Generalised Equilibrium Reduced-Density Operator

Abstract

Equilibrium reduced-density operators for subsystems of the universe are derived from the generalised Schrödinger variational principle, additionally assuming that the values of von Neumann entropy for the subsystems are fixed and higher than zero. The obtained reduced-density operator may be useful for description of the properties of an arbitrary system (macroscopic, mesoscopic, nanoscopic, or microscopic) IntroductionUsually, when investigating theoretically macroscopic bulk systems, we do not have to take into account their interaction with environment, regarding the short-range character of intermolecular interactions. Any effective interaction among the molecules of the system and its environment could only occur through the atoms on or near the system's surface. The number of interacting atoms is usually an insignificant part of all atoms of the system. However, considering ultrathin films and similar systems, the number of atoms interacting with environment is frequently of the same order of magnitude as the total number of atoms in the system. Moreover, thin films are deposited on bulk substrates whose structure affects that of the films, so the interaction between the substrate and the film has to be taken into account. A common procedure to realise it is to introduce phenomenological surface parameters. The values of these parameters significantly influence physical properties of mesoscopic systems or thin films, which for example is evidenced in their spectra of collective excitations.For such system the postulates of the quantum statistical thermodynamics (QST) are not exactly fulfilled as this theory assumes that the effect of environment can only be a mixing of quantum states in the system studied and it has no effect on the spectrum of eigenstates of its Hamiltonian.In this paper we propose a general method which takes into account the interaction between the nanoscopic or mesoscopic system or thin film and the substrate on a microscopic level. In this method, the interactions are included already in the construction of the reduced density operator. The method proposed is more general than the QST and is an extension of the ideas presented earlier [1][2][3].