Dual ensemble and fluctuations for systems with random elements

A double averaging is required for computing expectation values when a sample can be considered as a mixture of components. First there is an average for a given component and then an average over the distribution of components. Different types of fluctuations are then possible. They are not given by expectations of operators in the same functional space. Each fluctuation is naturally described, in the usual canonical form, in a functional space with which a density operator and an entropy are associated. A general formalism to deal with double averaging is given, and the limitations of the reduced density operator and its dynamics are discussed.