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I study the statistical description of a small quantum system, which iscoupled to a large quantum environment in a generic form and with a genericinteraction strength, when the total system lies in an equilibrium statedescribed by a microcanonical ensemble. The focus is on the difference betweenthe reduced density matrix (RDM) of the central system in this interacting caseand the RDM obtained in the uncoupled case. In the eigenbasis of the centralsystem's Hamiltonian, it is shown that the difference between diagonal elementsis mainly confined by the ratio of the maximum width of the eigenfunctions ofthe total system in the uncoupled basis to the width of the microcanonicalenergy shell; meanwhile, the difference between off-diagonal elements is givenby the ratio of certain property of the interaction Hamiltonian to the relatedlevel spacing of the central system. As an application, a sufficient conditionis given, under which the RDM may have a canonical Gibbs form undersystem-environment interactions that are not necessarily weak; this Gibbs stateusually includes certain averaged effect of the interaction. For centralsystems that interact locally with many-body quantum chaotic systems, it isshown that the RDM usually has a Gibbs form. I also study the RDM which iscomputed from a typical state of the total system within an energy shell.

Closeness of the reduced density matrix of an interacting small system to the Gibbs state