Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Q 8.28 (page 354)

Modify the ising program to compute the total magnetisation (that is, the sum of all the s values) for each iteration, and to tally how often each possible magnetisation value occurs during a run, plotting the results as a histogram. Run the program for a 5 x 5 lattice at a variety of temperatures, and discuss the results. Sketch a graph of the most likely magnetisation value as a function of temperature. If your computer is fast enough, repeat for a 10 x 10 lattice.

Short Answer

Expert verified

Hence the codes and diagrams are given.

Step by step solution

01

Given information

The total magnetisation (that is, the sum of all the s values) for each iteration, and to tally how often each possible magnetisation value occurs during a run, plotting the results as a histogram. Run the program for a 5 x 5 lattice at a variety of temperatures,

02

Explanation

This exercise's code can be seen below.

03

Conclusion

As can be seen in the diagram above, the system becomes highly magnetic as the temperature is lowered. That is, the M = 0 mean value flips to a non-zero value.

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

Consider a gas of "hard spheres," which do not interact at all unless their separation distance is less than r0, in which case their interaction energy is infinite. Sketch the Mayer f-function for this gas, and compute the second virial coefficient. Discuss the result briefly.

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.

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.

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.)

Keeping only the first two diagrams in equation 8.23, and approximating N≈N-1≈N-2≈..... expand the exponential in a power series through the third power. Multiply each term out, and show that all the numerical coefficients give precisely the correct symmetry factors for the disconnected diagrams.
See all solutions

Recommended explanations on Physics Textbooks

Electric Charge Field and Potential

Read Explanation

Work Energy and Power

Read Explanation

Classical Mechanics

Read Explanation

Oscillations

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Further Mechanics and Thermal 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