Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Q 8.30 (page 355)

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.

Short Answer

Expert verified

Therefore, the codes and graphs are given.

Step by step solution

01

Given information

Modify the Ising program to simulate a one-dimensional Ising model.

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

02

Explanation

a) For temperatures of 1K, 1.5K, 2K, 2.5K, 3K, 3.5K, and 4K, some typical states (three states per temperature). As you scroll down, the temperature rises.

The program code is:

03

Explanation

b) Program to compute task b):

The graph is:

04

Conclusion

c) Program for task c):

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

By changing variables as in the text, express the diagram in equation 8.18 in terms of the same integral as in the equation8.31. Do the same for the last two diagrams in the first line of the equation8.20. Which diagrams cannot be written in terms of this basic integral?

Consider an Ising model of just two elementary dipoles, whose mutual interaction energy is ±ϵ. Enumerate the states of this system and write down their Boltzmann factors. Calculate the partition function. Find the probabilities of finding the dipoles parallel and antiparallel, and plot these probabilities as a function of kT/ϵ. Also calculate and plot the average energy of the system. At what temperatures are you more likely to find both dipoles pointing up than to find one up and one down?

Consider an Ising model of 100 elementary dipoles. Suppose you wish to calculate the partition function for this system, using a computer that can compute one billion terms of the partition function per second. How long must you wait for the answer?

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.)

You can estimate the size of any diagram by realizing that fr is of order 1 out to a distance of about the diameter of a molecule, andf≈0 beyond that. Hence, a three-dimensional integral of a product of f's will generally give a result that is of the order of the volume of a molecule. Estimate the sizes of all the diagrams shown explicitly in equation8.20 and explain why it was necessary to rewrite the series in exponential form.
See all solutions

Recommended explanations on Physics Textbooks

Astrophysics

Read Explanation

Physical Quantities And Units

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation

Fields in Physics

Read Explanation

Radiation

Read Explanation

Fundamentals of Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free