Quantal methods for classical dynamics in Liouville space

S. Twareque Ali, Eduard Prugovečki

Dipak K. Saraswati, Hans Primas

A system theoretic representation of mechanical systems is proposed. It is shown that mechanical systems, classical or quantal, described by the Liouville equation, can be realized as a stationary linear input–output system. This state space representation of mechanical systems is then extended to decompose a mechanical system into a hierarchy of orthogonal stationary linear dynamical systems.

Sara Bonella, Giovanni Ciccotti, Raymond Kapral

C. Syros, George P. Flessas

Irène Burghardt, Gérard Parlant

A new approach to the coupling of quantum and classical dynamics is developed, by combining a hydrodynamic, Bohmian description for the quantum subsystem with a Liouville-space description for the classical subsystem. To this end, partial hydrodynamic moments are introduced, the dynamics of which is determined by a hierarchy of equations derived from the quantum Liouville equation. We focus on pure states (wave functions) and introduce a trajectory representation in a hybrid hydrodynamic-Liouvillian phase space. The interleaved trajectory dynamics is guided by a new type of quantum force. For illustration, we consider a pair of bilinearly coupled harmonic oscillators, for which the method is exact.

Meghan E. Halse, Jean‐Nicolas Dumez, Lyndon Emsley

The quasi-equilibrium behaviour of isolated nuclear spin systems in full and reduced Liouville spaces is discussed. We focus in particular on the reduced Liouville spaces used in the low-order correlations in Liouville space (LCL) simulation method, a restricted-spin-space approach to efficiently modelling the dynamics of large networks of strongly coupled spins. General numerical methods for the calculation of quasi-equilibrium expectation values of observables in Liouville space are presented. In particular, we treat the cases of a time-independent Hamiltonian, a time-periodic Hamiltonian (with and without stroboscopic sampling) and powder averaging. These quasi-equilibrium calculation methods are applied to the example case of spin diffusion in solid-state nuclear magnetic resonance. We show that there are marked differences between the quasi-equilibrium behaviour of spin systems in the full and reduced spaces. These differences are particularly interesting in the time-periodic-Hamiltonian case, where simulations carried out in the reduced space demonstrate ergodic behaviour even for small spins systems (as few as five homonuclei). The implications of this ergodic property on the success of the LCL method in modelling the dynamics of spin diffusion in magic-angle spinning experiments of powders is discussed.