compasses
This program, compasses, searches maximum, minimum and saddle-point states of
finite size dipole arrays (i.e. sets of compasses).
Copying
compasses
Copyright (C) 2014 Takeshi Nishimatsu
compasses is distributed in the hope that
it will be useful, but WITHOUT ANY WARRANTY.
You can copy, modify and redistribute compasses,
but only under the conditions described in
the GNU General Public License (the "GPL").
For more detail, see COPYING.
Installation instructions
Quick start
$ ./configure --help
$ ./configure
$ make
$ make test
$ evince examples/square/4x4-2.1669993154.eps
$ gv examples/square/4x4-2.1669993154.eps # If you do not have evince, use gv.
$ sudo make install # Install compasses into /usr/local/bin/.
Hints for ./configure
$ ./configure FC=g95
$ ./configure FC=ifort --with-lapack=mkl
Requirements
Preferables for developers
- autoconf (version 2.61 or higher)
- automake (version 1.10 or higher)
How to use compasses
compasses can read parameters from the standard input (stdin) as
well as from the file(s) given by the argument(s).
$ compasses < an_input_file
$ cat an_input_file | compasses
$ compasses input_file_1 input_file_2 input_file_3 ...
examples:
$ compasses < 4x4
$ compasses hex_with_center/hex03 hex_without_center/hex03
$ compasses 4x4 4x4dia
$ ls -1 4x4*
4x4
4x4+0.0814637120
4x4+0.0814637120.gp
4x4+0.0814637120.rotational
4x4+0.0814637120.xy
4x4+0.3772358527
4x4+0.3772358527.gp
4x4+0.3824498212
:
4x4.forcematrix
When your input file was foo, you will get foo and foo.gp, where
is the interaction energy per dipole. foo has data of position, strength
and direction of dipoles. foo.gp is a GNUPLOT script. When it was a z=0
problem, i.e. a problem of compasses on a table, you will also get
foo.rotational, foo.xy. foo.rotational contains eigenvalues
and eigenvectors of the rotational-angular expression.
foo.xy has interaction energy around the extremum along two
"directions". In the file foo.forcematrix, the forcematrix and
its eigenvectors and eigenvalues are written. Note that eigenvectors
and eigenvalues of the forcematrix are MEANINGLESS. But the eigenvectors
are used as initial guesses.
You can visualize the results by executing the GNUPLOT scripts, then you will
get EPS files. To get beautiful EPS files, you have to edit and adjust some
plotting parameters in your GNUPLOT scripts.
$ compasses 4x4dia
$ gnuplot 4x4dia-1.8693735707.gp
How to write an input file
The input file for compasses is a text file consisting of comment lines and 'tag = value(s)' lines.
Comment
Lines beginning with '#' are ignored.
Blank lines are also ignored.
# This is a comment line.
# Here are two more
# comment lines.
Tags
tag = value(s)
You must put ' = ', space-equal-space, between tag and value(s) as:
tag = 1.0
tag = -2.0 -3.0 -4.0
tag = 5.0 6.0 7.0
Followings are currently available tags.
Descriptions of tags
title
title = 'Example Title'
title = 'Example for the enhanced postscript terminal in gnuplot: {/Times-Italic a}_1'
dipole
position, [strength], [initial direction] of a dipole.
example:
dipole = 1.0 0.0 0.0
dipole = 1.0 1.0 0.0 2.0
dipole = 1.0 2.0 0.0 1.0 1.0 0.0 0.0
dumping
Dumping factor for steady minimization.
Lager positive dumping factor may achieve faster convergence to the extrema,
but may possibly fall into an oscillation in the updating scheme.
Its default is 0.1.
example:
dumping = 0.2
Output files
STDOUT
To STDOUT, compasses outputs input parameters, the number of dipoles (n_dipoles), and
calculated N*3+1 kinds of interaction energy. If positions and directions of dipoles are
restricted in a z=0 flat 2-dimensional area, compasses reports whether the system is
minimum, maximum or saddle state. Otherwise 'z' will be reported.
