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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

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.

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.

For each of the diagrams shown in equation 8.20, write down the corresponding formula in terms of f-functions, and explain why the symmetry factor gives the correct overall coefficient.

Problem 8.10. Use a computer to calculate and plot the second virial coefficient for a gas of molecules interacting via the Lennard-Jones potential, for values of kT/u0 ranging from 1to 7. On the same graph, plot the data for nitrogen given in Problem 1.17, choosing the parameters r0 and u0so as to obtain a good fit.

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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