Study of sphaleron transitions by means of the real-time Langevin equation

Microscopic Langevin evolution is introduced in Abelian gauge field theory. It is studied numerically in the (1 + 1)-dimensional Abelian Higgs model at finite temperature. The Brownian behavior of the Chern-Simons number is established. Sharp transitions between the classical vacua with different Chern-Simons numbers are observed. The rate of these transitions depends exponentially on the temperature, indicating the relevance of the thermal activation theory involving the sphaleron saddle point.