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Advanced Statistical Physics UniversityofCologne InstituteforTheoreticalPhysics Advanced Statistical Physics Lectures: Johannes Berg Exercises: Simon Dettmer and Chau Nguyen Sheet 10: Christmas review Due: No hand-in required (Mailbox: Advanced Statistical Physics) Discussion: 12:00-13:30 Monday, 13 January, 2014 20. An intermediate review This is a good time to look back at what we have archived. The short questions below are intended to help you review some of the important concepts, and link them to some historical developments. For a reference, we recommend that you read N. Goldenfeld, “Lectures on phase transitions and the renormalisation group”, chapter 1 (Addison-Wesley Publishing Company, 1992), if you have not yet done so. At the beginning of the course, we started with a discussion on the derivation of the Boltz- mann distribution based understandings in stochastic processes. We sketched the conditions under which the Boltzmann distributions emerge as an invariant measure of a stochastic pro- cess. A by-product of the discussion was Markov chain Monte Carlo (MCMC) simulation, which is a very important technique well beyond statistical mechanics. a) Can you give a physical example of a stochastic system that does not obey the Boltzmann distribution? We further illustrated the method of partition functions in statistical mechanics, Ising models were taken as paradigms (Ising models are said to be the physicists’ Drosophila). We showed that the partition functions of one-dimensional systems are usually solvable by the recursion method or the transfer matrix method. The method of low temperature series and high temperature series were also introduced, which found very important applications later on. When we studied the two-dimensional Ising model, for the first time (in this course) we encountered the concept of phase transition. b) The fact that for a system described by the same Hamiltonian, different phases exist (ultimately, water and ice are described by the same Hamiltonian!), is associated with the singularity of the free energy with respect to its parameters. Note that the free energy is just the logarithm of the partition function, which is just a sum over all states (seemingly analytic); how does the free energy develop singularities then16? The special structure of the two-dimensional Ising model allowed us to map exactly the high temperature series into the low temperature series (Kramers–Wannier duality), and thereby the critical temperature at which the phase transition happens could be determined exactly. 16Historically, this was a very controversial question, for which the answer is credited to Kramers. One can find a discussion at length in Goldenfeld’s book, or in Huang’s book. 19 Although Onsager’s exact solution for the partition function of the two-dimensional Ising model is available, we skipped this calculation. c) What is the difference between first order phase transitions and continuous phase transitions17? d) We learned about critical exponents. Why are they interesting anyway? (This question motivates all our further studies!) Much of the effort of statistical physicists in the twentieth century was spent to understand continuous phase transitions, in particular the mystery of the critical exponents18. It was soon recognised that in order to maintain the thermodynamic stability of the system, the critical exponents need to satisfy different critical exponent inequalities. e) Using the fact that the free energy is a concave function of h and T , show that α + 2β + γ 2, (60) ≥ which is known as Rushbrooke’s inequality. A number of similar inequalities be- tween critical exponents can be derived in a similar way19. It came as a surprise that within experimental precision, the critical exponent inequalities seemed to be always satisfied as equalities. This implies that of the critical exponents are not all independent, they obey equalities later-known-as scaling relations. (Historically, this promoted Widom’s scaling hypothesis, which remained much of a mystery until Kanadoff’s block-spin construction and Wilson’s renormalisation group theory (RG).) We attempted to calculate the critical exponents using the variational mean-field approxi- mation (β = 1/2, γ = 1, ν = 1/2, η = 0 for any dimension), which were apparently not in agreement with experiments. We will learn that Landau, and later Ginzburg, developed a general phenomenological theory of continuous phase transitions, which on one hand laid the foundation for statistical field theory, on the other hand turned out to be of the mean-field nature, yielding the above mean-field critical exponents. Efforts to incorporate fluctuations (Gaussian model) and (naive) perturbative expansions (φ4-theory) failed to change the values of the critical exponents (with the special exception of α): it was shown that (naive) dimensional analysis of statical field theory always yields the mean-field critical exponents20! We are essentially stuck: no classical method can give such strange values of the critical exponents. Then comes RG to solve the mystery. We already started an introduction to RG in real space. By “iteratively tracing out” the degrees of freedom of a system, we introduced a self-similar transformation (RG transformation) of the system between different scales. RG transformations were usually rather complicated; but the key idea was that, there were fixed points, near where the transformations locally look like scaling transformations. This local scaling behaviour of the system near the critical fixed points (linearised RG) explained the scaling behaviours and allowed the critical exponents to be calculated. 17Sometimes continuous phase transitions are historically called second order phase transitions, due to the early classification by P. Ehrenfest. 18See N. Goldenfeld’s book, chapter 1. 19For curious readers, see H. Eugene Stanley, “Introduction to phase transitions and critical phenomena”, chapter 4 (Oxford University Press, 1971.) 20See N. Goldenfeld’s book, chapter 7. 20.
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