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400 chapter 157. Make a plot of the magnetization M as a function of k B T and compare it tothe analytic result. Does this agree with how you expect a heated magnet tobehave?8. Compute the energy fluctuations U 2 (15.17) and the specific heat C (15.18).Compare the simulated specific heat to the analytic result (15.8).15.4.3 Beyond Nearest Neighbors and 1-D (Exploration)• Extend the model so that the spin–spin interaction (15.3) extends to nextnearestneighbors as well as nearest neighbors. For the ferromagnetic case,this should lead to more binding and less fluctuation because we haveincreased the couplings among spins and thus increased the thermal inertia.• Extend the model so that the ferromagnetic spin–spin interaction (15.3)extends to nearest neighbors in two dimensions, and for the truly ambitious,three dimensions (the code Ising3D.java is available for instructors). Continueusing periodic boundary conditions and keep the number of particlessmall, at least to start [G,T&C 06].1. Form a square lattice and place √ N spins on each side.2. Examine the mean energy and magnetization as the system equilibrates.3. Is the temperature dependence of the average energy qualitatively differentfrom that of the 1-D model?4. Identify domains in the printout of spin configurations for small N.5. Once your system appears to be behaving properly, calculate the heat capacityand magnetization of the 2-D Ising model with the same technique used forthe 1-D model. Use a total number of particles of 100 ≤ N ≤ 2000.6. Look for a phase transition from ordered to unordered configurations byexamining the heat capacity and magnetization as functions of temperature.The former should diverge, while the latter should vanish at the phasetransition (Figure 15.4).Exercise: Three fixed spin- 1 2particles interact with each other at temperatureT =1/k b such that the energy of the system isE = −(s 1 s 2 + s 2 s 3 ).The system starts in the configuration ↑↓↑. Do a simulation by hand that usesthe Metropolis algorithm and the series of random numbers 0.5, 0.1, 0.9, 0.3 todetermine the results of just two thermal fluctuations of these three spins.15.5 Unit II. Magnets via Wang–Landau Sampling ⊙In Unit I we used a Boltzmann distribution to simulate the thermal properties ofan Ising model. There we described the probabilities for explicit spin states α with−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 400