Official webpage of this course at the University of São Paulo. (in Portuguese)
Classes: Tuesday 16:00 - 18:00
Thursday 16:00 - 18:00
Room 267 Bloco A
This is an introductory course, we will try cover the following topics:
- Graphs and amenability.
- Canonical and Grand Canonical Ensembles.
- Lattice Gases.
- Ising Type Models: Short and Long range, with and without external fields, random external fields, correlations, Phase Transitions, Pressure, Thermodynamic Limits.
- Correlations Inequalities: FKG, Griffiths inequalities, GKS, GHS and Wells’ inequality.
- Gibbs Measures: Dobrushin's criterion for uniqueness, Phase Transitions and Ground states.
- Cluster Expansions: The Peierls Argument, Polymer Gases, Pirogov-Sinai and Kirkwood-Salzburg equations.
- Renormalization Group.
Some references:
Books, thesis and lectures notes:
- Statistical Mechanics A Short Treatise. Giovanni Gallavotti. Springer Verlag, (1999).
- Statistical Mechanics of Disorder Systems - A Mathematical Perspective. Anton Bovier. Cambridge Series in Statistical and Probabilistic Mathematics, (2006).
- Lectures notes: Gibbs measures and phase transitions. Anton Bovier. Part 1, Part 2.
- Statistical Mechanics: Rigorous Results. David Ruelle. World Scientific, (1999).
- Gibbs Measures and Phase Transitions(Second Edition). Hans-Otto Georgii. De Gruyter Studies in Mathematics; 9. Walter de Gruyter & Co; (2011).
- The statistical mechanics of lattice gases. B. Simon. Vol. I. Princeton Series in Physics. Princeton University Press, Princeton, NJ, (1993).
- Introduction to (generalized) Gibbs Measures. Arnaud Le Ny. Ensaios Matemáticos. Vol. 15, 1-126. SBM, (2008).
- Quantum Physics: A Functional Integral Point of View. J. Glimm & A. Jaffe. Springer-Verlag, (1981).
- Pirogov-Sinai Theory and Singularity of the Ising Model on Zˆ2. (clique em textos) Sacha Friedli, (2005).
- Entropy, Large Deviations, and Statistical Mechanics. Richard S. Ellis. Classics in Mathematics. Springer-Verlag, (2006).
- Entropy and equilibrium states in Classical Statistical Mechanics. O.E. Lanford. Lecture notes in Physics, vol. 20, Springer-Verlag, (1973).
- The Theory of Large Deviations and Applications to Statistical Mechanics. Les Houches 2008 Session XC. Richard S. Ellis. Oxford University Press, (2010).
- A Course on Large Deviations with an Introduction to Gibbs Measures. Firas Rassoul-Agha and Timo Seppalainen, (2010).
- Interacting particle systems on graphs. Nazim Hikmet Tekmen. PhD thesis, Bielefeld University, (2010).
- Properties and applications of Bernoulli random Fields with strong dependency graphs. Christoph Temmel. PhD thesis, (2012).
- Le modèle d'Ising. Yvan Velenik, (2009).
- Convexity in the Theory of Lattice Gases. Robert B. Israel. Princeton University Press, (1979).
- Introdução a Teoria das Medidas de Gibbs. Rodrigo Bissacot e Leandro Cioletti. Notas de aula, (2012).
- Cluster expansion methods in rigorous statistical mechanics. Aldo Procacci, (2005).
- Técnicas para convergência da Expansão do Gás de Polímeros e uma aplicação ao Método Probabilístico. Rodrigo Bissacot. PhD thesis, UFMG, (2009).
- O Teorema de Lee-Yang. Paulo Cupertino, (2004).
- On the Non-Analytic Behaviour of Thermodynamic Potentials at First Order Phase Transitions. Sacha Friedli, PhD thesis, EPFL, (2003).
- O Lema Local De Lovász: do Método Mágico de Erdös à teoria dos gases de rede. Rodrigo Bissacot II Colóquio de Matemática da região Sul., (2012).
Papers:
- Absence of First-Order Phase Transitions for Antiferromagnetic Systems. David Klein and Wei-Shih Yang. JSP. vol 70, (1993).
- Phase Transitions with Four-Spin Interactions. Joel L. Lebowitz and David Ruelle. CMP. vol. 304, 711–722 (2011).
- A Finite-Volume Version of Aizenman-Higuchi Theorem for the 2d Ising model. L. Coquille and Y. Velenik. Probab. Theory Relat. Fields 153, 25-44, (2012).
- Translation invariant Gibbs states for the Ising model. T. Bodineau. Probab. Theory Relat. Fields 135, 153-168, (2006).
- Limit Theorems and Coexistence Probabilities for the Curie-Weiss Potts Model with an external field. D. Gandolfo, J. Ruiz, M. Wouts SPTA-120, 84–104, (2010).
- Caracterização da Fase Desordenada do Modelo de Ising d-dimensional via Desigualdades de Correlações . G. Braga e F. Fontenele. Mat. Univ., 9-37, (2002).
- O Limite Termodinâmico e Independência das Condições de Contorno para o Modelo de Ising d-Dimensional. G. Braga e F. Fontenele. Mat. Univ, 101-125, (2001).
- O Teorema de Perron-Frobenius e Ausência de Transição de Fase em Modelos Unidimensionais da Mecânica Estatística . G. Braga e M. R. Hilário (2005).
