Classical fields near thermal equilibrium

We discuss the classical limit for the long-distance (“soft”) modes of a quantum field when the hard modes of the field are in thermal equilibrium. We address the question of the correct semiclassical dynamics when a momentum cutoff |p|<~k_c≪T is introduced. Higher order contributions lead to a stochastic interpretation for the effective action in analogy with quantum Brownian motion, resulting in dissipation and decoherence for the evolution of the soft modes. Particular emphasis is put on the understanding of dissipation. Our discussion focuses mostly on scalar fields, but we make some remarks on the extension to gauge theories.