From 27589713566e8a2069c4c790b8f46d045f166e20 Mon Sep 17 00:00:00 2001 From: JerryChen97 <582020970@qq.com> Date: Fri, 9 Feb 2018 19:51:16 +1100 Subject: [PATCH] Polarization Cases I have finished the pimary analysis and calculation of cases with polarization and imbalances. Mainly differences lie in the additional terms of imbalance parts in the thermaldynamical potential, which is due to both the different of masses and chemical potentials. --- Polarization_Imbalance.nb | 1888 +++++++++++++++++++++++++++++++++++++ 1 file changed, 1888 insertions(+) create mode 100644 Polarization_Imbalance.nb diff --git a/Polarization_Imbalance.nb b/Polarization_Imbalance.nb new file mode 100644 index 0000000..91e18ba --- /dev/null +++ b/Polarization_Imbalance.nb @@ -0,0 +1,1888 @@ +(* Content-type: application/vnd.wolfram.mathematica *) + +(*** Wolfram Notebook File ***) +(* http://www.wolfram.com/nb *) + +(* CreatedBy='Mathematica 11.1' *) + +(*CacheID: 234*) +(* Internal cache information: +NotebookFileLineBreakTest +NotebookFileLineBreakTest +NotebookDataPosition[ 158, 7] +NotebookDataLength[ 78477, 1880] +NotebookOptionsPosition[ 76347, 1817] +NotebookOutlinePosition[ 76704, 1833] +CellTagsIndexPosition[ 76661, 1830] +WindowFrame->Normal*) + +(* Beginning of 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{3.7269676776622972`*^9, + 3.7269677077302637`*^9}, {3.726967803847849*^9, 3.726967804159703*^9}, + 3.7269679399186735`*^9, {3.7269681375367637`*^9, 3.7269681406269813`*^9}, { + 3.7269682456512547`*^9, 3.7269682542972884`*^9}, {3.72696842073483*^9, + 3.7269684724594035`*^9}, {3.726968518019517*^9, 3.7269685890353975`*^9}, { + 3.726968668992016*^9, 3.726968961052644*^9}, {3.726969012571667*^9, + 3.7269690936614943`*^9}, {3.726969131289547*^9, 3.7269691359018335`*^9}, + 3.726969169977436*^9, {3.7269692281871834`*^9, 3.7269692303589597`*^9}, { + 3.726969264061517*^9, 3.7269692914046507`*^9}, {3.7269693480949965`*^9, + 3.726969398551165*^9}, {3.726974353606646*^9, 3.726974356628274*^9}, + 3.726974810837315*^9, {3.726975112405623*^9, 3.726975143147767*^9}}, + ExpressionUUID -> "55714beb-69d6-4cb2-b076-66327c6ea8e2"], + +Cell[BoxData[ + RowBox[{ + StyleBox[ + RowBox[{"NIntegrate", "::", "inumr"}], "MessageName"], + RowBox[{ + ":", " "}], "\<\"The integrand \[NoBreak]\\!\\(\\(\\(k\\^2\\\\ \ +\\(\\((\\(\\(1.` \[VeryThinSpace]\\)\\) + k\\^2)\\)\\)\\)\\) + \ +\[CapitalDelta]\\^2\\/2 - \\(\\(k\\^2\\\\ \\@\\(\\((\\(\\(1.` \ +\[VeryThinSpace]\\)\\) + k\\^2)\\)\\^2 + \[CapitalDelta]\\^2\\)\\)\\)\\)\ +\[NoBreak] has evaluated to non-numerical values for all sampling points in \ +the region with boundaries \[NoBreak]\\!\\({\\({0, 5}\\)}\\)\[NoBreak]. \ +\\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", ButtonStyle->\\\"Link\\\", \ +ButtonFrame->None, ButtonData:>\\\"paclet:ref/message/NIntegrate/inumr\\\", \ +ButtonNote -> \\\"NIntegrate::inumr\\\"]\\)\"\>"}]], "Message", "MSG", + CellChangeTimes->{3.7269743582217712`*^9, 3.726974813382347*^9, + 3.7269751434921303`*^9}], + +Cell[BoxData[ + RowBox[{ + StyleBox[ + RowBox[{"NIntegrate", "::", "inumr"}], "MessageName"], + RowBox[{ + ":", " "}], "\<\"The integrand \[NoBreak]\\!