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Introduction: MDLAB, a Fortran-based code, is designed to analyze extensive molecular dynamics trajectories generated by common simulation software (LAMMPS, VASP, DL-POLY, GPUMD). It calculates a wide range of thermodynamic properties, including mean squared displacements, phonon density of states, pair-distribution functions, atomic probability density functions, Van Hove correlation functions, bond-angle distributions, diffraction patterns, dynamic structure factors (powder-averaged, single-crystal, incoherent, and coherent), quasi-elastic neutron spectra, phonon spectral energy density, and acoustic dispersions in various material states (liquid, amorphous, crystalline). The following section details the mathematical underpinnings of these quantities.