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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 85308, 3079]*) (*NotebookOutlinePosition[ 85995, 3104]*) (* CellTagsIndexPosition[ 85951, 3100]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Model Partition Function", "Subtitle", TextAlignment->Center, TextJustification->0], Cell["\<\ In this notebook, we calculate the asymptotic expansion of a \ one-dimensional integral, which serves as a model for a path integral \ partition function. \ \>", "Text"], Cell["\<\ \tProgrammer: Dick Furnstahl furnstahl.1@osu.edu \tRevision history: \t21-Jan-2003 --- original version for 880.05 \t23-Jan-2003 --- added checks and documentation\ \>", "Text"], Cell[CellGroupData[{ Cell["Definitions", "Subsection"], Cell[BoxData[ \(\(\(Off[General::spell1]\)\(\ \ \)\( (*\ turn\ off\ spelling\ warnings\ *) \)\)\)], "Input"], Cell["Define the integrand:", "Text"], Cell["\<\ integrand[a_, g_, x_] := Exp[(-a x^2/2 - g x^4/4)]/Sqrt[(2*Pi)/a]\ \>", "Input"], Cell["\<\ We can get an exact analytic result for the integral (note that we \ tell both Integrate and Simplify that a>0 and g>0 here).\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(exact[a_, g_] = Simplify[Integrate[integrand[a, g, x], {x, \(-Infinity\), Infinity}, Assumptions \[Rule] {a > 0, g > 0}], {a > 0, g > 0}]\)], "Input"], Cell[BoxData[ \(\(a\ \[ExponentialE]\^\(a\^2\/\(8\ g\)\)\ BesselK[1\/4, a\^2\/\(8\ g\)]\ \)\/\(2\ \@g\ \@\[Pi]\)\)], "Output"] }, Open ]], Cell["Now we define the full integral calculated numerically:", "Text"], Cell[BoxData[ \(exactN[a_, g_] := NIntegrate[integrand[a, g, x], {x, \(-Infinity\), Infinity}]\)], "Input"], Cell["\<\ The approximations follow from expanding the integrand in a Taylor \ series about g=0 out to n terms. (Note that we use a dummy value gp in doing \ the expansion and then substitute the passed value of g, which may be \ numerical.)\ \>", "Text"], Cell[BoxData[ \(approxN[a_, g_, n_]\ := \ NIntegrate[ Normal[Series[integrand[a, gp, x], {gp, 0, n}]] /. gp \[Rule] g, {x, \(-Infinity\), Infinity}]\)], "Input"], Cell[BoxData[ \(approx[a_, g_, n_] := Integrate[ Normal[Series[integrand[a, gp, x], {gp, 0, n}]] /. {gp \[Rule] g}, {x, \(-Infinity\), Infinity}, Assumptions \[Rule] {a > 0, g > 0}]\)], "Input"], Cell["\<\ R0 is a reference value: the difference of the exact result and \ the result with g=0.\ \>", "Text"], Cell[BoxData[ \(R0[a_, \ g_]\ := \ Abs[exact[a, g] - exact[a, 0]]\)], "Input"], Cell[BoxData[ \(R0N[a_, g_] := Abs[exactN[a, g] - exactN[a, 0]]\)], "Input"], Cell["Define absolute and relative error functions:", "Text"], Cell[BoxData[ \(R[a_, \ g_, \ n_]\ := \ Abs[approx[a, g, n] - exact[a, g]]\)], "Input"], Cell[BoxData[ \(RN[a_, g_, n_] := Abs[approxN[a, g, n] - exactN[a, g]]\)], "Input"], Cell[BoxData[ \(RelN[a_, g_, n_] := RN[a, g, n]/R0N[a, g]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Checks", "Subsection"], Cell["\<\ Here we make some spot checks of the numerical vs. analytic \ calculations. We turn off the underflow messages from the numerical \ calculations (but we shouldn't forget they are there!).\ \>", "Text"], Cell[BoxData[ \(\(\(Off[General::unfl]\)\(\ \ \)\( (*\ turn\ off\ underflow\ messages\ *) \)\)\)], "Input"], Cell[BoxData[ \(checkR[a_, g_, n_]\ := \ Abs[1 - RN[a, g, n]/R[a, g, n]]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(checkR[1, .2, 20]\)], "Input"], Cell[BoxData[ \(1.887379141862766`*^-14\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(checkR[1, .1, 10]\)], "Input"], Cell[BoxData[ \(8.815170815523743`*^-14\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(checkR[1, .01, 10]\)], "Input"], Cell[BoxData[ \(0.000029674457075246075`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(checkR[1, .01, 25]\)], "Input"], Cell[BoxData[ \(0.2136494252873563`\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(checkR[1, .01, 40]\)], "Input"], Cell[BoxData[ \(0.6531643989514417`\)], "Output"] }, Open ]], Cell["\<\ The relative error (calculated by checkR) is very small for larger \ values of the coupling, but degrades for the smallest value we will consider \ (g=0.01) and becomes particularly poor for the largest values of n we will \ consider (up to n=40). However, since we will look at these on a log scale, \ the accuracy of the numerical calculations is good enough and they are much \ faster. \ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Tables ", "Subsection"], Cell["\<\ In this section we generate tables of the absolute difference \ between the exact (numerical) answer and the result from a perturbative \ expansion with n terms, for a =1 and values of g ranging from 0.01 to 0.2, \ and also for this difference normalized to the difference between the exact \ result and the g=0 result. Output is suppressed (by ;'s).