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We define a Banach space $\mathcal{M}_{1}$ of models for fermions or quantumspins in the lattice with long range interactions and explicit the structure of(generalized) equilibrium states for any $\mathfrak{m}\in \mathcal{M}_{1}$. Inparticular, we give a first answer to an old open problem in mathematicalphysics - first addressed by Ginibre in 1968 within a different context - aboutthe validity of the so-called Bogoliubov approximation on the level of states.Depending on the model $\mathfrak{m}\in \mathcal{M}_{1}$, our method provides asystematic way to study all its correlation functions and can thus be used toanalyze the physics of long range interactions. Furthermore, we show that thethermodynamics of long range models $\mathfrak{m}\in \mathcal{M}_{1}$ isgoverned by the non-cooperative equilibria of a zero-sum game, called here thethermodynamic game.

Non-cooperative Equilibria of Fermi Systems With Long Range Interactions