Q 8.29 To quantify the clustering of al... [FREE SOLUTION]

91影视

An Introduction to Thermal Physics

Daniel V. Schroeder

Physics

1st Edition 路 356 pages

ISBN: 9780201380279

Chapter 8: Q 8.29 (page 354)

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: s_is_j. 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
$c (r) = \overset{炉}{s_{i} s_{j}} - {\overset{炉}{s_{i}}}^{2}$
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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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: s_is_j. 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

$c (r) = \overset{炉}{s_{i} s_{j}} - {\overset{炉}{s_{i}}}^{2}$

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.