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Chapter 8: Q 8.26 (page 353)

Implement the ising program on your favourite computer, using your favourite programming language. Run it for various lattice sizes and temperatures and observe the results. In particular:

(a) Run the program with a 20 x 20 lattice at T = 10, 5, 4, 3, and 2.5, for at least 100 iterations per dipole per run. At each temperature make a rough estimate of the size of the largest clusters.

(b) Repeat part (a) for a 40 x 40 lattice. Are the cluster sizes any different? Explain. (c) Run the program with a 20 x 20 lattice at T = 2, 1.5, and 1. Estimate the average magnetisation (as a percentage of total saturation) at each of these temperatures. Disregard runs in which the system gets stuck in a metastable state with two domains.

(d) Run the program with a 10x 10 lattice at T = 2.5. Watch it run for 100,000 iterations or so. Describe and explain the behaviour.

(e) Use successively larger lattices to estimate the typical cluster size at temperatures from 2.5 down to 2.27 (the critical temperature). The closer you are to the critical temperature, the larger a lattice you'll need and the longer the program will have to run. Quit when you realise that there are better ways to spend your time. Is it plausible that the cluster size goes to infinity as the temperature approaches the critical temperature?

Short Answer

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Step by step solution

01

Given information and explanation

Implement the ising program on your favourite computer, using your favourite programming language. Run it for various lattice sizes and temperatures and observe the results.

A Python application is written, and this is how it looks:

02

Explanation

a)We can observe that when the temperature rises, the size of the domain shrinks.

b)We can see that the clusters are smaller for the larger lattice.

03

Explanation

c) We have the following scenarios for these three temperatures. The average magnetisation (number of black squares) is around 50% or higher, as can be shown.

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Most popular questions from this chapter

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鈮�32NkT+N2V路2蟺鈭�0鈭�r2u(r)e-尾u(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.

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.

Consider an Ising model of just two elementary dipoles, whose mutual interaction energy is 卤系. Enumerate the states of this system and write down their Boltzmann factors. Calculate the partition function. Find the probabilities of finding the dipoles parallel and antiparallel, and plot these probabilities as a function of kT/系. Also calculate and plot the average energy of the system. At what temperatures are you more likely to find both dipoles pointing up than to find one up and one down?

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?

Problem 8.13. 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 approximately
U鈮�32NkT+N2V路2蟺鈭�0鈭�r2u(r)e-尾u(r)dr
Use 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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