Thermodynamic restrictions on statistics of molecular random walks
  It is shown that time reversibility of Hamiltonian microscopic dynamics and Gibbs canonical statistical ensemble of initial conditions for it together produce an exact virial expansion for probability distribution of path of molecular Brownian particle in a fluid. This expansion leads to inequality connecting logarithmic derivative of the distribution with respect to fluid density and characteristic volume occupied by pair statistical correlation between the path and fluid molecules. Due to the inequality, finiteness of this volume means that asymptotic of the distribution is essentially non-Gaussian. For principal case when fluid is dilute gas it is demonstrated that the pair correlation volume is actually finite and bounded above. Therefore even under the Boltzmann-Grad limit the path distribution possess power-law long tails (cut off at distances of ballistic flight). 
Yu.E.Kuzovlev