Entropy production from chaoticity in Yang-Mills field theory with use of the Husimi function

Abstract

We investigate possible entropy production in Yang-Mills (YM) field theory by using a quantum distribution function called the Husimi function ${f}_{\mathrm{H}}(A,E,t)$ for the YM field, which is given by a coarse graining of the Wigner function and non-negative. We calculate the Husimi-Wehrl entropy ${S}_{\mathrm{HW}}(t)=\ensuremath{-}\mathrm{Tr}{f}_{\mathrm{H}}\mathrm{log}{f}_{\mathrm{H}}$ defined as an integral over the phase space, for which two adaptations of the test-particle method are used combined with Monte Carlo method. We utilize the semiclassical approximation to obtain the time evolution of the distribution functions of the YM field, which is known to show chaotic behavior in the classical limit. We also make a simplification of the multidimensional phase-space integrals by making a product ansatz for the Husimi function, which is found to give a 10--20% overestimate of the Husimi-Wehrl entropy for a quantum system with a few degrees of freedom. We show that the quantum YM theory does exhibit the entropy production and that the entropy production rate agrees with the sum of positive Lyapunov exponents or the Kolmogorov-Sinai entropy, suggesting that the chaoticity of the classical YM field causes the entropy production in the quantum YM theory.