Molecular random walk and a symmetry group of the Bogoliubov equation

We consider the statistics of molecular random walks in fluids using the Bogoliubov equation for the generating functional of the distribution functions. We obtain the symmetry group of this equation and its solutions as functions of the medium density. It induces a series of exact relations between the probability distribution of the total path of a walking test particle and its correlations with the environment and consequently imposes serious constraints on the possible form of the path distribution. In particular, the Gaussian asymptotic form of the distribution is definitely forbidden (even for the Boltzmann–Grad gas), but the diffusive asymptotic form with power-law tails (cut off by the ballistic flight length) is allowed.