The Role of Gaussian Domination and Sum Rules in Phase Transitions - An Unpedagogical lntroduction

The purpose o f t h i s paper i s t o show how, from the study o f two m t i v a t i n g examples: the Free Bose Gas and the Spher ica l Model, i t isposs i b l e t a ahs t rac t some general features o f the phase t r a n s i t i o n s associated t o a la rge c lass o f c l a s s i c a l and quantum systems. These models r e f e r t o q u i t e d i f f e r e n t phys ica l s i t u a t i o n s ( t h e Free Bose Gas describes quantum p a r t i c l e s , the Spher ica l Model incorporates some features of c l a s s i ca l ferromagnets o r c l a s s i c a l I a t t i c e gases). In s p i t e o f tha t , there a re s t r i k i n g s i m i l a r i t i e s i n t h a i r behavior, s p e c i a l l y w i t h respect t o the associated phase t r a n s i t i o n s . The comnon features we want t o s t ress are: