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Notebook[{
Cell[CellGroupData[{
Cell["The package phasePlot3D", "Subtitle"],
Cell["\[Copyright] 2000 Heiko Feldmann", "Text"],
Cell["\<\
The package phasePlot3D serves to create three-dimensional plots of \
a function with discontinuities (e.g phase boundaries).
This may be for example some thermodynamic parameter plotted in the phase \
diagram,
but also any function with branch cuts in the complex plane.\
\>", "Text"],
Cell[CellGroupData[{
Cell["Introduction", "Section"],
Cell["\<\
First, one needs to define one function which gives information \
about the \"phase diagram\":
which function to plot in which region. It is typically called \
phaseSelection[{x,y}] and
returns integer values 1,2,3...n, where n is the number of different \"phases\
\". Typically,
it will consist of an \"If\"-command or a \"Which\"-command.
An example is shown here, defining two circles around {0,0} and {2,0}
as two phases and the rest as the third phase:\
\>", "Text"],
Cell[BoxData[
\(phaseSelection[{x_, y_}] :=
Which[\n\t\t{x, y} . {x, y} < 0.8, 1, \n
\t\t{x - 2, y} . {x - 2, y} < 0.8, 2, \n\t\tTrue, 3]\)], "Input",
Evaluatable->False],
Cell["\<\
The second ingredient are the functions to be plotted in the \
different regions. I called them phase1[{x,y}] etc.
As usual, one has to define the ranges of the variables in x- and y- \
direction.
Finally, phasePlot3D uses the same options as Graphics3D.
In total, the complete call has the form\
\>", "Text"],
Cell[BoxData[
\(phasePlot3D[
phaseSelection, {phase1, phase2\ , ... }, {x, xmin, xmax}, {y, ymin,
ymax}]\)], "Input",
Evaluatable->False],
Cell["\<\
Two things should be noticed:
1, The functions are used as \"pure functions\", i.e. without explicitely \
giving their arguments
2, The functions are expected to be of the form function[{x,y}], with the \
position {x,y} as a list of length 2 as the only argument.
Apart from this, phasePlot3D can be used as an intuitive extention of \
Mathematica.\
\>", "Text"]
}, Closed]],
Cell[CellGroupData[{
Cell["Example 1: What dithering can do to a cylinder", "Section"],
Cell[BoxData[
\(\(Needs["\", "\"]; \)\)],
"Input"],
Cell[CellGroupData[{
Cell[BoxData[
\(phase1[{x_, y_}] := 1; \nphase2[{x_, y_}] := \(-1\); \n
phase3[{x_, y_}] := 0; \n
phaseSelection[{x_, y_}] :=
Which[\n\t\t{x, y} . {x, y} < 0.8, 1, \n
\t\t{x - 2, y} . {x - 2, y} < 0.8, 2, \n\t\tTrue, 3]; \n
phasePlot3D[
phaseSelection, {phase1, phase2, phase3}, {x, \(-1\), 3}, {y, \(-1\),
1}, BoxRatios\ -> \ Automatic]; \)], "Input"],
Cell[GraphicsData["PostScript", "\<\
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behavior of ",
Cell[BoxData[
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" mixtures at low temperatures.\nIt was one of the first systems discovered \
that have a tricritical point. Details can be found in\nM. Blume, V. J. Emery \
and R. B. Griffiths, Phys. Rev. A 4, p.1071\nThe external parameters are \
T=Temperature and \[CapitalDelta]=chemical potential.\nShown in the (T,\
\[CapitalDelta]).plane is the concentration of ",
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Cell["The equations and a tiny bit of physics", "Subsection"],
Cell[TextData[
"To determine the first order line, we need the free energy f and the \
magnetisation m.\nThe transition line is obtained by equating the two \
solutions with M=0 and M\[NotEqual]0 below the tricritical point, while for \
the second order transition line there exists an analytic formula."], "Text"],
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M\ /. FindRoot[
M == 2\ Sinh[M/T]\/\(Exp[\[CapitalDelta]/T] + 2 Cosh[M/T]\), {M,
1}]; \nf[T_, \[CapitalDelta]_, M_] :=
T\ Log[Exp[\[CapitalDelta]/T] + 2 Cosh[M/T]] - M\^2/2; \n
firstOrderPoints =
Table[{T,
\[CapitalDelta]\ /.
FindRoot[
f[T, \[CapitalDelta], m[T, \[CapitalDelta]]] -
f[T, \[CapitalDelta], 0], {\[CapitalDelta], 0.4, 0.46}]}, {T,
1. /60, 1. /3 - 1. /60, 1. /60}]; \n
firstOrderLine\ = \
Interpolation[
Join[firstOrderPoints, {{0. , 0.5}, {1. /3, 2/3\ Log[2]}}]]; \n
secondOrderLine[T_] := T\ Log[2 \((1 - T)\)/T]; \n
transition[T_] :=
If[T\ < \ 1. /3, firstOrderLine[T], secondOrderLine[T]]; \)], "Input"],
Cell["This function determines the thermodynamic phase:", "Text"],
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Cell[BoxData[
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}], "Text"],
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\(phase1[{\[CapitalDelta]_, T_}] :=
1 - 2\/\(Exp[\[CapitalDelta]/T] + 2\); \n
phase2[{\[CapitalDelta]_, T_}] :=
1 - m[T, \[CapitalDelta]]\ Coth[m[T, \[CapitalDelta]]/T]; \)], "Input"]
}, Closed]],
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.5 Mabswid
.623 .733 .925 r
.69842 .75836 .6521 .74737 .63402 .76919 .67974 .77898 Metetra
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.71762 .73703 .67067 .72454 .6521 .74737 .69842 .75836 Metetra
.612 .71 .914 r
.73735 .7149 .68973 .70052 .67067 .72454 .71762 .73703 Metetra
.606 .694 .906 r
.75764 .69184 .70929 .67513 .68973 .70052 .73735 .7149 Metetra
.769 .572 .632 r
.7631 .43589 .71728 .43356 .70929 .67513 .75764 .69184 Metetra
.685 .78 .923 r
.78333 .4097 .73687 .40538 .71728 .43356 .7631 .43589 Metetra
.678 .771 .921 r
.80415 .38294 .75707 .37718 .73687 .40538 .78333 .4097 Metetra
.673 .764 .919 r
.82558 .35553 .77787 .34867 .75707 .37718 .80415 .38294 Metetra
.669 .76 .918 r
.84766 .32738 .79931 .31963 .77787 .34867 .82558 .35553 Metetra
.667 .756 .917 r
.8704 .29843 .82141 .28995 .79931 .31963 .84766 .32738 Metetra
.665 .754 .917 r
.89384 .26863 .8442 .25951 .82141 .28995 .8704 .29843 Metetra
.663 .752 .916 r
.91801 .23793 .8677 .22824 .8442 .25951 .89384 .26863 Metetra
.662 .751 .916 r
.94294 .20627 .89195 .19606 .8677 .22824 .91801 .23793 Metetra
.662 .75 .915 r
.96868 .17361 .91699 .16291 .89195 .19606 .94294 .20627 Metetra
.61 .697 .905 r
.6521 .74737 .6053 .73189 .58784 .75536 .63402 .76919 Metetra
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.659 .723 .896 r
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.66 .726 .898 r
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.668 .736 .901 r
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.67 .739 .902 r
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0 g
.25 Mabswid
.96795 .16753 m
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