Wigner's [E. P. Wigner, Phys. Rev., 1932, 40, 749] representation of the density operator as a c-number quasiprobablity distribution in phase space allowing quantum mechanical averages involving the density matrix to be calculated as phase space averages just as classical averages originally used by him to obtain quantum mechanical corrections to classical thermodynamic equilibrium i.e. to the Maxwell-Boltzmann distribution so applying to closed quantum systems is extended to open quantum systems comprising a canonical ensemble of Brownian particles in a potential. This is accomplished via an idea of Gross and Lebowitz [E. P. Gross and J. L. Lebowitz, Phys. Rev. 1956, 104, 1528]. They suggested that using Wigner's representation the connection between classical and quantum collision kernels, (i.e. in classical mechanics the Stosszahlanzatz describing the bath-particle interaction in the open system in the Boltzmann equation for the single particle distribut...