Evolution of Quantum Phase Space Distribution： a Trajectory-Density Approach

Corey Jason Trahan, Róbert E. Wyatt

Nuno Costa Dias, João Nuno Prata

Hai-Woong Lee

Corey Jason Trahan, Róbert E. Wyatt

Recently, Donoso and Martens described a method for evolving both classical and quantum phase-space distribution functions, W(q,p,t), that involves the propagation of an ensemble of correlated trajectories. The trajectories are linked into a unified whole by spatial and momentum derivatives of density dependent terms in the equations of motion. On each time step, these nonlocal terms were evaluated by fitting the density around each trajectory to an assumed functional form. In the present study, we develop a different trajectory method for propagating phase-space distribution functions. A hierarchy of coupled analytic equations of motion are derived for the q and p derivatives of the density and a truncated set of these are integrated along each trajectory concurrently with the equation of motion for the density. The advantage of this approach is that individual trajectories can be propagated, one at a time, and function fitting is not required to evaluate the nonlocal terms. Regional nonlocality can be incorporated at various levels of approximation to “dress” what would otherwise be “thin” locally propagating trajectories. This derivative propagation method is used to obtain trajectory solutions for the Klein–Kramers equation, the Husimi equation, and for a smoothed version of the Caldeira–Leggett equation derived by the Diosi. Trajectory solutions are obtained for the relaxation of an oscillator in contact with a thermal bath and for the decay of a metastable state.

Xuefeng Zhang, Zheng Yu-Jun

The trajectory-density method of a quantum system is developed by using local Koopman and Frobenius–Perron operators. We propose a new scheme of approximation from two sets of trajectory-density mixed equations. By examining the local generation and termination of trajectories, we show how they can be adopted to the propagation of negative values of the Wigner function even if it starts off positive everywhere.

Fangyuan Wang, Yingxun Zhang, Z. P. Li

In this work, we derived a transport equation based on a generalized equation of time-dependent generator coordinate method (TDGCM) under the Gaussian overlap approximation (GOA). The transport equation is obtained by using quantum-mechanics phase space distributions under a ``quasi-particle" picture and strategy of Bogoliubov-Born-Green-Kirkood-Yvon (BBGKY) hierarchy. The theoretical advantage of this transport equation is that time evolution of $s$-body phase space density distribution is coupled with $s+1$-body phase space density distributions, and thus, non-adiabatic effects and dynamical fluctuations could be involved by more collective degrees and entanglement of phase space trajectories. In future, we will perform the numerical calculations for fission nuclei after obtaining collective inertia and potential energy surface (PES).