We treat the non-equilibrium evolution of an open one-particle statistical system, subject to a potential and to an external “heat bath” (

*hb*) with negligible dissipation. For the classical equilibrium Boltzmann distribution,

*W*_{c,eq}, a non-equilibrium three-term hierarchy for moments

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We treat the non-equilibrium evolution of an open one-particle statistical system, subject to a potential and to an external “heat bath” (

*hb*) with negligible dissipation. For the classical equilibrium Boltzmann distribution,

*W*_{c,eq}, a non-equilibrium three-term hierarchy for moments fulfills Hermiticity, which allows one to justify an approximate long-time thermalization. That gives partial dynamical support to Boltzmann’s

*Wc*_{,eq}, out of the set of classical stationary distributions, Wc;st, also investigated here, for which neither Hermiticity nor that thermalization hold, in general. For closed classical many-particle systems without

*hb* (by using W

_{c,eq}), the long-time approximate thermalization for three-term hierarchies is justified and yields an approximate Lyapunov function and an arrow of time. The largest part of the work treats an open quantum one-particle system through the non-equilibrium Wigner function, W. Weq for a repulsive finite square well is reported.

*W*’s (< 0 in various cases) are assumed to be quasi-definite functionals regarding their dependences on momentum (

*q*). That yields orthogonal polynomials,

*H*_{Q,n}(

*q*), for Weq (and for stationary

*W*_{st}), non-equilibrium moments,

*W*_{n}, of

*W* and hierarchies. For the first excited state of the harmonic oscillator, its stationary

*W*_{st} is a quasi-definite functional, and the orthogonal polynomials and three-term hierarchy are studied. In general, the non-equilibrium quantum hierarchies (associated with Weq) for the

*W*_{n}’s are not three-term ones. As an illustration, we outline a non-equilibrium four-term hierarchy and its solution in terms of generalized operator continued fractions. Such structures also allow one to formulate long-time approximations, but make it more difficult to justify thermalization. For large thermal and de Broglie wavelengths, the dominant

*W*_{eq} and a non-equilibrium equation for W are reported: the non-equilibrium hierarchy could plausibly be a three-term one and possibly not far from Gaussian, and thermalization could possibly be justified.

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