Memoirs of the American Mathematical Society 2013; 155 pp; softcover Volume: 224 ISBN10: 0821889761 ISBN13: 9780821889763 List Price: US$80 Member Price: US$64 Order Code: MEMO/224/1052
 The authors define a Banach space \(\mathcal{M}_{1}\) of models for fermions or quantum spins in the lattice with long range interactions and make explicit the structure of (generalized) equilibrium states for any \(\mathfrak{m}\in \mathcal{M}_{1}\). In particular, the authors give a first answer to an old open problem in mathematical physicsfirst addressed by Ginibre in 1968 within a different contextabout the validity of the socalled Bogoliubov approximation on the level of states. Depending on the model \(\mathfrak{m}\in \mathcal{M}_{1}\), the authors' method provides a systematic way to study all its correlation functions at equilibrium and can thus be used to analyze the physics of long range interactions. Furthermore, the authors show that the thermodynamics of long range models \(\mathfrak{m}\in \mathcal{M}_{1}\) is governed by the noncooperative equilibria of a zerosum game, called here thermodynamic game. Table of Contents Part 1. Main Results and Discussions  Fermi systems on lattices
 Fermi systems with longrange interactions
Part 2. Complementary Results  Periodic boundary conditions and Gibbs equilibrium states
 The set \(E_{\vec{\ell}}\) of \(\vec{\ell}.\mathbb{Z}^{d}\)invariant states
 Permutation invariant Fermi systems
 Analysis of the pressure via t.i. states
 Purely attractive longrange Fermi systems
 The maxmin and minmax variational problems
 Bogoliubov approximation and effective theories
 Appendix
 Bibliography
 Index of notation
 Index
