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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

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.)

Consider a gas of molecules whose interaction energy u(r)u is infinite for r<r0and negative for r>r0, with a minimum value of -u0. Suppose further that kT鈮�u0, so you can approximate the Boltzmann factor forr>r0using ex鈮�1+x. Show that under these conditions the second virial coefficient has the form B(T)=b-(a/kT), the same as what you found for a van der Waals gas in Problem 1.17. Write the van der Waals constants aand b in terms of r0and u(r), and discuss the results briefly.

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?

Consider an Ising model of 100 elementary dipoles. Suppose you wish to calculate the partition function for this system, using a computer that can compute one billion terms of the partition function per second. How long must you wait for the answer?

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.
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