On statistics and 1/f noise of Brownian motion in Boltzmann-Grad gas and finite gas on torus. I. Infinite gas

An attempt is made to compare statistical properties of self-diffusion of particles constituting gases in infinite volume and on torus. In this first part, equations are derived which represent roughened but solvable variant of the collisional approximation to exact BBGKY equations. With their help, statistics of Brownian motion in infinite gas is considered, under the Boltzmann-Grad limit, and shown to be essentially non-Gaussian, involving 1/f fluctuations in diffusivity.