  We study the dynamics of a quantum-coherent thermally isolated Luttinger
liquid with noisy Luttinger parameter. To characterize the fluctuations of the
absorbed energy in generic noise-driven systems, we first identify two types of
energy moments, which can help tease apart the effects of classical
(sample-to-sample) and quantum sources of fluctuations. One type of moment
captures the total fluctuations due to both sources, while the other one
captures the effect of the classical source only. We then demonstrate that in
the Luttinger liquid case, the two types of moments agree in the thermodynamic
limit, indicating that the classical source dominates. In contrast to
equilibrium thermodynamics, in this driven system the relative fluctuations of
energy do not decay with the system size. Additionally, we study the deviations
of equal-time correlation functions from their ground-state value, and find a
simple scaling behavior.
