  We deal with the power-law q-distribution functions, so-called q-exponentials
in nonextensive statistics. The system considered is a many-body Hamiltonian
system with arbitrary interacting potentials. We find that the usual form of
power-law q-distribution function employed in nonextensive statistics for the
system may be not to stand by the probabilistically independent postulate and
it only represents a dynamical isothermal situation. The probabilistically
independent postulate is validated by the nonextensivity (or pseudoadditivity)
in nonextensive statistics and thus the usual forms of power-law q-distribution
function have to be modified. In this letter, we present a new power-law
q-distribution based on the probabilistically independent postulate, which can
represent nonequilibrium stationary state of the system away from equilibrium
and with the nonextensive energy.