\\(k\\^2\\\\ \ +\\(\\((\\(\\(\\(\\((\\(\\(1.` \[VeryThinSpace]\\)\\) - \ +\\(\\(0.3333333333333333`\\\\ k\\^2\\)\\) + \\@\\(\\(Plus[\\(\\(\ +\[LeftSkeleton] 2 \[RightSkeleton]\\)\\)]\\)\\^2 + \[CapitalDelta]\\^2\\))\\)\ +\\)\\\\ \\(\\(If[\\(\\(\\(\\(\\(\\(\\(\\(1.` \[VeryThinSpace]\\)\\) - \ +\\(\\(0.3333333333333333`\\\\ \\(\\(Power[\\(\\(\[LeftSkeleton] 2 \ +\[RightSkeleton]\\)\\)]\\)\\)\\)\\) + \\@\\(Plus[\\(\\(\[LeftSkeleton] 2 \ +\[RightSkeleton]\\)\\)]\\)\\)\\) < 0\\)\\), 1, 0\\)\\)]\\)\\)\\)\\) + \ +\\(\\(\\(\\((\\(\\(-1.`\\)\\) + \\(\\(0.3333333333333333`\\\\ k\\^2\\)\\) + \ +\\@\\(\\(Plus[\\(\\(\[LeftSkeleton] 2 \[RightSkeleton]\\)\\)]\\)\\^2 + \ +\[CapitalDelta]\\^2\\))\\)\\)\\\\ \\(\\(If[\\(\\(\\(\\(\\(\\(\\(\\(-1.`\\)\\) \ ++ \\(\\(0.3333333333333333`\\\\ \\(\\(Power[\\(\\(\[LeftSkeleton] 2 \ +\[RightSkeleton]\\)\\)]\\)\\)\\)\\) + \\@\\(Plus[\\(\\(\[LeftSkeleton] 2 \ +\[RightSkeleton]\\)\\)]\\)\\)\\) < 0\\)\\), 1, 0\\)\\)]\\)\\)\\)\\))\\)\\)\\)\ +\[NoBreak] has evaluated to non-numerical values for all sampling points in \ +the region with boundaries \[NoBreak]\\!\\({\\({0, 5}\\)}\\)\[NoBreak]. \ +\\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", ButtonStyle->\\\"Link\\\", \ +ButtonFrame->None, ButtonData:>\\\"paclet:ref/message/NIntegrate/inumr\\\", \ +ButtonNote -> \\\"NIntegrate::inumr\\\"]\\)\"\>"}]], "Message", "MSG", + CellChangeTimes->{3.7269743582217712`*^9, 3.726974813382347*^9, + 3.726975143513546*^9}], + +Cell[BoxData[ + RowBox[{ + StyleBox[ + RowBox[{"NIntegrate", "::", "inumr"}], "MessageName"], + RowBox[{ + ":", " "}], "\<\"The integrand \[NoBreak]\\!\\(\\(\\(k\\^2\\\\ \ +\\(\\((\\(\\(1.` \[VeryThinSpace]\\)\\) + k\\^2)\\)\\)\\)\\) + \ +\[CapitalDelta]\\^2\\/2 - \\(\\(k\\^2\\\\ \\@\\(\\((\\(\\(1.` \ +\[VeryThinSpace]\\)\\) + k\\^2)\\)\\^2 + \[CapitalDelta]\\^2\\)\\)\\)\\)\ +\[NoBreak] has evaluated to non-numerical values for all sampling points in \ +the region with boundaries \[NoBreak]\\!\\({\\({0, 5}\\)}\\)\[NoBreak]. \ +\\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", ButtonStyle->\\\"Link\\\", \ +ButtonFrame->None, ButtonData:>\\\"paclet:ref/message/NIntegrate/inumr\\\", \ +ButtonNote -> \\\"NIntegrate::inumr\\\"]\\)\"\>"}]], "Message", "MSG", + CellChangeTimes->{3.7269743582217712`*^9, 3.726974813382347*^9, + 3.726975143536277*^9}], + +Cell[BoxData[ + RowBox[{ + StyleBox[ + RowBox[{"General", "::", "stop"}], "MessageName"], + RowBox[{ + ":", " "}], "\<\"Further output of \ +\[NoBreak]\\!