Statistical mechanics of a collisionless system based on the maximum entropy principle

The statistical equilibrium state of a collisionless system has been derived based on the maximum entropy principle. The result shows that the equilibrium distribution function is a Gaussian distribution in the nondegenerate limit. This result contradicts the Lynden-Bell statistics, which predict a superposition of Fermi distributions as the equilibrium distribution. It is shown that the Lynden-Bell statistics are equivalent to maximizing the entropy calculated from the probability of particle transition, whereas the probability of particle existence should be used to obtain the correct answer.