Green–Kubo formula for Boltzmann and Fermi–Dirac statistics

Shear viscosity of nuclear matter is extracted via the Green–Kubo formula and the Gaussian thermostated SLLOD algorithm (the shear rate method) in a periodic box by using an improved quantum molecular dynamic (ImQMD) model without mean field, also it is calculated by a Boltzmann-type equation. Here a new form of the Green–Kubo formula is put forward in the present work. For classical limit at nuclear matter densities of $$0.4\rho _{0}$$ and $$1.0\rho _{0}$$ , shear viscosity by the traditional and new form of the Green–Kubo formula as well as the SLLOD algorithm are coincident with each other. However, for non-classical limit, shear viscosity by the traditional form of the Green–Kubo formula is higher than those obtained by the new form of the Green–Kubo formula as well as the SLLOD algorithm especially in low temperature region. In addition, shear viscosity from the Boltzmann-type equation is found to be less than that by the Green–Kubo method or the SLLOD algorithm for both classical and non-classical limits.