\\(\\*StyleBox[\\(NIntegrate :: inumr\\), \ +\\\"MessageName\\\"]\\)\[NoBreak] will be suppressed during this calculation. \ +\\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", ButtonStyle->\\\"Link\\\", \ +ButtonFrame->None, ButtonData:>\\\"paclet:ref/message/General/stop\\\", \ +ButtonNote -> \\\"General::stop\\\"]\\)\"\>"}]], "Message", "MSG", + CellChangeTimes->{3.7269743582217712`*^9, 3.726974813382347*^9, + 3.726975143550954*^9}], + +Cell[BoxData[ + RowBox[{ + StyleBox[ + RowBox[{"NIntegrate", "::", "slwcon"}], "MessageName"], + RowBox[{ + ":", " "}], "\<\"Numerical integration converging too slowly; suspect one \ +of the following: singularity, value of the integration is 0, highly \ +oscillatory integrand, or WorkingPrecision too small. \\!\\(\\*ButtonBox[\\\"\ +\[RightSkeleton]\\\", ButtonStyle->\\\"Link\\\", ButtonFrame->None, \ +ButtonData:>\\\"paclet:ref/message/NIntegrate/slwcon\\\", ButtonNote -> \ +\\\"NIntegrate::slwcon\\\"]\\)\"\>"}]], "Message", "MSG", + CellChangeTimes->{3.7269743582217712`*^9, 3.726974813382347*^9, + 3.726975144049362*^9}], + +Cell[BoxData[ + RowBox[{ + StyleBox[ + RowBox[{"NIntegrate", "::", "ncvb"}], "MessageName"], + RowBox[{ + ":", " "}], "\<\"NIntegrate failed to converge to prescribed accuracy after \ +\[NoBreak]\\!\\(9\\)\[NoBreak] recursive bisections in \[NoBreak]\\!\\(k\\)\ +\[NoBreak] near \[NoBreak]\\!\\({k}\\)\[NoBreak] = \ +\[NoBreak]\\!\\({4.835445152341417`15.954589770191005}\\)\[NoBreak]. \ +NIntegrate obtained \[NoBreak]\\!\\(6.491708354947283`*^-10\\)\[NoBreak] and \ +\[NoBreak]\\!\\(5.64180641227194`*^-15\\)\[NoBreak] for the integral and \ +error estimates. \\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", ButtonStyle->\\\ +\"Link\\\", ButtonFrame->None, \ +ButtonData:>\\\"paclet:ref/message/NIntegrate/ncvb\\\", ButtonNote -> \ +\\\"NIntegrate::ncvb\\\"]\\)\"\>"}]], "Message", "MSG", + CellChangeTimes->{3.7269743582217712`*^9, 3.726974813382347*^9, + 3.726975144075665*^9}], + +Cell[BoxData[ + RowBox[{ + StyleBox[ + RowBox[{"NIntegrate", "::", "slwcon"}], "MessageName"], + RowBox[{ + ":", " "}], "\<\"Numerical integration converging too slowly; suspect one \ +of the following: singularity, value of the integration is 0, highly \ +oscillatory integrand, or WorkingPrecision too small. \\!\\(\\*ButtonBox[\\\"\ +\[RightSkeleton]\\\", ButtonStyle->\\\"Link\\\", ButtonFrame->None, \ +ButtonData:>\\\"paclet:ref/message/NIntegrate/slwcon\\\", ButtonNote -> \ +\\\"NIntegrate::slwcon\\\"]\\)\"\>"}]], "Message", "MSG", + CellChangeTimes->{3.7269743582217712`*^9, 3.726974813382347*^9, + 3.726975144252795*^9}], + +Cell[BoxData[ + RowBox[{ + StyleBox[ + RowBox[{"NIntegrate", "::", "ncvb"}], "MessageName"], + RowBox[{ + ":", " "}], "\<\"NIntegrate failed to converge to prescribed accuracy after \ +\[NoBreak]\\!\\(9\\)\[NoBreak] recursive bisections in \[NoBreak]\\!\\(k\\)\ +\[NoBreak] near \[NoBreak]\\!\\({k}\\)\[NoBreak] = \ +\[NoBreak]\\!\\({4.835445152341417`15.954589770191005}\\)\[NoBreak]. \ +NIntegrate obtained \[NoBreak]\\!\\(6.491708354947283`*^-10\\)\[NoBreak] and \ +\[NoBreak]\\!