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thermodynamic simulations & feynman quantum path integration 3990.3CE1M0.2–0.40.50.1–0.80.000 2 4 0 2 4kTkTFigure 15.3 Simulation results for the energy, specific heat, and magnetization of a 1-D latticeof 100 spins as a function of temperature.energy occurring during M trials and then determine the specific heat from thefluctuations:U 2 = 1 MM∑(E t ) 2 , (15.17)t=1C = 1N 2 U 2 − (U) 2k B T 2 = 1N 2 〈E 2 〉−〈E〉 2k B T 2 . (15.18)1. Extend your program to calculate the internal energy U and the magnetizationM for the chain. Do not recalculate entire sums when only one spin changes.2. Make sure to wait for your system to equilibrate before you calculate thermodynamicquantities. (You can check that U is fluctuating about its average.)Your results should resemble Figure 15.3.3. Reduce statistical fluctuations by running the simulation a number of timeswith different seeds and taking the average of the results.4. The simulations you run for small N may be realistic but may not agree withstatistical mechanics, which assumes N ≃∞(you may assume that N ≃ 2000is close to infinity). Check that agreement with the analytic results for thethermodynamic limit is better for large N than small N.5. Check that the simulated thermodynamic quantities are independent of initialconditions (within statistical uncertainties). In practice, your cold and hot startresults should agree.6. Make a plot of the internal energy U as a function of k B T and compare itto the analytic result (15.7).−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 399