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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

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).

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.

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?

In this section I've formulated the cluster expansion for a gas with a fixed number of particles, using the "canonical" formalism of Chapter 6. A somewhat cleaner approach, however, is to use the "grand canonical" formalism introduced in Section 7.1, in which we allow the system to exchange particles with a much larger reservoir.

(a) Write down a formula for the grand partition function (Z) of a weakly interacting gas in thermal and diffusive equilibrium with a reservoir at fixed T and碌. Express Z as a sum over all possible particle numbers N, with each term involving the ordinary partition function Z(N).

(b) Use equations 8.6 and 8.20 to express Z(N) as a sum of diagrams, then carry out the sum over N, diagram by diagram. Express the result as a sum of similar diagrams, but with a new rule 1 that associates the expression (>./vQ) J d3ri with each dot, where >. = e13碌,. Now, with the awkward factors of N(N - 1) 路 路 路 taken care of, you should find that the sum of all diagrams organizes itself into exponential form, resulting in the formula

Note that the exponent contains all connected diagrams, including those that can be disconnected by removal of a single line.

(c) Using the properties of the grand partition function (see Problem 7.7), find diagrammatic expressions for the average number of particles and the pressure of this gas.

(d) Keeping only the first diagram in each sum, express N(碌) and P(碌) in terms of an integral of the Mayer /-function. Eliminate 碌 to obtain the same result for the pressure (and the second virial coefficient) as derived in the text.

(e) Repeat part (d) keeping the three-dot diagrams as well, to obtain an expression for the third virial coefficient in terms of an integral of /-functions. You should find that the A-shaped diagram cancels, leaving only the triangle diagram to contribute to C(T).

To quantify the clustering of alignments within an Ising magnet, we define a quantity called the correlation function, c(r). Take any two dipoles i and j, separated by a distance r, and compute the product of their states: sisj. This product is 1 if the dipoles are parallel and -1 if the dipoles are antiparallel. Now average this quantity over all pairs that are separated by a fixed distance r, to |obtain a measure of the tendency of dipoles to be "correlated" over this distance. Finally, to remove the effect of any overall magnetisation of the system, subtract off the square of the average s. Written as an equation, then, the correlation function is

c(r)=sisj炉-si炉2

where it is understood that the first term averages over all pairs at the fixed distance r. Technically, the averages should also be taken over all possible states of the system, but don't do this yet.

(a) Add a routine to the ising program to compute the correlation function for the current state of the lattice, averaging over all pairs separated either vertically or horizontally (but not diagonally) by r units of distance, where r varies from 1 to half the lattice size. Have the program execute this routine periodically and plot the results as a bar graph.

(b) Run this program at a variety of temperatures, above, below, and near the critical point. Use a lattice size of at least 20, preferably larger (especially near the critical point). Describe the behaviour of the correlation function at each temperature.

(c) Now add code to compute the average correlation function over the duration of a run. (However, it's best to let the system "equilibrate" to a typical state before you begin accumulating averages.) The correlation length is defined as the distance over which the correlation function decreases by a factor of e. Estimate the correlation length at each temperature, and plot graph of the correlation length vs.
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