# 780.20: 2082 Session 12

Handouts: Excerpt from Biner/Heermann, printouts of sampling_test.cpp and Ising model codes

In this session, we'll extend our study of Monte Carlo methods. We'll compare importance sampling to ordinary random sampling and then explore a standard example, the two-dimensional Ising model. Please read the background notes first for an introduction and then use the Biner/Heermann handout as you go.

## Monte Carlo Sampling

In the sampling_test.cpp code, three distributions of energy for a one-dimensional Ising model [equation (2.1.1) in the handout] are generated. The first is the exact distribution at temperature kT in the canonical ensemble; that is, the distribution of energies considering every possible configuration of spins weighted by a normalized Boltzmann factor. The second is the energy distribution if a large number of spin configurations are chosen at random. The third is the energy distribution from a Metropolis Markov process.

1. Here are some questions to get you familiar with the model and its implementation in the code sampling_test.cpp:
• The "J" in the Ising model doesn't appear anywhere. What is its implicit value in the code? If you were to add a global variable J_Ising, where would it appear in the code? (You'd also have to change how the energies are binned.) Bonus: What is the physical origin of J for a ferromagnetic?

• How would you add an external magnetic field? [See (2.1.1) for the Hamiltonian.]

• How many total "microstates" are there for a 1-D Ising model with the number of sites in the code? (I.e., how many possible "configurations" of spins are there?) If calculating the energy of each configuration takes the same amount of time, how much longer would it take to calculate a lattice with twice the number of sites?

• What are the minimum and maximum energies given a number of lattice sites? How many different energies are possible? (Can you have any integer energy between the minimum and maximum?)

• What are the boundary conditions for the line of spins? Find where it is used in the code. Can you think of a different set of boundary conditions that could be used?

• Find the calculation of the exact partition function in the code. How is this the same thing as (2.1.5)? Can you figure out how the complete set of configurations are constructed in the code using the "next_configuration" function?

2. Compile and link sampling_test.cpp (using make_sampling_test) and run it to see what the output looks like. Did you get the correct answer for the number of configurations? (If not, rethink!)
• Look through the code and make sure you understand how each distribution is generated.
3. Generate gnuplot graphs of the probability of energy E, P(E), vs. the energy (this is what is output to the screen) for kT = 10. and kT = 1. [There is a plot file to help.]
• Why is the distribution for ordinary random sampling centered at E=0? What kind of temperature does ordinary random sampling correspond to? [Hint: What Boltzmann factor?]
• Compare the exact P(E) for a canonical distribution to those of random sampling and the Metropolis algorithm. Will the latter work for importance sampling? Why will the random sampling be a problem for evaluating thermal averages [see figure 2.3 and the discussion after equation (2.1.33)]?
4. Verify that the transition probabilities in (2.1.39a) and (2.1.39b) both satisfy the condition (2.1.38). Which one is implemented in the code? Switch to the other and check (bonus: make it an option which one is used).
5. Modify the code to calculate the average energy at kT = 1. and kT = 10. using the two sampling methods and compare to the exact average energy (according to the canonical Boltzmann distribution).

## The Two-D Ising Model

In this section, we explore some aspects of Monte Carlo simulations that are discussed in section 2.2 of the handout. We use the two-dimensional Ising model with an "anti-ferromagnetic" interaction (J < 0) as our example.

1. Take a look at ising_model.cpp and note the use of #ifdef, #else, and #endif to enable one to switch between the one-d and two-d Ising models. We've implemented it here as a "compile switch" signaled by -D on the g++ command line. The default is the two-d model. To get the one-d version, you can either compile by hand with:
g++ -c -Done_dim -Wall ising_model.cpp
or by setting DFLAGS=-Done_dim in make_ising_model. What are the differences between the one-d and two-d versions?
2. Modify the code to change it from ferromagnetic to anti-ferromagnetic (in the calculate_energy function).
3. Equilibration. Compile and link ising_model.cpp (use make_ising_model) and run for several temperatures.
• Make a gnuplot graph of the energy vs. time for kT = 2.0, 1.0, and 0.5 (all on the same graph, so rename your files appropriately). Run several times at each temperature. Do you always get the same qualitative result? Explain.
• Look at the small time region. How long does it (apparently) take at each temperature to reach "equilibrium"? If you were going to use the configurations generated here to calculate thermal averages, would you want to use the ones at the beginning? How can you deal with this?
• BONUS: Modify the code so that the output file names automatically have the temperature in their names. (Recall filename_test.cpp from Session 9.)
4. Cooling. At present, the code starts from a random configuration. Modify the code to implement "cooling" by looping through temperatures kT = 2.0, 1.0, then 0.5 but start the simulation at each successive temperature using the final configuration of the higher temperature as the initial configuration of the lower temperature. Generate a gnuplot graph of the energy vs. temperature for each of the temperatures and compare to your previous results. Conclusions?
5. Efficiency. The code at present has several inefficiencies. Here are some ways to speed it up:
1. The current approach compares energies of new and old configurations by calculating the full energy of each and subtracting. Can you devise a (much) more efficient approach? (You don't need to implement it here.)
2. During one MCS, we can update each spin sequentially, which saves random number calls but also leads to shorter equilibration times.
3. We can reduce the random number calls when deciding if a spin flips or not.
4. We can use a table of nearest neighbors of each spins to save calculation time.
These optimizations speed up the code by a factor of about six! A new version is ising_opt.cpp (with make_ising_opt). Look at how the optimizations were implemented. [Note: The new code has different boundary conditions and is ferromagnetic.]
6. Phase Transition
• Modify the optimized code ising_opt.cpp to calculate the absolute value of the magnetization of the system (with linear size L=5) for various temperatures. Starting at kT=4.0, cool down the system by Delta kT =0.2 until kT=1.0. What do you observe about the behavior of magnetization? (Make a plot.)
• When we calculate the magnetization, we use the absolute value of it, why? It may help to look at the time dependence of the magnetization around kT=2 with size L=5 or less. Can you explain your observation in terms of "spontaneous symmetry breaking"? Also, can you explain what will happen when you increase the system size? [Hint: Would it be possible for an infinite system to change from a state with all up spins to one with all down spins?]
• Now change the system size (try L=10, 20, 40). By changing the system size, can you observe a change in the behavior of magnetization? If so, can you make an argument why this happens?