I am reading a paper, and I came across the Green-Kubo formulation, where the conductivity $\sigma$ of charged particles is related to the time correlation function of the $z$-component of the collective ionic current $J_z(t)$:
$$\sigma_{GK} = \frac{1}{V k_B T} \int_0^{\infty} dt \; C_{JJ}(t)$$
where $C_{JJ}(t) = \langle J_z(0) J_z(t) \rangle$ and the collective current along the $z$ direction is $J_z(t) = \sum_{i=1}^N q_i v_{z, i}(t)$. $V$ is the volume of the system and $q_i$ and $v_{z, i}$ are the charge and $z$-component of the velocity of the $i$th charged particle. $\langle … \rangle$ is an equilibrium ensemble average.
My question is, what is the time correlation function? Is the time correlation function $C_{JJ}(t) = \langle J_z(0) J_z(t) \rangle$? Or is the time correlation function the integral: $\int_0^{\infty} dt \; C_{JJ}(t)$?
Best Answer
Your first answer is the correct one. The temporal auto-correlation function (or the time correlation function) of a fluctuating quantity $A(t)$ is $$C_{AA}(\tau) = \langle A(t) A(t+\tau)\rangle$$ This is a measure of how correlated $A(t)$ is to its value at another time $t+\tau$.
The Green-Kubo formula that connects a response function (the electrical conductivity in your case) to the time-integral of a correlation function (the current correlation function in your case) is a general result of linear response theory.
See this as well.