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Chapter 8: Q. 8.4 (page 333)

Draw all the connected diagrams containing four dots. There are six diagrams in total; be careful to avoid drawing two diagrams that look superficially different but are actually the same. Which of the diagrams would remain connected if any single dot were removed?

Short Answer

Expert verified

All the connected diagrams that contain 4 dots have been drawn.

Step by step solution

01

Step 1. Given information

Configuration integral:- The configuration integral is used in probability theory, information theory and dynamical systems, it's a generalization of the definition of a partition function in statistical mechanics.

02

Step 2. Drawing all the diagrams in which four dots present.

(1)

(2)

(3)

(4)

(5)

(6)

03

Step 3. If a single dot is removed, these 3 will still remain connected.

(1)

(2)

(3)

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Most popular questions from this chapter

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?

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.

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≈32NkT+N2V·2π∫0∞r2u(r)e-βu(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.

The critical temperature of iron is 1043K. Use this value to make a rough estimate of the dipole-dipole interaction energy ε, in electron-volts.

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.)
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