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This study overviews and extends a recently developed stochasticfinite-temperature Kohn-Sham density functional theory to study warm densematter using Langevin dynamics, specifically under periodic boundaryconditions. The method's algorithmic complexity exhibits nearly linear scalingwith system size and is inversely proportional to the temperature.Additionally, a novel linear-scaling stochastic approach is introduced toassess the Kubo-Greenwood conductivity, demonstrating exceptional stability forDC conductivity. Utilizing the developed tools, we investigate the equation ofstate, radial distribution, and electronic conductivity of Hydrogen at atemperature of 30,000K. As for the radial distribution functions, we reveal atransition of Hydrogen from gas-like to liquid-like behavior as its densityexceeds $4 g/cm^3$. As for the electronic conductivity as a function of thedensity, we identified a remarkable isosbestic point at frequencies around 7eV,which may be an additional signature of a gas-liquid transition in Hydrogen at30,000K.

Stochastic density functional theory combined with Langevin dynamics for warm dense matter