# Shear viscosity of classical fields in scalar theory

## TL;DR

This paper calculates the shear viscosity of massless classical scalar fields in $^4$ theory using the Green-Kubo formula, revealing a slow decay in the correlation function that dominates viscosity and may be a universal feature of dense matter.

## Contribution

It introduces a scaling function approach to shear viscosity in classical scalar fields and identifies a slow decay process influencing viscosity, connecting to molecular dynamics findings.

## Key findings

- Two distinct damping time scales in correlation functions.
- Slow decay dominates shear viscosity in the massless classical field.
- Presence of damped oscillations in early correlation behavior.

## Abstract

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. 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 monotonous 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 the molecular dynamics simulation, then it may be a generic feature of dense matter.