Non-Equilibrium Liouville and Wigner Equations: Moment Methods and Long-Time Approximations
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 W c,eq , out of the set of classical stationary distributions, W c,st ,
... ributions, W c,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 . W eq 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 W eq (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 W eq ) 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.