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Non–cooperative equilibria of Fermi systems with long range interactions

About this Title

J.-B. Bru, Departamento de Matemáticas, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain, and IKERBASQUE, Basque Foundation for Science, 48011, Bilbao, Spain and W. de Siqueira Pedra, Institut für Mathematik, Universität Mainz, Staudingerweg 9, 55099 Mainz, Germany

Publication: Memoirs of the American Mathematical Society
Publication Year: 2013; Volume 224, Number 1052
ISBNs: 978-0-8218-8976-3 (print); 978-1-4704-1003-2 (online)
Published electronically: October 18, 2013
Previous version: Posted November 14, 2012
Corrected version: Corrects some unlinked Definitions in the original version
Keywords:Long range interaction, Choquet theorem, equilibrium state, non–cooperative equilibrium, two–person zero–sum game, Bogoliubov approximation, approximating Hamiltonian method, fermion system, quantum spin system
MSC: Primary 82B10, 91A40; Secondary 46A55, 58E30

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Table of Contents


  • Preface
  • Part 1. Main results and discussions
  • Part 2. Complementary results


We 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, 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 at equilibrium 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 thermodynamic game.

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