Since the days of Boltzmann and Gibbs there has been an ongoing
discussion about if and how thermodynamical behavior might be
explained from an underlying theory, such as, e.g., classical Hamiltonian
dynamics. Especially the origen of the second law has always been an
issue , for the microscopical equations of motion (Hamilton equations,
Schrödinger equation) do not feature any attractive fixpoint.
Nevertheless, we find that thermodynamical behavior might be established on the basis of the Schr?dinger equation for systems beeing in contact with some environment. To those ends we transfer some of the traditional ideas about thermodynamical behavior from phase space to Hilbert space. This method differs from the well-known open quantum system approach in so far as neither an equilibrium environment is needed, nor must systems be Markovian.
Within this approach no minimum size or particle number is required for a system to show thermodynamicval behavior.