Lecture Summary: Written from memory and may not be completely faithful to the sequence followed in class
Lecture 1 [January 2, 2025, Thursday]: Overview of the equilibrium thermodynamic formalism--Equilibrium states, Entropy, fundamental relation, Euler relation, Gibbs-Duhem relation, Legendre transformation, Free energies (Helmholtz, Gibbs, Enthalpy, Grand potential,..), Entropy maximazation=> energy minimization, free energy minimization conditions [Refs:Callen]. Connection to statistical mechanics: Boltzmann entropy formula, Ensembles: Legendre transform (thermodynamics)-->Laplace transform (statistical mechanics). Ensemble equivalence [Pathria]. Negligible fluctuations in the thermodynamic limit [Pathria 3.6 pg 58-61]. Optional further reading: (i) For the physical interpretation of free energies, see for example Schroeder [Elementary], (ii) RKP Zia, EF Redish, SR McKay. Making sense of the Legendre transform, American Journal of Physics 77, 614 (2009). [Intermediate]
Lecture 2 [January 7, Tuesday]: [Chapter 8, Callen. Chapter 2, Stanley] Thermodynamic stability. Concave (Convex down) entropy vs internal energy, convex up Free energy. Specific heat(c_v) and isothermal comprehensibility positive definite, Helmholtz free energy minimum. Le Chatlier's principle (qualitative. 8.4 Callen)
Lecture 3 [January 8, Wednesday]: Geometrical interpretation of Gibbs' and Helmholtz potential (Stanley pg 30, Section 2.5)
Lecture 4 [January 9, Thursday]:Van der Waals equation of state (crude derivation from Reif)
Lecture 5 [January 14, Tuesday]: van der Waals gas (cont.) Grand canonical understanding of the phase coexistence and the occurrence of the phase transition in the thermodynamic limit [Kardar(particles) Section 5.5 pg 141-143]. This is also discussed in Huang in the context of the pressure ensemble
Lecture 6 [January 15, Wednesday]: Yang-Lee theory of Phase transitions. [Ref: Huang, Yang-Lee paper (first 3 pages)]
---------------Midsem Syllabus begins-----------------
Lecture 7 [January 16, Thursday]: van der Waal gas (cont). Determination of the critical point. Universal equation for all fluids in terms of (p_c, V_c, T_c). Law of corresponding states. The equation does not describe the observations quantitatively but the condensation in many gases seems to be universal. Law of rectilinear diameter (historically significant but can be skipped). Guggenheim plot (e.g., Stanley: Fig. 1.8 p10). Section 5.7 Kardar (Particles). Critical exponents \beta, gamma, and \delta for the van der waals equation. Kardar Section 5.8 particles. David Tong- Stat Mech Notes pg. ...., Many of the topics are also discussed in Goldenfeld Chapter 4 and Stanley Chapter 5. (Note that the vdW equation is sometimes defined in molar units.)
Lecture 8 [January 21, Tuesday]: The Ising model. Introduction and mean field solution. Argument that the mean field solution is equavalent to ignoring fluctuations [e..g, Greinger, Stat Mech Example 18.2, p443]. Determination of the free energy A(T, h, N). [Bragg Williams approximation not discussed. 1 D exact solution not discussed].
Lecture 9 [January 22, Wednesday]: Mean field Ising model (cont). Transcendental equation for magnetization. Energy entropy argument for absence of spontaneous magnetization at finite temperature in 1D and its presence in 2D. [Plischke and Bergersen p69, Section 3.3] (Note: Huang has a more sophisticated version of the argument which we did not follow.)
Lecture 10 [January 23, Thursday (2 hours)]: Critical exponents in the mean field Ising model (\beta, \gamma, \delta). Antiferromagnetic Ising model (only sketchly discussed) [See e.g., Nishimori-Ortiz 2.7, p34]. Fluid magnet analogy, Lattice gas [Huang] [Stauffer-Chowdhury 7.4.2 p270]
Lecture 11 [January 28, Tuesday]: Infinite range Ising model, mapping spin to continuous variable by Gaussian identity ("Hubbard-Stratonovich transformation") and equivalence to the mean field theory. [Refs. Nishimori-Ortiz Section 2.5, p30]
Lecture 12 [January 29, Wednesday]: Spontaneous symmetry breaking. Spontaneous symmetry breaking in the infinite range Ising model. Importance of the order of limits. [Ref. Nishimori-Ortiz Section 5.6, p116. Suggested reading: Goldenfeld Section 2.7-2.9 p40-54]. Toward the Landau free energy via the small m expansion of the infinite range Ising model free energy.
