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thermodynamic simulations & feynman quantum path integration 39715.4.1 Metropolis Algorithm Implementation1. Write a program that implements the Metropolis algorithm, that is, thatproduces a new configuration α k+1 from the present configuration α k .(Alternatively, use the program Ising.java shown in Listing 15.1.)2. Make the key data structure in your program an array s[N] containing thevalues of the spins s i . For debugging, print out + and − to give the spin ateach lattice point and examine the pattern for different trial numbers.3. The value for the exchange energy J fixes the scale for energy. Keep it fixedat J =1. (You may also wish to study antiferromagnets with J = −1, but firstexamine ferromagnets whose domains are easier to understand.)4. The thermal energy k B T is in units of J and is the independent variable. Usek B T =1for debugging.5. Use periodic boundary conditions on your chain to minimize end effects. Thismeans that the chain is a circle with the first and last spins adjacent to eachother.6. Try N ≃ 20 for debugging, and larger values for production runs.7. Use the printout to check that the system equilibrates fora. a totally ordered initial configuration (cold start); your simulation shouldresemble Figure 15.2.b. a random initial configuration (hot start).15.4.2 Equilibration, Thermodynamic Properties (Assessment)1. Watch a chain of N atoms attain thermal equilibrium when in contact with aheat bath. At high temperatures, or for small numbers of atoms, you shouldsee large fluctuations, while at lower temperatures you should see smallerfluctuations.2. Look for evidence of instabilities in which there is a spontaneous flipping ofa large number of spins. This becomes more likely for larger k B T values.3. Note how at thermal equilibrium the system is still quite dynamic, withspins flipping all the time. It is this energy exchange that determines thethermodynamic properties.4. You may well find that simulations at small k B T (say, k B T ≃ 0.1 for N = 200)are slow to equilibrate. Higher k B T values equilibrate faster yet have largerfluctuations.5. Observe the formation of domains and the effect they have on the total energy.Regardless of the direction of spin within a domain, the atom–atom interactionsare attractive and so contribute negative amounts to the energy of thesystem when aligned. However, the ↑↓ or ↓↑ interactions between domainscontribute positive energy. Therefore you should expect a more negativeenergy at lower temperatures where there are larger and fewer domains.6. Make a graph of average domain size versus temperature.−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 397