Shear viscosity of massless classical fields in scalar theory

We investigate the shear viscosity of massless classical scalar fields in the $\phi^4$ theory on a lattice by using the Green–Kubo formula. Based on the scaling property of the classical field, the shear viscosity is represented using a scaling function. The equilibrium expectation value of the time-correlation function of the energy–momentum tensor is evaluated as the ensemble average of the classical field configurations, whose time evolution is obtained by solving the classical equation of motion starting from the initial condition in thermal equilibrium. It is found that there are two distinct damping time scales in the time-correlation function, which is found to show damped oscillation behavior in the early stage around a slow monotonic decay with an exponential form, and the slow decay part is found to dominate the shear viscosity in the massless classical field theory. This kind of slow decay is also known to exist in molecular dynamics simulations, so it may be a generic feature of dense matter.