\\(5.64180641227194`*^-15\\)\[NoBreak] for the integral and \ +error estimates. \\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", ButtonStyle->\\\ +\"Link\\\", ButtonFrame->None, \ +ButtonData:>\\\"paclet:ref/message/NIntegrate/ncvb\\\", ButtonNote -> \ +\\\"NIntegrate::ncvb\\\"]\\)\"\>"}]], "Message", "MSG", + CellChangeTimes->{3.7269743582217712`*^9, 3.726974813382347*^9, + 3.726975144278961*^9}], + +Cell[BoxData[ + RowBox[{ + StyleBox[ + RowBox[{"NIntegrate", "::", "slwcon"}], "MessageName"], + RowBox[{ + ":", " "}], "\<\"Numerical integration converging too slowly; suspect one \ +of the following: singularity, value of the integration is 0, highly \ +oscillatory integrand, or WorkingPrecision too small. \\!\\(\\*ButtonBox[\\\"\ +\[RightSkeleton]\\\", ButtonStyle->\\\"Link\\\", ButtonFrame->None, \ +ButtonData:>\\\"paclet:ref/message/NIntegrate/slwcon\\\", ButtonNote -> \ +\\\"NIntegrate::slwcon\\\"]\\)\"\>"}]], "Message", "MSG", + CellChangeTimes->{3.7269743582217712`*^9, 3.726974813382347*^9, + 3.726975144454548*^9}], + +Cell[BoxData[ + RowBox[{ + StyleBox[ + RowBox[{"General", "::", "stop"}], "MessageName"], + RowBox[{ + ":", " "}], "\<\"Further output of \ +\[NoBreak]\\!\\(\\*StyleBox[\\(NIntegrate :: slwcon\\), \ +\\\"MessageName\\\"]\\)\[NoBreak] will be suppressed during this calculation. \ +\\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", ButtonStyle->\\\"Link\\\", \ +ButtonFrame->None, ButtonData:>\\\"paclet:ref/message/General/stop\\\", \ +ButtonNote -> \\\"General::stop\\\"]\\)\"\>"}]], "Message", "MSG", + CellChangeTimes->{3.7269743582217712`*^9, 3.726974813382347*^9, + 3.726975144468851*^9}], + +Cell[BoxData[ + RowBox[{ + StyleBox[ + RowBox[{"NIntegrate", "::", "ncvb"}], "MessageName"], + RowBox[{ + ":", " "}], "\<\"NIntegrate failed to converge to prescribed accuracy after \ +\[NoBreak]\\!\\(9\\)\[NoBreak] recursive bisections in \[NoBreak]\\!\\(k\\)\ +\[NoBreak] near \[NoBreak]\\!\\({k}\\)\[NoBreak] = \ +\[NoBreak]\\!\\({4.94748847128778389270475912553592934273183345794677734375`\ +65.954589770191}\\)\[NoBreak]. NIntegrate obtained \ +\[NoBreak]\\!\\(6.485387624179541`*^-10\\)\[NoBreak] and \ +\[NoBreak]\\!\\(6.454580101038746`*^-15\\)\[NoBreak] for the integral and \ +error estimates. \\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", ButtonStyle->\\\ +\"Link\\\", ButtonFrame->None, \ +ButtonData:>\\\"paclet:ref/message/NIntegrate/ncvb\\\", ButtonNote -> \ +\\\"NIntegrate::ncvb\\\"]\\)\"\>"}]], "Message", "MSG", + CellChangeTimes->{3.7269743582217712`*^9, 3.726974813382347*^9, + 3.7269751444908943`*^9}], + +Cell[BoxData[ + RowBox[{ + StyleBox[ + RowBox[{"General", "::", "stop"}], "MessageName"], + RowBox[{ + ":", " "}], "\<\"Further output of \ +\[NoBreak]\\!