  We discuss spontaneous symmetry breaking of open classical and quantum
systems. When a continuous symmetry is spontaneously broken in an open system,
a gapless excitation mode appears corresponding to the Nambu-Goldstone mode.
Unlike isolated systems, the gapless mode is not always a propagation mode, but
it is a diffusion one. Using the Ward-Takahashi identity and the effective
action formalism, we establish the Nambu-Goldstone theorem in open systems, and
derive the low-energy coefficients that determine the dispersion relation of
Nambu-Goldstone modes. Using these coefficients, we classify the
Nambu-Goldstone modes into four types: type-A propagation, type-A diffusion,
type-B propagation, and type-B diffusion modes.

High Energy Physics - Theory;Mesoscale and Nanoscale Physics;Quantum Gases;High Energy Physics - Phenomenology