Boltzmann Equation for Relativistic Neutral Scalar Field in Non-equilibrium Thermo Field Dynamics

TL;DR

This paper derives a relativistic Boltzmann equation for a neutral scalar field within non-equilibrium thermo field dynamics, demonstrating its application through numerical solutions of the distribution function's evolution.

Contribution

It presents a novel derivation of the relativistic Boltzmann equation from the Schwinger-Dyson equation in non-equilibrium thermo field dynamics, specifically for the $^4$ interaction model at 2-loop level.

Findings

Derived the Boltzmann equation for a relativistic scalar field.

Numerically solved the equation showing the distribution function's time evolution.

Analyzed relaxation times in the scalar field system.

Abstract

A relativistic neutral scalar field is investigated on the basis of the Schwinger-Dyson equation in the non-equilibrium thermo field dynamics. A time evolution equation for a distribution function is obtained from a diagonalization condition for the Schwinger-Dyson equation. An explicit expression of the time evolution equation is calculated in the $\lambda\phi^4$ interaction model at the 2-loop level. The Boltzmann equation is derived for the relativistic scalar field. We set a simple initial condition and numerically solve the Boltzmann equation and show the time evolution of the distribution function and the relaxation time.