Lecture 13 [January 30, Thursday]: Landau free energy from the partition function of the infinite range model.
-----------------------------------------------------------
Final Exam Syllabus Begins
Lecture 14 [February 04, Tuesday]: Systematic construction of the Landau free energy as a low order polynomial of the partition function. Goldenfeld Section 5.3
Lecture 15 [February 05, Wednesday]: Determination of the critical exponents (\alpha, \beta, \gamma, \delta) for the Landau theory. (Goldenfeld Section 5.4)
Lecture 16 [February 06, Thursday]: Discussion of the first-order phase transition in Landau theory: [\eta^3] theory. Various stable solutions, The concept of spinodal and binodal temperatures, Bifurcation diagrams
Lecture 17 [February 11, Tuesday]: Discussion of the first-order phase transition in Landau theory [\eta^3] and [\eta^6] Tricritical point (when the coefficient of the 4th order term is zero) and the critical exponents for the triticritial point [e.g. Chaikin Lubensky only page 175-176]
[References for Lecture 16, 17, 18: Stauffer-Chowdhury parts of Chapter 12 and Arovas parts of Chapter 7]
--------------------------Midsem Syllabus ends----------------------------------------Class Test 1 syllabus Begins-------------
Lecture 18 [February 12, Wednesday]: Compressible Landau model => \eta^6 free energy [Ref]. Introduction to Landau Ginzburg free theory. Coarse-graining and introducing a spatial variation of the order paparemeter and gradient term in the free energy [Chaikin-Lubensky section 5.2.1 and 5.2.2].
Lecture 19 [February 13, Thursday]: Allowing for spatial variation of the order parameter, construction of the partition function as a path integral. Ref. Goldenfeld Section 5.7, Chaikin and Lubensky section 5.2.1 and 5.2.2]
Lecture 20 [February 18, Tuesday]: Functional derivatives (quick and dirty introduction), Euler Lagrange equation=mean field equation of state. Fluctuation-response relationship. [Closely following the notation of Goldenfeld.] [Functional derivatives are discussed in e.g., Chaikin and Lubensky, pg 140, Appendix 3A]
Lecture 21 [February 19, Wednesday]: Introduction to correlation functions, their physical meaning. The notion of the correlation length. Critical exponent \nu. Solution of the correlation equation in real and Fourier space, Ornstein-Zernike relation. The critical exponent \eta (=0 in mean field theory).
Lecture 22 [February 20, Thursday]: Topics of Lecture 21 continued. Goldenfeld. pg 158 onwards. Also see Tong's Stat Field Theory Notes [pg 42-22] for the saddle point approximation of the Fourier integral.
------------Midsem (5/3/25) and Spring Break----------------------
Lecture 23 [March 18, Tuesday]: Revision after the midsem: Overview of the Ginzburg-Landau theory. Divergence of the correlation length. Physical meaning of the correlation length. Asymptotic forms of the correlation function (Power law at T=Tc and an Yukawa-like exponential decay away from the critical point)
March 19--No class: DPS day!
Lecture 24 [March 20, Thursday] [8-10 am]: Ginzburg-Levanyuk Criterion and Ginzburg temperature. Dimensional analysis of Landau theory. [Goldenfeld: Section 6.2 p 169-173. Section 7.1, pg 189-192]
Lecture 25 [March 21, Friday] [Extra Class, 8-10 am]: Divergence of the specific heat due to Gaussian fluctuations and its dimensionality dependence. [Ref. Goldenfeld Section 6.3.3 pg 176-178, Section 6.4, pg 181-185 or David Tong [Stat Field Theory: pg 32-29]
----------Class Test 1 syllabus ends-----------------------------------Class test 27, March 2025 Thursday 8.30 am-------------
Lecture 26 [April 1, Tuesday]: A self-consistent calculation for the change in the mean field transition temperature [Chaikin Lubensky 5.3, pg 226-227]
Lecture 27 [April 2, Wednesday]: Continuous Symmetry: Goldenfeld Chapter 11.1 See here for a clearer discussion of transverse susceptibility from Parisi.