$ compasses 4x4
title = '4x4 dipole square lattice'
dipole = 0.0000000000000000 0.0000000000000000 0.0000000000000000 1.00 0.00 0.00 0.00
dipole = 1.0000000000000000 0.0000000000000000 0.0000000000000000 1.00 0.00 0.00 0.00
dipole = 2.0000000000000000 0.0000000000000000 0.0000000000000000 1.00 0.00 0.00 0.00
dipole = 3.0000000000000000 0.0000000000000000 0.0000000000000000 1.00 0.00 0.00 0.00
dipole = 0.0000000000000000 1.0000000000000000 0.0000000000000000 1.00 0.00 0.00 0.00
dipole = 1.0000000000000000 1.0000000000000000 0.0000000000000000 1.00 0.00 0.00 0.00
dipole = 2.0000000000000000 1.0000000000000000 0.0000000000000000 1.00 0.00 0.00 0.00
dipole = 3.0000000000000000 1.0000000000000000 0.0000000000000000 1.00 0.00 0.00 0.00
dipole = 0.0000000000000000 2.0000000000000000 0.0000000000000000 1.00 0.00 0.00 0.00
dipole = 1.0000000000000000 2.0000000000000000 0.0000000000000000 1.00 0.00 0.00 0.00
dipole = 2.0000000000000000 2.0000000000000000 0.0000000000000000 1.00 0.00 0.00 0.00
dipole = 3.0000000000000000 2.0000000000000000 0.0000000000000000 1.00 0.00 0.00 0.00
dipole = 0.0000000000000000 3.0000000000000000 0.0000000000000000 1.00 0.00 0.00 0.00
dipole = 1.0000000000000000 3.0000000000000000 0.0000000000000000 1.00 0.00 0.00 0.00
dipole = 2.0000000000000000 3.0000000000000000 0.0000000000000000 1.00 0.00 0.00 0.00
dipole = 3.0000000000000000 3.0000000000000000 0.0000000000000000 1.00 0.00 0.00 0.00
n_dipoles = 16
-2.1669993154 [interaction energy / dipole] min 1 th eigenvalue = -4.3659566912
-1.9673746064 [interaction energy / dipole] saddle 2 th eigenvalue = -4.3359322790
-1.9757644501 [interaction energy / dipole] min 3 th eigenvalue = -4.1086111417
-1.9757644501 [interaction energy / dipole] min 4 th eigenvalue = -4.1086111417
-1.9839701948 [interaction energy / dipole] min 5 th eigenvalue = -4.0806985789
:
-1.0931513536 [interaction energy / dipole] z 12 th eigenvalue = -2.3121735492
:
+2.3863435230 [interaction energy / dipole] max 49 th eigenvalue = 0.0000000000
foo.forcematrix
The force matrix.
foo-x.xxxxxxxxxx
Calculated configurations of dipoles, where -x.xxxxxxxxxx or +x.xxxxxxxxxx is the
interaction energy per dipole.
foo-x.xxxxxxxxxx.gp
A gnuplot script for that state.
foo-x.xxxxxxxxxx.rotational
The NxN interaction energy matrix in rotational-angular expression
around the extremum is given, if positions and directions of
dipoles are restricted in a z=0 flat 2-dimensional area.
foo-x.xxxxxxxxxx.xy
interaction energy along two "directions" around the extremum.
Examples
compasses can also minimize interaction energy of a
0-dimensional finite-size system consisting of
fixed-amplitude dipoles.
4x4 dipole square array
Input file:
examples/square/4x4max
examples/square/4x4dia
examples/square/4x4mirror
examples/square/4x4
Figure: The maximum state (a), saddle-point states (b) and (c), and minimum
states (d)-(f) for the 4 × 4 square array. (f) is the ground states.
Below each panel, averaged interaction energy per dipole is indicated.
In the idealized system, the dipoles are points at each center of
circle. Here, we emphasize the directions of dipoles and their packings
with arrows and circles.
Interaction energy per dipole around the saddle-point state (b) along two
"directions" are plotted.
20x20 dipole square array
Input file: examples/square/20x20
Figure: Ground state of the 20x20 dipole square array.
finite-size triangular arrays
Input file:
examples/hex_with_center/hex12 and examples/hex_without_center/hex12
(a) 
(b) 
Figure: Dipoles in two kinds of finite-size triangular arrays;
(a) with a central dipole and (b) without a central dipole.
The existence of central dipole breaks the 6-fold symmetry.
2x2x2 dipole array
Input file: examples/square/2x2x2
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Author
Takeshi Nishimatsu
(t-nissie{at}imr.tohoku.ac.jp or takeshi{at}physics.rutgers.edu)
Copyright © 2014 Takeshi Nishimatsu