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 (Delle Site and Klein 2020 J. Math. Phys. 61 083102) 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.