  We study nonequilibrium thermodynamic properties of a driven one-dimensional
quantum fluid by combining nonlinear Luttinger liquid theory with the quantum
kinetic equation. In particular, we derive an entropy production consistent
with the laws of thermodynamics for a system subject to an arbitrary
perturbation varying slowly in space and time. Working in a basis of weakly
interacting fermionic quasiparticles, we show that the leading contribution to
the entropy production results from three-particle collisions, and we derive
its scaling law at low temperatures.
