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Chapter 8: Q. 8.18 (page 343)

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.

Short Answer

Expert verified

The average energy =-N蔚tanh尾蔚

Sketch for the average energy as function of temperature

Step by step solution

01

Step 1. Given information

The partition function of the system:

Z=鈭�se-蔚i/r

Z=鈭�se-蔚i/r

02

Step 2. To find the expression for average energy 

We substitute尾for1/蟿

Z=鈭�se-蔚n尾

The average energy of the system is,

U=鉄�蔚鉄�

=1Z鈭�s蔚se-尾cs

=-1Z蔚s鈭�se-尾ss

Substituting the value of dZd尾=蔚s鈭�se-尾cs

U=-1ZdZd尾

=-鈭�鈭�尾(lnZ)

The expression for partition function for final sum of Ndipole is,

Z=(2cosh尾蔚)

lnZ=ln(2cosh尾蔚)

U=-1ZdZd尾

=-鈭�鈭�尾(ln(2cosh尾蔚))

=1(2cosh尾蔚)((2sinh尾蔚))伪

=-N蔚tanh尾蔚

The average energy ==-N蔚tanh尾蔚

03

Step 3. Sketch for the following function

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Most popular questions from this chapter

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.

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.

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.

For a two-dimensional Ising model on a square lattice, each dipole (except on the edges) has four "neighbors"-above, below, left, and right. (Diagonal neighbors are normally not included.) What is the total energy (in terms of 蔚) for the particular state of the 4脳4square lattice shown in Figure 8.4?

Figure 8.4. One particular state of an Ising model on a 4脳4square lattice (Problem 8.15).

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?
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