By K. P. N. Murthy

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**Additional info for An Introduction to Monte Carlo Simulations in Statistical Physics**

**Example text**

FOR, written and tested by my colleague V. Sridhar, MSD, IGCAR, Kalpakkam. in . 34 K. P. N. 5 1 0 0 Figure 5: Magnetization per spin ( |M |/L2 ) vs. J/kB T for various system sizes L. 5 1 0 0 Figure 6: Magnetic susceptibility per spin (χ/L2 ) v s. 5 1 0 0 Figure 7: Specific heat per spin (CV /L2 ) v s. J/kB T for various system sizes L So far so good; we know now how to assemble, employing random numbers, a Markov chain of microstates each produced from its predecessor through an attempted single flip.

5 1 0 0 Figure 5: Magnetization per spin ( |M |/L2 ) vs. J/kB T for various system sizes L. 5 1 0 0 Figure 6: Magnetic susceptibility per spin (χ/L2 ) v s. 5 1 0 0 Figure 7: Specific heat per spin (CV /L2 ) v s. J/kB T for various system sizes L So far so good; we know now how to assemble, employing random numbers, a Markov chain of microstates each produced from its predecessor through an attempted single flip. 19 The system has a natural tendency to remain stay put in a microstate step after step after step.

Murthy How do we calculate Monte Carlo averages and error bars? We calculate the required macroscopic property by averaging over the ensemble constructed as per the Metropolis rejection technique described earlier. For example, the energy given by, E(C) = −J Si (C)Sj (C) , (70) i,j when the Ising spin system is in the configuration C. Similarly, the magnetization in the microstate C is given by Si (C) . M(C) = (71) i The above and any other macroscopic quantity of interest can be averaged over an ensemble generated by the Monte Carlo algorithm.

### An Introduction to Monte Carlo Simulations in Statistical Physics by K. P. N. Murthy

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