\ \>", "Text"], Cell[BoxData[ \(\(t1\ = \ Table[RN[1, .01, n], {n, 0, 40}];\)\)], "Input"], Cell[BoxData[ \(\(t1b = Table[RelN[1, .01, n], {n, 0, 40}];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(t2 = Table[RN[1, .02, n], {n, 0, 40}];\)\)], "Input"], Cell[BoxData[ \(NIntegrate::"ncvb" \(\(:\)\(\ \)\) "NIntegrate failed to converge to prescribed accuracy after \!\(7\) \ recursive bisections in \!\(x\) near \!\(x\) = \ \!\(-12.473684210526315`\)."\)], "Message"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(t2b = Table[RelN[1, .02, n], {n, 0, 40}];\)\)], "Input"], Cell[BoxData[ \(NIntegrate::"ncvb" \(\(:\)\(\ \)\) "NIntegrate failed to converge to prescribed accuracy after \!\(7\) \ recursive bisections in \!\(x\) near \!\(x\) = \ \!\(-12.473684210526315`\)."\)], "Message"] }, Open ]], Cell[BoxData[ \(\(t3 = Table[RN[1, .05, n], {n, 0, 40}];\)\)], "Input"], Cell[BoxData[ \(\(t3b = Table[RelN[1, .05, n], {n, 0, 40}];\)\)], "Input"], Cell[BoxData[ \(\(t4 = Table[RN[1, .1, n], {n, 0, 40}];\)\)], "Input"], Cell[BoxData[ \(\(t4b = Table[RelN[1, .1, n], {n, 0, 40}];\)\)], "Input"], Cell[BoxData[ \(\(t5 = Table[RN[1, .2, n], {n, 0, 40}];\)\)], "Input"], Cell[BoxData[ \(\(t5b = Table[RelN[1, .2, n], {n, 0, 40}];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Graphs", "Subsection"], Cell["\<\ Log plots are made for each of the tables, with the display \ suppressed (using DisplayFunction->Identity). \"Show\" is used to combine the plots to one graph.\ \>", "Text"], Cell[BoxData[ \(<< Graphics`Graphics`\)], "Input"], Cell[BoxData[ \(\(LL1\ = \ LogListPlot[t1, PlotStyle \[Rule] {PointSize[ .015]}, DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell[BoxData[ \(\(LL2 = LogListPlot[t2, PlotStyle \[Rule] {PointSize[ .015]}, DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell[BoxData[ \(\(LL3 = LogListPlot[t3, PlotStyle \[Rule] {PointSize[ .015]}, DisplayFunction \[Rule] Identity];\)\)], "Input"], Cell[BoxData[ 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"\"]\)\(\[IndentingNewLine]\)\)\)], "Input"], Cell[BoxData[ \("Z_int_error.dat"\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Export["\", TableForm[{Table[i, {i, 0, 40}], t1b, t2b, t3b, t4b, t5b}, TableDirections \[Rule] {Row, Column}, TableSpacing \[Rule] {1, 0}], "\"]\)], "Input"], Cell[BoxData[ \("Z_int_relerr.dat"\)], "Output"] }, Open ]], Cell["\<\ Here is a sample gnuplot plotfile for the relative error. If it is \ called Z_integral_relerr.plt, invoke it from within gnuplot using: gnuplot> load \"Z_integral_relerr.plt \tset logscale y \tset title \"Relative Error for Model Partition Function\" \tset xlabel \"# of terms\" \tset ylabel \"relative error\" \tset xrange [0:40] \tset yrange [1e-11:1] \tplot \\ \t \"Z_int_relerr.dat\" using 1:2 title \"lambda = 0.01\" ,\\ \t\"Z_int_relerr.dat\" using 1:3 title \"lambda = 0.02\" ,\\ \t\"Z_int_relerr.dat\" using 1:4 title \"lambda = 0.05\" ,\\ \t\"Z_int_relerr.dat\" using 1:5 title \"lambda = 0.10\" ,\\ \t\"Z_int_relerr.dat\" using 1:6 title \"lambda = 0.20\" \tset out \"Z_int_relerr1.ps\" \tset term postscript \treplot \tset xrange [0:15] \tset yrange [1e-2:10] \tset out \"Z_int_relerr2.ps\" \tset term postscript \treplot \t\ \>", "Text"] }, Open ]] }, Open ]] }, Open ]] }, FrontEndVersion->"4.0 for X", ScreenRectangle->{{0, 1280}, {0, 1024}}, WindowToolbars->"EditBar", WindowSize->{968, 867}, WindowMargins->{{74, Automatic}, {Automatic, 14}}, Magnification->1.5 ] (*********************************************************************** Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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