Derivation of Liouville-like equations for the n-state probability density of an open system with thermalized particle reservoirs and its link to molecular simulation

Abstract

A physico-mathematical model of open {systems} {proposed in a previous paper [ L.Delle Site and R.Klein, J.Math.Phys. 61, 083102 (2020)] can represent a guiding reference in designing an accurate simulation scheme for an open molecular system embedded in a reservoir of energy and particles. The derived equations and the corresponding boundary conditions are obtained without assuming the action of an external source of heat that assures thermodynamic consistency of the open system with respect to a state of reference. However, in numerical schemes the temperature in the reservoir must be controlled by an external heat bath otherwise thermodynamic consistency cannot be achieved. In this perspective, the question to address is whether the explicit addition of an external heat bath in the theoretical model modifies the equations of the open system and its boundary conditions. In this work we consider this aspect and explicitly describe the evolution of the reservoir employing the Bergmann-Lebowitz statistical model of thermostat. It is shown that the resulting equations for the open system itself are not affected by this change and an example of numerical application is reviewed where the current result shows its conceptual relevance.} Finally, a list of pending mathematical and modelling problems is discussed the solution of which would strengthen the mathematical rigour of the model and offer new perspectives for the further development of a new multiscale simulation scheme.

An effective Hamiltonian for the simulation of open quantum molecular systems

“…Since n is a variable quantity it follows that each ρn represents one of the possible statistical realizations; in the limit of stationary equilibrium we recover a known solution, that is the grand canonical density matrix as it would be expected for naturally fluctuating open system in equilibrium. The procedure employed for the derivation of the hierarchy of equations is similar to the procedure employed for a system of classical particles [16,17]; such a procedure was used as a theoretical platform to design algorithms of (classical) molecular simulation [22]. We have pointed out that our derivation can be also considered as the explicit formal derivation of a result that was empirically conjectured within the field of superconductivity and that, in addition, now can be extended to the larger field of quantum molecular simulations.…”

“…In the following, instead of a pure dynamical view, as done for example in the derivation of the Lindblad equation, we consider a statistical view, that is consider a partitioning of the total system with N particles in a subsystem S with n particles and a reservoir B with N − n particles, as done for the Liouville equation in the case of classical systems [16,17]. Next we trace out the degrees of freedom of B in the Von Neumann equation, considering all the partitioning of N particles in n and N − n sets.…”

“…This field is, in our view, still in its infancy, since most of the current open-system ab initio algorithms are built on empirical basis and thus are often numerically not reliable [14] due to conceptual inconsistencies that lead to incorrect statistical sampling [1,15]. This means that the construction of a solid theoretical framework would assure physical consistency and computational robustness to the underlying numerical methods, as it occurred for molecular dynamics of classical and semi-classical open systems with a system-reservoir exchange of particles [16][17][18][19][20].…”

“…Next, having in mind systems typically treated in molecular simulations, we integrate out the degrees of freedom of the bath region and derive an n-hierarchy of equations for the density matrix of the subsystem of interest, ρn . This procedure is conceptually equivalent to the procedure followed for the Liouville equation in the classical case [16,17] developed within the framework of molecular dynamics [22]. The novelty of our proposal lies in the explicit (step-by-step) formal derivation of the equation from first principles, having in mind a molecular system for numerical simulations.…”

Coupled Multiphysics Modelling of Sensors for Chemical, Biomedical, and Environmental Applications with Focus on Smart Materials and Low-Dimensional Nanostructures

“…As mentioned, the primary step in the approach advocated here is the density operator, where the density matrix is its possible representation that allows to describe the state of a quantum system and to calculate the probabilities of measurement outcomes carried out upon the system, once the Born rules are imposed. The approach to the analysis of open systems based on the quantum Liouville model continues to be under active developments [ 25 , 26 ] with advanced applications stimulating further progress in phase space formulations [ 27 ], including the Wigner–Weyl formulation as highlighted above.…”