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Chapter 8: Q. 8.2 (page 333)

Draw all the diagrams, connected or disconnected, representing terms in the configuration integral with four factors of fij. You should find 11 diagrams in total, of which five are connected.

Short Answer

Expert verified

All the diagrams representing terms in configuration integral with four factors of fij 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 representing terms in configuration integral with four factors of fij 

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

03

Step 3. All the 5 connected  diagrams are

(1)

(2)

(3)

(4)

(5)

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Most popular questions from this chapter

Show that, if you don't make too many approximations, the exponential series in equation 8.22 includes the three-dot diagram in equation 8.18. There will be some leftover terms; show that these vanish in the thermodynamic limit.

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?

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?

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.

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.
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