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Chapter 8: Q 8.28 (page 354)

Modify the ising program to compute the total magnetisation (that is, the sum of all the s values) for each iteration, and to tally how often each possible magnetisation value occurs during a run, plotting the results as a histogram. Run the program for a 5 x 5 lattice at a variety of temperatures, and discuss the results. Sketch a graph of the most likely magnetisation value as a function of temperature. If your computer is fast enough, repeat for a 10 x 10 lattice.

Short Answer

Expert verified

Hence the codes and diagrams are given.

Step by step solution

01

Given information

The total magnetisation (that is, the sum of all the s values) for each iteration, and to tally how often each possible magnetisation value occurs during a run, plotting the results as a histogram. Run the program for a 5 x 5 lattice at a variety of temperatures,

02

Explanation

This exercise's code can be seen below.

03

Conclusion

As can be seen in the diagram above, the system becomes highly magnetic as the temperature is lowered. That is, the M = 0 mean value flips to a non-zero value.

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Most popular questions from this chapter

Modify the ising program to simulate a one-dimensional Ising model.

(a) For a lattice size of 100, observe the sequence of states generated at various temperatures and discuss the results. According to the exact solution (for an infinite lattice), we expect this system to magnetise only as the temperature goes to zero; is the behaviour of your program consistent with this prediction? How does the typical cluster size depend on temperature?

(b) Modify your program to compute the average energy as in Problem 8.27. Plot the energy and heat capacity vs. temperature and compare to the exact result for an infinite lattice.

(c) Modify your program to compute the magnetisation as in Problem 8.28. Determine the most likely magnetisation for various temperatures and sketch a graph of this quantity. Discuss.

Use a computer to plot s¯ as a function of kT/ε, as predicted by mean field theory, for a two-dimensional Ising model (with a square lattice).

For a two-dimensional Ising model on a square lattice, each dipole (except on the edges) has four "neighbors"-above, below, left, and right. (Diagonal neighbors are normally not included.) What is the total energy (in terms of ε) for the particular state of the 4×4square lattice shown in Figure 8.4?

Figure 8.4. One particular state of an Ising model on a 4×4square lattice (Problem 8.15).

Starting from the partition function, calculate the average energy of the one-dimensional Ising model, to verify equation 8.44. Sketch the average energy as a function of temperature.

Use the cluster expansion to write the total energy of a monatomic nonideal gas in terms of a sum of diagrams. Keeping only the first diagram, show that the energy is approximatelyU≈32NkT+N2V·2π∫0∞r2u(r)e-βu(r)drUse a computer to evaluate this integral numerically, as a function of T, for the Lennard-Jones potential. Plot the temperature-dependent part of the correction term, and explain the shape of the graph physically. Discuss the correction to the heat capacity at constant volume, and compute this correction numerically for argon at room temperature and atmospheric pressure.
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