The exam and its solution can be downloaded from this link.
You may want to take a look at exams from previous years:
A. 2012 exam
B. 2011 exam
C. 2009 exam
D. 2007 exam
Take into account that the material taught in previous years was slightly different (especially 2009,2007).
Tutorial 13- 28.01.2013, Maximal Entropy principle – Problem 2.6 from Kardar:
A. The tutorial notes can be downloaded from here.
Tutorial 12- 21.01.2013, Linear Response and Onsager Reciprocal Relations:
A. The tutorial notes can be downloaded from here
Tutorial 11- 14.01.2013, Brownian motion:
A. The tutorial notes can be downloaded from here
B. Exercise 6, due to 02.04.13 can be downloaded from here
Tutorial 10 – 07.01.2013, Renormalization Group analysis of 2D Ising model:
A. The tutorial notes can be downloaded from here
Tutorial 9 – 31.12.2012, Two dimensional Coulomb gas:
A. The tutorial notes can be downloaded from here
B. Here is the new version of exercise 5 , which includes now only 3 questions.
Tutorial 8 – 24.12.2012, Mapping between Ising model and a lattice gas model
Tutorial 7 - 17.12.2012, Solution of the 2D Ising model:
A. Home exercise 4 – due to 31.12.2012.
B. Reading material:
* Landau & Lifshitz, Section 151
* Peliti, Statistical Mechanics in a Nutshell , Section 5.20
Tutorial 6 - 10.12.2012, Transfer matrices and soultion of the 1D Ising model:
A. Reading material:
* Pathria, Section 12
Tutorial 5 - 03.12.2012, Chemical reactions:
A. Tutorial note can be downloaded from here.
B. Home exercise 3 – due to 17.12.2012.
C. Reading material:
* Landau & Lifshitz, Sections 101-104
Tutorial 4 - 26.11.2012, Diamagnetism and De Haas van Alphen effect:
A. Tutorial note can be downloaded from here.
B. Reading material:
* Diamagnetism : Peierls, Surprises in Theoretical Physics, sect 4.3 and Landau & Lifshitz 59.
* De Haas van Alphen effect: Calculation for T=0 from Huang Statistical Mechanics 11.4. Full calculation found in Landau & Lifshitz 60.
Tutorial 3 - 19.11.2012, Inner degrees of freedom & BEC in harmonic trap:
A. Home exercise 2 – due to 03.12.2012. Updated
on 25.11.2012 with some clarification of question 2.
B. The tutorial about diatomic molecules is taken from Grisha’s lecture notes, page 34.
Tutorial 2 - 12.11.2012, Monte Carlo simulation of Markov chains:
A. You may download the sample code, written in C++, from here . You can use it start writing your simulation of the Ising model. It can be compiled using visual studio installed on the computers found in the computer lab. To compile press F5. Before you do that, make sure you have the correct input variables set by right clicking the project -> Properties -> Configuration Properties -> Debugging -> Command Arguments. You may also run the program after compiling it outside the visual studio environment. You can use for that the *.bat files such as the “run.bat” sample file found in the ZIP file you’ve just downloaded. Good luck !
B.Just to clarify how the Metropolis algorithm goes :
1. Choose a site at random out of the L^2 sites.
2. If E_j <= E_i , flip the spin.
3. If E_j > E_i, generate a random number r between 0 and 1. If r<exp(-\beta*(E_j-E_i)) flip the spin.
C. Because W_ij scale as 1/L^2, a time step of 1 in reality corresponds to L^2 Metropolis steps. You should therefore rescale you time appropriately in your plots.
D. In the homework you need to consider periodic boundary conditions (Ising model on a sphere).
E. In question 3B you’re asked to compute \tau_{corr} and not \tau_{relax}.
F. Regarding 3C, the point of the question is to practice how to check the scaling of your simulation results with L. Try to think why you get this specific scaling. You can understand it from thermodynamics or from the central limit theorem. For the latter you need to think of the system as squares of size \zeta*\zeta sites that fluctuate (almost) independently. \zeta here is the correlation length of the model which you will discuss in the coming lectures.
G. Reading material :
* Landau and Binder, “A guide to Monte Carlo simulations in statistical physics”
* Newman and Barkema, “Monte Carlo methods in statistical physics”
* Krauth, “Statistical Mechanics: Algorithms and Computations”
* Krauth, “Introduction to Monte Carlo Algorithms”.
* You’ll find many examples in Gould and Tobochnick.
H. If you plan do Monte Carlo for a living you might want to read about “the dark side of simulations”
Tutorial 1 - 05.11.2012, Saddle point approximation and harmonic oscillator :
A. Homework policy
B. Home exercise 1 – due to 19.11.2012
C. Here is a short exercise sheet for anyone who doesn’t feel comfortable with expanding functions into power series (this is for your benefit only, there is of course no need to submit it). You will be expected to know how to expand such series in the next homework exercises (as well as in your scientific career).