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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?

Show that, if you don't make too many approximations, the exponential series in equation 8.22 includes the three-dot diagram in equation 8.18. There will be some leftover terms; show that these vanish in the thermodynamic limit.

The Ising model can be used to simulate other systems besides ferromagnets; examples include anti ferromagnets, binary alloys, and even fluids. The Ising model of a fluid is called a lattice gas. We imagine that space is divided into a lattice of sites, each of which can be either occupied by a gas molecule or unoccupied. The system has no kinetic energy, and the only potential energy comes from interactions of molecules on adjacent sites. Specifically, there is a contribution of -u0to the energy for each pair of neighbouring sites that are both occupied.

(a) Write down a formula for the grand partition function for this system, as a function of u0, T, and p.

(b) Rearrange your formula to show that it is identical, up to a multiplicative factor that does not depend on the state of the system, to the ordinary partition function for an Ising ferromagnet in the presence of an external magnetic field B, provided that you make the replacements u0鈫�4系and 渭鈫�2渭BB-8系. (Note that is the chemical potential of the gas while uB is the magnetic moment of a dipole in the magnet.)

(c) Discuss the implications. Which states of the magnet correspond to low density states of the lattice gas? Which states of the magnet correspond to high-density states in which the gas has condensed into a liquid? What shape does this model predict for the liquid-gas phase boundary in the P-T plane?

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 蔚) for the particular state of the 4脳4square lattice shown in Figure 8.4?

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

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鈮�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.
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