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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 =-Nεtanhβε

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=∑se-εi/r

Z=∑se-εi/r

02

Step 2. To find the expression for average energy 

We substituteβfor1/τ

Z=∑se-εnβ

The average energy of the system is,

U=⟨ε⟩

=1Z∑sεse-βcs

=-1Zεs∑se-βss

Substituting the value of dZdβ=εs∑se-βcs

U=-1ZdZdβ

=-∂∂β(lnZ)

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

Z=(2coshβε)

lnZ=ln(2coshβε)

U=-1ZdZdβ

=-∂∂β(ln(2coshβε))

=1(2coshβε)((2sinhβε))α

=-Nεtanhβε

The average energy ==-Nεtanhβε

03

Step 3. Sketch for the following function

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Most popular questions from this chapter

In this problem you will use the mean field approximation to analyse the behaviour of the Ising model near the critical point.

(a) Prove that, when x≪1,tanhx≈x-13x3

(b) Use the result of part (a) to find an expression for the magnetisation of the Ising model, in the mean field approximation, when T is very close to the critical temperature. You should find M∝Tc-Tβ¯,whereβ(not to be confused with 1/kT) is a critical exponent, analogous to the f defined for a fluid in Problem 5.55. Onsager's exact solution shows that β=1/8in two dimensions, while experiments and more sophisticated approximations show that β≈1/3in three dimensions. The mean field approximation, however, predicts a larger value.

(c) The magnetic susceptibility χis defined as χ≡(∂M/∂B)T. The behaviour of this quantity near the critical point is conventionally written as χ∝T-Tc-γ , where y is another critical exponent. Find the value of in the mean field approximation, and show that it does not depend on whether T is slightly above or slightly below Te. (The exact value of y in two dimensions turns out to be 7/4, while in three dimensions γ≈1.24.)

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.

In Problem 8.15 you manually computed the energy of a particular state of a 4 x 4 square lattice. Repeat that computation, but this time apply periodic boundary conditions.

Problem 8.8. Show that the nthvirial coefficient depends on the diagrams in equation 8.23 that have ndots. Write the third virial coefficient, C(T), in terms of an integral of f-functions. Why it would be difficult to carry out this integral?

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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