# Non-cooperative Equilibria of Fermi Systems with Long Range Interactions

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*J.-B. Bru; W. de Siqueira Pedra*

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 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 \(\mathfrak{m}\in \mathcal{M}_{1}\), 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 non–cooperative equilibria of a zero–sum game, called here thermodynamic game.

#### Table of Contents

# Table of Contents

## Non-cooperative Equilibria of Fermi Systems with Long Range Interactions

- Preface vii8 free
- Part 1 . Main Results and Discussions 114 free
- Chapter 1. Fermi Systems on Lattices 316
- Chapter 2. Fermi Systems with Long–Range Interactions 1528
- 2.1. Fermi systems with long–range interactions 1730
- 2.2. Examples of applications 1932
- 2.3. Free–energy densities and existence of thermodynamics 2437
- 2.4. Generalized t.i. equilibrium states 2841
- 2.5. Structure of the set \varOmega_{𝔪}^{♯} of generalized t.i. equilibrium states 3144
- 2.6. Gibbs states versus generalized equilibrium states 3548
- 2.7. Thermodynamics and game theory 3851
- 2.8. Gap equations and effective theories 4457
- 2.9. Long–range interactions and long–range order (LRO) 5164
- 2.10. Concluding remarks 5467

- Part 2 . Complementary Results 5972
- Chapter 3. Periodic Boundary Conditions and Gibbs Equilibrium States 6174
- Chapter 4. The Set 𝐸_{⃗ℓ} of ⃗ℓ.ℤ^{𝕕}–Invariant States 6982
- 4.1. GNS representation and the von Neumann ergodic theorem 6982
- 4.2. The set ℰ_{⃗ℓ} of extreme states of ℰ_{⃗ℓ} 7285
- 4.3. Properties of the space–averaging functional Δ_{𝐴} 7689
- 4.4. Von Neumann entropy and entropy density of ⃗ℓ–periodic states 7992
- 4.5. The set 𝐸₁ as a subset of the dual space 𝒲₁* 8194
- 4.6. Well–definiteness of the free–energy densities on 𝐸_{⃗ℓ} 8396

- Chapter 5. Permutation Invariant Fermi Systems 8598
- Chapter 6. Analysis of the Pressure via t.i. States 93106
- Chapter 7. Purely Attractive Long–Range Fermi Systems 101114
- Chapter 8. The max–min and min–max Variational Problems 105118
- Chapter 9. Bogoliubov Approximation and Effective Theories 113126
- Chapter 10. Appendix 121134
- 10.1. Gibbs equilibrium states 121134
- 10.2. The approximating Hamiltonian method 122135
- 10.3. ℒ^{𝓅}–spaces of maps taking values in a Banach space 125138
- 10.4. Compact convex sets and Choquet simplices 126139
- 10.5. Γ–regularization of real functionals 131144
- 10.6. The Legendre–Fenchel transform and tangent functionals 137150
- 10.7. Two–person zero–sum games 139152

- Bibliography 143156
- Index of Notation 147160 free
- Index 153166 free