{xy}(t)$ be the off-diagonal stress tensor component. Then

\eta = \frac{1}{k_B T\,V}\int_0^\infty \langle P_{xy}(0)\,P_{xy}(t)\rangle_{\mathrm{eq}}\,dt.

Diffusion constant (Einstein–Green–Kubo form)

For a tagged particle velocity $v(t)$ in $d$ dimensions,

D = \frac{1}{d}\int_0^\infty \langle v(0)\cdot v(t)\rangle_{\mathrm{eq}}\,dt.

Conditions (what is implicitly required)

Stationarity: the equilibrium measure is invariant under the dynamics (e.g., Gibbs equilibrium).
Decay of correlations / mixing: ensures the integral over $t\in[0,\infty)$ converges.
Microreversibility: under time-reversal symmetry (or, in stochastic settings, detailed balance ), one obtains symmetry properties such as Onsager reciprocity.

Relationship to other linear-response statements

Green–Kubo is a concrete evaluation of linear-response coefficients and is closely tied to the fluctuation–dissipation theorem .
In quantum systems, the equilibrium state is often a Gibbs state and correlation/response relations are governed by the KMS condition .