Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Q. 8.4 (page 333)

Draw all the connected diagrams containing four dots. There are six diagrams in total; be careful to avoid drawing two diagrams that look superficially different but are actually the same. Which of the diagrams would remain connected if any single dot were removed?

Short Answer

Expert verified

All the connected diagrams that contain 4 dots have been drawn.

Step by step solution

01

Step 1. Given information

Configuration integral:- The configuration integral is used in probability theory, information theory and dynamical systems, it's a generalization of the definition of a partition function in statistical mechanics.

02

Step 2. Drawing all the diagrams in which four dots present.

(1)

(2)

(3)

(4)

(5)

(6)

03

Step 3. If a single dot is removed, these 3 will still remain connected.

(1)

(2)

(3)

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

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?

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¯-si¯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.

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.

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 andµ. 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 >. = e13µ,. Now, with the awkward factors of N(N - 1) · · · 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(µ) and P(µ) in terms of an integral of the Mayer /-function. Eliminate µ 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).

Use a computer to plot s¯ as a function of kT/ε, as predicted by mean field theory, for a two-dimensional Ising model (with a square lattice).
See all solutions

Recommended explanations on Physics Textbooks

Translational Dynamics

Read Explanation

Electricity and Magnetism

Read Explanation

Fields in Physics

Read Explanation

Atoms and Radioactivity

Read Explanation

Magnetism

Read Explanation

Linear Momentum

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