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Chapter 8: Q 8.30 (page 355)

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.

Short Answer

Expert verified

Therefore, the codes and graphs are given.

Step by step solution

01

Given information

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

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

02

Explanation

a) For temperatures of 1K, 1.5K, 2K, 2.5K, 3K, 3.5K, and 4K, some typical states (three states per temperature). As you scroll down, the temperature rises.

The program code is:

03

Explanation

b) Program to compute task b):

The graph is:

04

Conclusion

c) Program for task c):

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Most popular questions from this chapter

Problem 8.8. Show that the nthvirial coefficient depends on the diagrams in equation 8.23 that have ndots. Write the third virial coefficient, C(T), in terms of an integral of f-functions. Why it would be difficult to carry out this integral?

By changing variables as in the text, express the diagram in equation 8.18 in terms of the same integral as in the equation8.31. Do the same for the last two diagrams in the first line of the equation8.20. Which diagrams cannot be written in terms of this basic integral?

You can estimate the size of any diagram by realizing that fr is of order 1 out to a distance of about the diameter of a molecule, andf≈0 beyond that. Hence, a three-dimensional integral of a product of f's will generally give a result that is of the order of the volume of a molecule. Estimate the sizes of all the diagrams shown explicitly in equation8.20 and explain why it was necessary to rewrite the series in exponential form.

Modify the ising program to compute the average energy of the system over all iterations. To do this, first add code to the initialise subroutine compute the initial energy of the lattice; then, whenever a dipole is flipped, change the energy variable by the appropriate amount. When computing the average energy, be sure to average over all iterations, not just those iterations in which a dipole is actually flipped (why?). Run the program for a 5 x 5 lattice for T values from 4 down to l in reasonably small intervals, then plot the average energy as a function of T. Also plot the heat capacity. Use at least 1000 iterations per dipole for each run, preferably more. If your computer is fast enough, repeat for a 10x 10 lattice and for a 20 x 20 lattice. Discuss the results. (Hint: Rather than starting over at each temperature with a random initial state, you can save time by starting with the final state generated at the previous, nearby temperature. For the larger lattices you may wish to save time by considering only a smaller temperature interval, perhaps from 3 down to 1.5.)

In this problem you will use the mean field approximation to analyse the behaviour of the Ising model near the critical point.

(a) Prove that, when x≪1,tanhx≈x-13x3

(b) Use the result of part (a) to find an expression for the magnetisation of the Ising model, in the mean field approximation, when T is very close to the critical temperature. You should find M∝Tc-Tβ¯,whereβ(not to be confused with 1/kT) is a critical exponent, analogous to the f defined for a fluid in Problem 5.55. Onsager's exact solution shows that β=1/8in two dimensions, while experiments and more sophisticated approximations show that β≈1/3in three dimensions. The mean field approximation, however, predicts a larger value.

(c) The magnetic susceptibility χis defined as χ≡(∂M/∂B)T. The behaviour of this quantity near the critical point is conventionally written as χ∝T-Tc-γ , where y is another critical exponent. Find the value of in the mean field approximation, and show that it does not depend on whether T is slightly above or slightly below Te. (The exact value of y in two dimensions turns out to be 7/4, while in three dimensions γ≈1.24.)
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