Large Deviations in Statistical Physics

Information

• Lecturer: Prof. Hugo Touchette
• Module information/syllabus

References

• [HT2011]
  H. Touchette, A basic introduction to large deviations: Theory, applications, simulations, in R. Leidl and A. K. Hartmann (eds), Modern Computational Science 11: Lecture Notes from the 3rd International Oldenburg Summer School, BIS-Verlag der Carl von Ossietzky Universitat Oldenburg, 2011. arxiv:1106.4146 | errata at the end of the arxiv version | summer school website
• [HT2009]
  H. Touchette, The large deviation approach to statistical mechanics, Phys. Rep. 478, 1-69, 2009. arxiv:0804.0327 | published version
• [HT2018]
  H. Touchette, Introduction to dynamical large deviations of Markov processes, Physica A 504, 5, 2018
  arxiv:1705.06492 | published version
• [DZ1998]
  A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Springer, 1998.
  Springer | Amazon | Google books
• [GS2001]
  G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes, Oxford, 2001.
  Amazon
  
• [J2010]
  K. Jacobs, Stochastic Processes for Physicists, Cambridge, 2010.
  Amazon
  

Content (2014; for 2017 we will skip some sections)

• Week 1: Elements of probability theory
  
• Lecture notes
• Notes: Probability (by Oscar Bandtlow at QMUL)
• Reading: Sections not covered in the notes, especially the section on the Central Limit Theorem
  
• Exercise sheet
  
• Week 2: Examples of large deviations and the large deviation principle
  
• Lecture notes
• Notes: Sections 2.1 and 2.2 HT2011; sections 2 and 3.1 of HT2009
  
• Reading: Other examples in notes above and Appendix B of HT2009
  
• Exercises: HT2011 exercises 2.7.1-2.7.5
  
• Week 3: Varadhan's Theorem and the Gärtner-Ellis Theorem
  
• Lecture notes
• Notes: Sections 2.3 and 2.4 of HT2011; sections 3.3 and 3.5.4 of HT2009
  
• Reading: Properties of SCGFs and rate functions, sections 3.5 of HT2009
• Further reading: Legendre-Fenchel transforms in a nutshell
• Exercises: HT2011 exercises 2.7.6-2.7.10
  
• Week 4: IID sample means and Sanov's Theorem
  
• Lecture notes
• Notes: Section 3.1 of HT2011; section 4.1 of HT2009
  
• Reading: Contraction principle, section 2.5 of HT2011 and section 3.6 of HT2009
  
• Exercises: HT2011 exercises 3.6.1, 3.6.8, 3.6.9
• Extra exercises: HT2011 exercises 3.6.2-3.6.7
  
• Week 5: Large deviations of Markov chains
  
• Lecture notes
• Notes: Section 3.2 of HT2011; section 4.3 of HT2009
  
• Reading: Continuous-time processes, section 3.3 of HT2011; GS Chapter 6
  
• Exercises: HT2011 exercises 3.6.10 and 3.6.12
  
• Week 6: Continuous-time Markov processes
  
• Lecture notes
• Other notes: Stochastic Models of Complex Systems (by S. Grosskinsky at Warwick University)
• Reading: GS2001 Chapters 8, 9 and 13 and; J2010 Chapters 3 and 8
  
• Exercises: Problems in GS001, Chapters 8, 9 and 13
  
• Week 7: Large deviations of continuous-time processes
  
• Lecture notes
• H. Touchette, Introduction to dynamical large deviations of Markov processes.
• Reading: Chap. 3 of DZ1998
  
• Exercises: Problems 3.6.11-3.6.13 in HT2011 (see errata list at the end for corrections) plus exercises mentioned in lecture notes
  
• Week 8: Physical applications
  
• Lecture notes
• Reading: HT2009 Sec. 6. S.
  
• Extra reading: Symmetrization of non-hermitian operators: see notes on Markov processes and diffusions and large deviation theory of Markov processes (password protected: ask me for access)
  
• Exercises: See lecture notes
  

Other sources

• Large deviation theory: History, sources, references, and pointers
• Notes: Legendre-Fenchel transforms in a nutshell. pdf
• Video: Large deviations and statistical mechanics
• Video: Large deviations in statistical mechanics. video (wma) | video (flash)