Partition Functions for Statistical Mechanics With MicroPartitions and   Phase Transitions
  The fundamentals of Statistical Mechanics require a fresh definition in the context of the developments in Classical Mechanics of integrable and chaotic systems. This is done with the introduction of Micro Partitions ; a union of disjoint components in phase space. Partition functions including the invariants, Kolmogorov Entropy and Euler number are introduced. The ergodic hypothesis for partial ergodicity is discussed.   In the context of Quantum Mechanics the presence of symmetry groups with irreducible representations gives rise to degenerate and non degenerate spectrum for the Hamiltonian. Quantum Statistical Mechanics is formulated including these two cases ; by including the multiplicity dimension of the group representation and the Casimir invariants into the Partition function. The possibility of new kinds of phase transitions is discussed.   The occurence of systems with non simply connected configuration spaces and Quantum Mechanics for them, also requires a possible generalisation of Statistical Mechanics. The developments of Quantum pure, mixed, and entangled states has made it neccessary to understand the Statistical Mechanics of the multipartite N particle system. And to obtain in terms of the density matrices, written in energy basis, the Trace of the Gibbs operator as the Partition function and use it to get statistical averages of operators. There are some issues of definition, interpretation and application that are discussed. 
Ajay Patwardhan