An anharmonic liquid-entropy functional from the Mori-Zwanzig memory kernel

We utilize the Mori-Zwanzig memory kernel to separate the density of states of a liquid into gas, cage, and harmonic-solid components. The cage entropy is computed self-consistently as the non-Markovian excess of the full velocity response over its Markovian counterpart, and is assigned an excluded-volume entropy reference rather than a harmonic one. Employing physically motivated constraints, the resulting Three-Phase Explicit Anharmonic Thermodynamics (3PT) method reproduces thermodynamic-integration entropies of liquid metals and aligns two water models with independent free-energy perturbation benchmarks. For monoatomic Lennard-Jones liquids, 3PT's accuracy follows a universal efficiency curve governed by the kernel's non-Markovianity. We thus establish a direct, trajectory-level mapping between memory, transient cage dynamics, and liquid entropy.