We present a fully microscopic framework for evaluating full quantum work statistics in inhomogeneous many-body systems driven out of equilibrium[1]. The approach combines generalized linear-response theory for quantum work distributions[2] with linear-response thermal time-dependent density functional theory[3], providing first-principles access to the relaxation function governing all cumulants of the dissipated-work distribution in linear response.

The relaxation function is expressed via finite-temperature density-density response functions and decomposed into adiabatic and nonadiabatic contributions. This enables separating irreversibility due to spectral deformations of the instantaneous Hamiltonian from genuine dissipative dynamics associated with nonadiabatic transitions.

The adiabatic component contributes only to the first two moments of the work distribution, while all higher cumulants arise from nonadiabatic processes, consistent with Gaussian statistics in the slow-driving limit. The formalism is implemented in the Kohn-Sham representation of thermal density functional theory and closed using the thermal adiabatic local-density approximation (thALDA). Despite its local and memory-free character, thALDA captures the dominant many-body contributions and yields quantitatively accurate results in the linear-response regime, as confirmed by benchmarking. As an application, we study the nonequilibrium quantum thermodynamics of the 1D Hubbard model under a staggered potential. By computing the first three cumulants of the dissipated-work distribution for finite-temperature finite-time driving protocols, we identify distinct thermodynamic signatures of its rich phase diagram[4].

This framework provides a transferable and scalable route to describe nonequilibrium quantum thermodynamics in correlated systems, offering a platform for quantitative investigation of irreversible processes in complex quantum materials and ultracold atomic systems.

[1] Quantum Sci. Technol. 11 025055 (2026)
[2] PRL.133.070405(2024)
[3] PRL.116, 233001(2016)
[4] PRL.83.2014(1999)