#### How to Order

For AMS eBook frontlist subscriptions or backfile collection purchases:

1a. To purchase any ebook backfile or to subscibe to the current year of Contemporary Mathematics, please download this required license agreement,

1b. To subscribe to the current year of Memoirs of the AMS, please download this required license agreement.

2. Complete and sign the license agreement.

3. Email, fax, or send via postal mail to:

Customer Services
American Mathematical Society
201 Charles Street Providence, RI 02904-2294  USA
Phone: 1-800-321-4AMS (4267)
Fax: 1-401-455-4046
Email: cust-serv@ams.org

Visit the AMS Bookstore for individual volume purchases.

# memo_has_moved_text(); 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 2012: Volume 224, Number 1052
ISBNs: 978-0-8218-8976-3 (print); 978-1-4704-1003-2 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-2012-00666-6
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 (2010): Primary 82B10, 91A40; Secondary 46A55, 58E30

View full volume PDF

View other years and numbers:

### Table of Contents

Chapters

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

### Abstract

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.