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

Intro -- Contents -- Preface -- Part 1 . Main Results and Discussions -- Chapter 1. Fermi Systems on Lattices -- 1.1. Local fermion algebras -- 1.2. States of Fermi systems on lattices -- 1.3. The space-averaging functional Δ_{ } -- 1.4. Local interactions and internal energies -- 1.5. Energy and entropy densities -- Chapter 2. Fermi Systems with Long-Range Interactions -- 2.1. Fermi systems with long-range interactions -- 2.2. Examples of applications -- 2.3. Free-energy densities and existence of thermodynamics -- 2.4. Generalized t.i. equilibrium states -- 2.5. Structure of the set \varOmega_{ }^{♯} of generalized t.i. equilibrium states -- 2.6. Gibbs states versus generalized equilibrium states -- 2.7. Thermodynamics and game theory -- 2.8. Gap equations and effective theories -- 2.9. Long-range interactions and long-range order (LRO) -- 2.10. Concluding remarks -- Part 2 . Complementary Results -- Chapter 3. Periodic Boundary Conditions and Gibbs Equilibrium States -- 3.1. Interaction kernels -- 3.2. Periodic boundary conditions -- 3.3. Pressure and periodic boundary conditions -- 3.4. Gibbs and generalized t.i. equilibrium states -- Chapter 4. The Set _{⃗ℓ} of ⃗ℓ.ℤ^{ }-Invariant States -- 4.1. GNS representation and the von Neumann ergodic theorem -- 4.2. The set ℰ_{⃗ℓ} of extreme states of ℰ_{⃗ℓ} -- 4.3. Properties of the space-averaging functional Δ_{ } -- 4.4. Von Neumann entropy and entropy density of ⃗ℓ-periodic states -- 4.5. The set ₁ as a subset of the dual space ₁* -- 4.6. Well-definiteness of the free-energy densities on _{⃗ℓ} -- Chapter 5. Permutation Invariant Fermi Systems -- 5.1. The set _{Π} of permutation invariant states -- 5.2. Thermodynamics of permutation invariant Fermi systems -- Chapter 6. Analysis of the Pressure via t.i. States -- 6.1. Reduction to discrete finite range models.

6.2. Passivity of Gibbs states and thermodynamics -- Chapter 7. Purely Attractive Long-Range Fermi Systems -- 7.1. Thermodynamics of approximating interactions -- 7.2. Structure of the set \mathit{ }_{ }^{♯}=\varOmega_{ }^{♯} of t.i. equilibrium states -- Chapter 8. The max-min and min-max Variational Problems -- 8.1. Analysis of the conservative values _{ }^{♭} and _{ }^{♯} -- 8.2. _{ }^{♭} and _{ }^{♯} as variational problems over states -- Chapter 9. Bogoliubov Approximation and Effective Theories -- 9.1. Gap equations -- 9.2. Breakdown of effective local theories -- Chapter 10. Appendix -- 10.1. Gibbs equilibrium states -- 10.2. The approximating Hamiltonian method -- 10.3. ℒ^{ }-spaces of maps taking values in a Banach space -- 10.4. Compact convex sets and Choquet simplices -- 10.5. Γ-regularization of real functionals -- 10.6. The Legendre-Fenchel transform and tangent functionals -- 10.7. Two-person zero-sum games -- Bibliography -- Index of Notation -- Index.

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