Phonion Limits and Macroscopic quasi-particle spectrum for the BCS-model

On the basis of non-commutative central limit theorems, starting from a dynamical system in equilibrium the corresponding macroscopic system of field fluctuations at momentum k is constructed. A mathematical mechanism for the appearance of a macroscopic quasi particle structure is developed in the BCS-model. Due to the spontaneous breaking of symmetry below the critical temperature the resulting dispersion relation shows an energy gap at zero momentum. The BCS-model is used as a toy model for explaining the general theory.