\\(\\*StyleBox[\\(NIntegrate :: ncvb\\), \\\"MessageName\\\"]\\)\ +\[NoBreak] will be suppressed during this calculation. \ +\\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", ButtonStyle->\\\"Link\\\", \ +ButtonFrame->None, ButtonData:>\\\"paclet:ref/message/General/stop\\\", \ +ButtonNote -> \\\"General::stop\\\"]\\)\"\>"}]], "Message", "MSG", + CellChangeTimes->{3.7269743582217712`*^9, 3.726974813382347*^9, + 3.726975144505975*^9}], + +Cell[BoxData[ + RowBox[{ + StyleBox[ + RowBox[{"FindMinimum", "::", "lstol"}], "MessageName"], + RowBox[{ + ":", " "}], "\<\"The line search decreased the step size to within the \ +tolerance specified by AccuracyGoal and PrecisionGoal but was unable to find \ +a sufficient decrease in the function. You may need more than \ +\[NoBreak]\\!\\(MachinePrecision\\)\[NoBreak] digits of working precision to \ +meet these tolerances. \\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", \ +ButtonStyle->\\\"Link\\\", ButtonFrame->None, \ +ButtonData:>\\\"paclet:ref/message/FindMinimum/lstol\\\", ButtonNote -> \ +\\\"FindMinimum::lstol\\\"]\\)\"\>"}]], "Message", "MSG", + CellChangeTimes->{3.7269743582217712`*^9, 3.726974813382347*^9, + 3.726975147614923*^9}], + +Cell[BoxData[ + RowBox[{ + StyleBox[ + RowBox[{"FindMinimum", "::", "lstol"}], "MessageName"], + RowBox[{ + ":", " "}], "\<\"The line search decreased the step size to within the \ +tolerance specified by AccuracyGoal and PrecisionGoal but was unable to find \ +a sufficient decrease in the function. You may need more than \ +\[NoBreak]\\!\\(MachinePrecision\\)\[NoBreak] digits of working precision to \ +meet these tolerances. \\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", \ +ButtonStyle->\\\"Link\\\", ButtonFrame->None, \ +ButtonData:>\\\"paclet:ref/message/FindMinimum/lstol\\\", ButtonNote -> \ +\\\"FindMinimum::lstol\\\"]\\)\"\>"}]], "Message", "MSG", + CellChangeTimes->{3.7269743582217712`*^9, 3.726974813382347*^9, + 3.726975151544942*^9}], + +Cell[BoxData[ + RowBox[{ + StyleBox[ + RowBox[{"FindMinimum", "::", "lstol"}], "MessageName"], + RowBox[{ + ":", " "}], "\<\"The line search decreased the step size to within the \ +tolerance specified by AccuracyGoal and PrecisionGoal but was unable to find \ +a sufficient decrease in the function. You may need more than \ +\[NoBreak]\\!\\(MachinePrecision\\)\[NoBreak] digits of working precision to \ +meet these tolerances. \\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", \ +ButtonStyle->\\\"Link\\\", ButtonFrame->None, \ +ButtonData:>\\\"paclet:ref/message/FindMinimum/lstol\\\", ButtonNote -> \ +\\\"FindMinimum::lstol\\\"]\\)\"\>"}]], "Message", "MSG", + CellChangeTimes->{3.7269743582217712`*^9, 3.726974813382347*^9, + 3.726975155511532*^9}], + +Cell[BoxData[ + RowBox[{ + StyleBox[ + RowBox[{"General", "::", "stop"}], "MessageName"], + RowBox[{ + ":", " "}], "\<\"Further output of \ +\[NoBreak]\\!