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