Lecture 28 [April 3, Thursday]: [2 hours] Continuous symmetry (cont).
Lecture 29 [April 8, Tuesday]: Review of the discussion on continuous symmetry
Lectures 27-28: Goldenfeld: Section 11.1.
Lecture 30 [April 9, Wednesday]: Scaling. Generalized homogeneous functions. Relationship between critical exponents [Stanley]
Lecture 31 [April 10, Thurssday] (2 hours): Kadanoff block spins--justification of the homogeneous function form for the free energy. Scaling of the correlation function. Hyperscaling relation. [Stanley] Introduction to Renormalisation group--Example discussing the fixed points of a simple logistic map and linear stability analysis [See e.g. here]: [Interesting aside: This logistic map shows a period doubling route to chaos that is itself amenable to RG analysis! See e.g., https://pubs.aip.org/aapt/ajp/article/67/1/52/1055222/ or https://www.damtp.cam.ac.uk/user/tong/mathbio/mathbio2.pdf]
Lecture 32 [April 15, Tuesday] Introduction to Renormalization group. RG based analysis of the 1D Ising model. (Schwabl 7.3.2). We are doing the h=0 case. For finite field case, see Pathria, Huang or Chaikin-Lubensky. [pdf of Schwabl Section 7.3].
Lecture 33 [April 16, Wednesday] RG of the 1D Ising model (continued). (Schwabl Section 7.3.2).
Lecture 33 [April 17, Thursday] (2 hours) Approximate RG treatment of the 2D Ising model on square lattice. We discussed the case with h=0 [Read pdf] Schwabl Section 7.3 (except perhaps 7.3.5)]. For conceptual clarity you can also see Pathria sections 14.3, 14.4C and Sections 9.2-9.4 from Goldenfeld which overlap with Schawabl section 7.3]
Syllabus for the final Exam: Lecture 14 to Lecture 33
Primary References
Continuous Symmetries: Goldenfeld Section 11.1
Scaling, Kadanoff block spins, exponent equalities: Stanley Chapter 11 and Chapter 12 (except Section 12.3)
Renormalization group : Schwabl (Section 7.3)
References
[Arovas 2025] Daniel Arovas, Lecture Notes on Thermodynamics and Statistical Mechanics (A Work in Progress) [link] (Feb 2025 update)
[Callen]: H. B. Callen, Thermodynamics, 2nd Edition
[Goldenfeld]: N. Goldenfeld, Lectures on Phase transitions and the renormalisation group
[Kardar-Particles]: M. Kardar, Statistical Physics of Particles (Cambridge Univ Press 2007)
[Kardar-Fields] M. Kardar, Statistical Physics of Fields (Cambridge)
[Pathria]: R. K. Pathria and Paul D. Beale, Statistical Mechanics, Third Edition Elsevier
[Stanley]: H. Eugene Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford University Press 1971)
[Shroeder]: Daniel V Schroeder, Introduction to Thermal Physics (Pearson)
[Schwabl] Franz Schwabl: Statistical Mechanics 1st or 2nd Edition (Springer)
[Plischke-Bergersen] Michael Plischke and Birger Bergersen, Equilibrium Statistical Mechanics, 3rd Edition (World Scientific 2006)
[Stauffer-Chowdhury] D. Stauffer and D. Chowdhury, Principles of Equilibrium Statistical Mechanics (Wiley-VCH,2000)
[Tong] David Tong, Lecture Notes on Statistical Physics, Lecture Notes on Statistical Field Theory
[Fisher 1983] Michael E. Fisher, Scaling, Universality and Renormalization Group Theory in Critical Phenomena --Proc Summer School .. Ed. FJW Hahne (Springer 1983)