\\(\\*StyleBox[\\(FindMinimum :: lstol\\), \ +\\\"MessageName\\\"]\\)\[NoBreak] will be suppressed during this calculation. \ +\\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", ButtonStyle->\\\"Link\\\", \ +ButtonFrame->None, ButtonData:>\\\"paclet:ref/message/General/stop\\\", \ +ButtonNote -> \\\"General::stop\\\"]\\)\"\>"}]], "Message", "MSG", + CellChangeTimes->{3.7269743582217712`*^9, 3.726974813382347*^9, + 3.726975155526389*^9}], + +Cell[BoxData[ + RowBox[{ + StyleBox[ + RowBox[{"NIntegrate", "::", "errprec"}], "MessageName"], + RowBox[{ + ":", " "}], "\<\"Catastrophic loss of precision in the global error \ +estimate due to insufficient WorkingPrecision or divergent integral. \ +\\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", ButtonStyle->\\\"Link\\\", \ +ButtonFrame->None, ButtonData:>\\\"paclet:ref/NIntegrate\\\", ButtonNote -> \ +\\\"NIntegrate::errprec\\\"]\\)\"\>"}]], "Message", "MSG", + CellChangeTimes->{3.7269743582217712`*^9, 3.726974813382347*^9, + 3.726975476730262*^9}], + +Cell[BoxData[ + RowBox[{ + StyleBox[ + RowBox[{"NIntegrate", "::", "errprec"}], "MessageName"], + RowBox[{ + ":", " "}], "\<\"Catastrophic loss of precision in the global error \ +estimate due to insufficient WorkingPrecision or divergent integral. \ +\\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", ButtonStyle->\\\"Link\\\", \ +ButtonFrame->None, ButtonData:>\\\"paclet:ref/NIntegrate\\\", ButtonNote -> \ +\\\"NIntegrate::errprec\\\"]\\)\"\>"}]], "Message", "MSG", + CellChangeTimes->{3.7269743582217712`*^9, 3.726974813382347*^9, + 3.7269754768763*^9}], + +Cell[BoxData[ + RowBox[{ + StyleBox[ + RowBox[{"NIntegrate", "::", "errprec"}], "MessageName"], + RowBox[{ + ":", " "}], "\<\"Catastrophic loss of precision in the global error \ +estimate due to insufficient WorkingPrecision or divergent integral. \ +\\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", ButtonStyle->\\\"Link\\\", \ +ButtonFrame->None, ButtonData:>\\\"paclet:ref/NIntegrate\\\", ButtonNote -> \ +\\\"NIntegrate::errprec\\\"]\\)\"\>"}]], "Message", "MSG", + CellChangeTimes->{3.7269743582217712`*^9, 3.726974813382347*^9, + 3.726975476920856*^9}], + +Cell[BoxData[ + RowBox[{ + StyleBox[ + RowBox[{"General", "::", "stop"}], "MessageName"], + RowBox[{ + ":", " "}], "\<\"Further output of \ +\[NoBreak]\\!\\(\\*StyleBox[\\(NIntegrate :: errprec\\), \\\"MessageName\\\"]\ +\\)\[NoBreak] will be suppressed during this calculation. \\!\\(\\*ButtonBox[\ +\\\"\[RightSkeleton]\\\", ButtonStyle->\\\"Link\\\", ButtonFrame->None, \ +ButtonData:>\\\"paclet:ref/message/General/stop\\\", ButtonNote -> \ +\\\"General::stop\\\"]\\)\"\>"}]], "Message", "MSG", + CellChangeTimes->{3.7269743582217712`*^9, 3.726974813382347*^9, + 3.726975476935334*^9}], + +Cell[BoxData[ + RowBox[{ + StyleBox[ + RowBox[{"FindMinimum", "::", "nrnum"}], "MessageName"], + RowBox[{ + ":", " "}], "\<\"The function value \ +\[NoBreak]\\!\\(4.3976623586642924`*^-17 + \ +\\(\\(\\(-0.04957418683145378`\\)\\) + \ +\\(\\(NIntegrate[\\(\\(\\(\\(Compile`$1\\/2 + \\(\\(Compile`$220\\\\ \\(\\(\ +\[Xi][\\(\\(k, \\(\\(-0.3999999999999999`\\)\\)\\)\\)]\\)\\)\\)\\) - 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