Non-cooperative Equilibria of Fermi Systems With Long Range Interactions

We define a Banach space ${\mathcal{M}}_1$ of models for fermions or quantum spins in the lattice with long range interactions and explicit the structure of (generalized) equilibrium states for any $\mathfrak{m}\in {\mathcal{M}}_1$. In particular, we give a first answer to an old open problem in mathematical physics - first addressed by Ginibre in 1968 within a different context - about the 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 a systematic way to study all its correlation functions and can thus be used to analyze the physics of long range interactions. Furthermore, we show that the thermodynamics of long range models $\mathfrak{m}\in {\mathcal{M}}_1$ is governed by the non-cooperative equilibria of a zero-sum game, called here the thermodynamic game.