- The repulsive lattice gas, the independent-set polynomial, and the Lovász local lemma. A. Scott and A. Sokal, J. Stat. Phys. 118, no. 5-6, 1151–1261, (2005).
- The Chemical Potential. T. A. Kaplan, Journal of Statistical Physics 122, n◦ 6, (2006).
- Gibbsianness and Non-Gibbsianess in Lattice Random Fields. R. Fernández. Les Houches, Session LXXXIII 2005 Mathematical Statistical Physics, (2006).
- Regularity properties and pathologies of position-space renormalization-group transformations: Scope and limitations of Gibbsian theory. Aernout C. D. van Enter, Roberto Fernández, Alan D. Sokal. J. Stat. Phys. 72, 879-1167, (1993).
- The phase transition in a general class of Ising-type models is sharp. Michael Aizenman, David Barsky, Roberto Fernández. J. Stat. Phys. 47, 343-74, (1987).
- The Phase Transition of the Quantum Ising Model is Sharp. J. E. Björnberg and G. R. Grimmett. J. Stat. Phys. 136, 231-273, (2009).
- The characterization of ground states. Jean Bellissard, Charles Radin and Senya Shlosman. J. Phys. A: Math. Theor. 43 (2010).
- Description of Some Ground States by Puiseux Techniques. E. Garibaldi and P. Thieullen. Journal of Statistical Physics, v. 146, p. 125-180, (2012).
- Percolation on infinite graphs and isoperimetric inequalities. R. G. Alves, A. Procacci and R. Sanchis. J. Stat. Phys, v. 149, p. 831-845, (2012).
- Amenability and Phase Transition in the Ising Model. Johan Jonasson and Jeffrey E. Steif. J. of Theoretical Probability v. 12, Number 2, (1999).
- Absence of phase transitions in a class of integer spin systems. T. Morais and A. Procacci. J. Stat. Phys, v. 136, 677-684, (2009).
- Counting Regions with Bounded Surface Area. P. N. Balister, B. Bollobás. Commun. Math. Phys. 273, 305–315 (2007).
- A Characterization of First-Order PhaseTransitions for Superstable Interactions in Classical Statistical Mechanics. D. Kleinaand and W.-S. Yang. J. Stat. Phys. 71, 1043-1062, (1993).
- Critical Temperature of Periodic Ising Models. Z. Li. CMP 15, (2012).
- The Random Geometry of equilibrium phases. H-O Georgii, O. Hagstrom and C. Maes. In: C. Domb and J.L. Lebowitz (eds.) Phase Transitions and Critical Phenomena Vol. 18, (2000).
- A Manifold of Pure Gibbs States of the Ising Model on a Tree. D. Gandolfo, J. Ruiz and S. Shlosman. J. Stat. Phys. v. 148, Issue 6, 999-1005 (2012).
- Renormalization Group Maps for Ising Models in Lattice-Gas Variables. T. Kennedy. J. Stat. Phys. v. 140, N. 3, 409-426, (2010).
- A Constructive Description of Ground States and Gibbs Measures for Ising Model with Two-Step Interactions on Cayley Tree. U.A. Rozikov, J. Stat. Phys. v. 122, n. 2, , 217-235, (2006).
- Interface free energy or surface tension: definition and basic properties. C.-E. Pfister (2009).
- Differentiability Properties of the Pressure in Lattice Systems. H.A. Daniels and A.C.D. van Enter. CMP 71, (1980).
- Generic Triviality of Phase Diagrams in Space of Long-Range Interactions. R.B. Israel, CMP 106, (1986).
- Some Generic Results in Mathematical Physics. R.B.Israel, Markov Proc. and Related Fields 10, (2004).
- Semi-infinite Ising model I. Thermodynamic functions and phase diagram in absence of magnetic field. J. Fröhlich , C-E Pfister. CMP 109, 493-523 (1987).
- Zero Temperature Limits of Gibbs Equilibrium States for Countable Markov Shifts. T. Kempton J. Stat. Phys. v. 143, (2011).
- On the General One-Dimensional XY Model: Positive and Zero Temperature, Selection and Non-Selection. Baraviera, A.T. ; Cioletti, L. M. ; Lopes, A.O. ; Mohr, J. ; Souza, R. R. . Rev. in Math. Phys., v. 23, p. 1063-1113, (2011).
- Multiplicity of Phase Transitions and Mean-Field Criticality on Highly Non-Amenable Graphs. Roberto H. Schonmann, CMP v. 219, (2001).
- Gibbs Measures on Cayley Trees: Results and Open Problems. U. A. Rozikov, Rev. Math. Phys., 25, (2013).
- Discontinuity of the magnetization in one-dimensional 1/¦x−y¦2 Ising and Potts models. M. Aizenman, J. T. Chayes, L. Chayes, C. M. Newman, JSP , v. 50, (1988).
- On Free Energies of the Ising Model on the Cayley Tree. D. Gandolfo. M.M. Rakhmatullaev, U.A. Rozikov and J. Ruiz, JSP (2013).
Evaluation: Homework's + (2 interviews about the classes and homework's) + (presentation of an article or topic).
Survey articles written by graduate students
Pages of some Mathematical physicists where you can find more material related to this course:
Stanislav Smirnov (Fields Medal 2010)