Entropy and canonical ensemble of hybrid quantum classical systems

Abstract

In this work we generalize and combine Gibbs and von Neumann approaches to build, for the first time, a rigorous definition of entropy for hybrid quantum-classical systems. The resulting function coincides with the two cases above when the suitable limits are considered. Then, we apply the MaxEnt principle for this hybrid entropy function and obtain the natural candidate for the hybrid canonical ensemble (HCE). We prove that the suitable classical and quantum limits of the HCE coincide with the usual classical and quantum canonical ensembles since the whole scheme admits both limits, thus showing that the MaxEnt principle is applicable and consistent for hybrid systems.

Hybrid Koopman C∗ –formalism and the hybrid quantum–classical master equation *

“…An important application of this result is the possibility to relate the von Neumann entropy associated with the state ρH and the hybrid entropy function introduced in [3]. Indeed, our group introduced a hybrid entropy function for states of the form ρ(ξ), based on the analysis of mutually exclusive hybrid events, which reads:…”

“…This is a natural way to define a consistent statistical mechanical system leading to a well defined thermodynamics (see [7]). This statistical description has relevant applications [4,6] but also some limitations, as the difficulty to write an entropy function and the corresponding notion of canonical ensemble [3] or, in more general terms, of an equilibrium thermodynamics. From the mathematical point of view, these limitations are associated with the incompatibility of the notion of hybrid state as a probability density on the phase space and the definition of hybrid entropy, which require an alternative notion of state.…”

About the computation of finite temperature ensemble averages of hybrid quantum-classical systems with molecular dynamics

“…The MD procedure (and in particular the thermostats) was designed to produce a phase space visitation consistent with this. In the hybrid phase space, due to the quantum character of one of its parts, not all distinct points are mutually exclusive events, 36 and therefore the ensemble targetted by the thermostats, determined by a Boltzmann weight over the phase space, does not match the HC ensemble.…”

“…Ensembles of hybrid quantum-classical systems can be described [34][35][36] by ξ-dependent density matrices, ρ(ξ), normalized as:…”

First-principles modelling for time-resolved ARPES under different pump-probe conditions

“…This allows to describe the effects of coherent phonon excitations on the tr-ARPES spectrum 22 . In such a quantum-classical coupled system it is not possible to accurately describe the exchange of energy between the lattice and electrons [30][31][32] , hence processes like thermalization and decoherence can not be described in this approach. Nevertheless, decoherence effects can be accounted for in TDDFT by phenomenological scattering times and coupling to an density matrix 33 , while the coupling of quantum nuclear dynamics to a first-principles approach is under development [34][35][36][37] .…”