Narrow Results

We review our approach to the second law of thermodynamics as a theorem asserting the growth of the mean (Gibbs–von Neumann) entropy of quantum spin systems undergoing automorphic (unitary) adiabatic transformations. Non-automorphic interactions with the environment, although known to produce on the average a strict reduction of the entropy of systems with finite number of degrees of freedom, are proved to conserve the mean entropy on the average. The results depend crucially on two properties of the mean entropy, proved by Robinson and Ruelle for classical systems and Lanford and Robinson for quantum lattice systems: upper semicontinuity and affinity.

This paper deals with the violation or retention of symmetries associated with the self-adjoint extension of the Hamiltonian for homogeneous but anisotropic Bianchi I cosmological model. This extension is required to make sure the quantum evolution is unitary. It is found that the scale invariance is lost, but the Noether symmetries are preserved.