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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

Consider an Ising model in the presence of an external magnetic field B, which gives each dipole an additional energy of -渭BB if it points up and +渭BB if it points down (where渭B is the dipole's magnetic moment). Analyse this system using the mean field approximation to find the analogue of equation 8.50. Study the solutions of the equation graphically, and discuss the magnetisation of this system as a function of both the external field strength and the temperature. Sketch the region in the T-B plane for which the equation has three solutions.

At T = 0, equation 8.50 says that s炉=1. Work out the first temperature-dependent correction to this value, in the limit 尾鈭�n鈮�1. Compare to the low-temperature behaviour of a real ferromagnet, treated in Problem 7.64.

Show that the Lennard-Jones potential reaches its minimum value at r=r0, and that its value at this minimum is -u0. At what value of rdoes the potential equal zero?

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鈮�1,tanhx鈮�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鈭�Tc-T尾炉,where尾(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 尾=1/8in two dimensions, while experiments and more sophisticated approximations show that 尾鈮�1/3in three dimensions. The mean field approximation, however, predicts a larger value.

(c) The magnetic susceptibility 蠂is defined as 蠂鈮�(鈭�M/鈭�B)T. The behaviour of this quantity near the critical point is conventionally written as 蠂鈭�T-Tc-纬 , 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 纬鈮�1.24.)

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?
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