GreenKubo

GreenKubo, in the context of statistical mechanics, refers to the Green-Kubo relations, a set of integral formulas that relate transport coefficients to time-correlation functions of spontaneous fluctuations in an equilibrium system. Developed within linear response theory and based on the fluctuation-dissipation theorem, these relations connect how a system responds to small perturbations with how its fluctuations decay in equilibrium. They are named after Melville Green and Ryogo Kubo.

The most common Green-Kubo expressions are for viscosity, thermal conductivity, and diffusion. The shear viscosity η is given by η = (V / (k_B T)) ∫_0^∞ dt ⟨P_xy(t) P_xy(0)⟩, where V is the system volume, k_B is Boltzmann’s constant, T the temperature, and P_xy is the off-diagonal component of the microscopic stress tensor. The thermal conductivity κ is κ = (1 / (k_B T^2 V)) ∫_0^∞ dt ⟨J_qx(t) J_qx(0)⟩, with J_q the energy (heat) flux. The self-diffusion coefficient D follows D = (1/3) ∫_0^∞ dt ⟨v_i(t)·v_i(0)⟩ for a given particle i in an isotropic fluid.

In practice, Green-Kubo calculations are performed with equilibrium molecular dynamics by sampling the appropriate flux autocorrelation functions and numerically integrating them to a finite cutoff. Success depends on adequate sampling, proper definitions of fluxes, and the system remaining near equilibrium. These relations provide a foundational, nonperturbative means to predict transport properties from microscopic dynamics across a range of materials and conditions.