(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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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[ 38770, 1030]*) (*NotebookOutlinePosition[ 39477, 1054]*) (* CellTagsIndexPosition[ 39433, 1050]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ "Special Problem Reading Summary\nSummer 2004, Dr. Predrag Cvitanovic\nby \ Odell Austin Collins\n\n\nThis summer's reading focused on two categories: \ reviewing two preprints to nonlinearity on relating Eulerian fluid flows and \ mappings to the Schrodinger equation, and independent readings on various \ information metrics and their relation to physical phenomena.\n\n", StyleBox["First, we review the two non-linearity pre-prints:", FontVariations->{"Underline"->True}], "\n\n", StyleBox["Mapping of the classical kinetic balance equations onto the \ Schrodinger equation ", FontWeight->"Bold"], "(Adriana I Pesci, Raymond E Goldstein), and ", StyleBox["Mapping of the classical kinetic balance equations onto the Pauli \ equation", FontWeight->"Bold"], " (Adriana I Pesci, Raymond E Goldstein, and Hermann Uys)", "\n\nIn this paper, the authors start with Liouville's equation and using \ what appears to be a standard procedure generate the \"BBKGY hierarchy\" of N \ coupled reduced probability distributions:\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(f\_j\), "(", RowBox[{\(x\^N\), ",", SuperscriptBox[ StyleBox["p", FontWeight->"Bold"], "N"]}], ")"}], " ", "=", " ", RowBox[{\(\[Integral]\_\[CapitalOmega]\), RowBox[{ RowBox[{\(f\_N\), "(", RowBox[{\(x\^N\), ",", SuperscriptBox[ StyleBox["p", FontWeight->"Bold"], "N"]}], ")"}], " ", RowBox[{\(\[Product]\+\(l = j + 1\)\%N\), RowBox[{\(\[DifferentialD]x\), RowBox[{"\[DifferentialD]", StyleBox["p", FontWeight->"Bold"]}]}]}]}]}]}], TraditionalForm]]] }], "Text", PageWidth->PaperWidth, TextAlignment->Left, FontSize->14], Cell[TextData[{ StyleBox["where ", FontSize->14], Cell[BoxData[ FormBox[ RowBox[{\(f\_1\), " ", "and", " ", SubscriptBox[ StyleBox["f", FontSlant->"Italic"], "2"]}], TraditionalForm]], FontSize->14], StyleBox["determine the kinetic and potential energy of some collection of \ particles.\n\nThese equations can be decoupled by varying choices of \ assumptions. In particular, one assumption credited as the \"Bogliobuv ansatz\ \" which corresponds to no rotational motion about the centroid of motion can \ decouple the above reduced probability functions to produce Boltzmann's \ equation for a single particle :", FontSize->14], "\n ", StyleBox["\n ", FontSize->18], Cell[BoxData[ FormBox[ StyleBox[\(\[PartialD]\ f\_1\/\[PartialD]t\), FontSize->18], TraditionalForm]]], StyleBox["+ ", FontSize->18], Cell[BoxData[ FormBox[ RowBox[{ FractionBox[ SubscriptBox[ StyleBox["p", FontWeight->"Bold"], "1"], "m"], "\[CenterDot]"}], TraditionalForm]], FontSize->18], Cell[BoxData[{ FormBox[ RowBox[{\(\[PartialD]\ f\_1\/\[PartialD]t\), " ", "-", " ", RowBox[{\(\[PartialD]\(V(x\_1)\)\/\[PartialD]x\_1\), "\[CenterDot]", " ", FractionBox[\(\[PartialD]f\_1\), RowBox[{"\[PartialD]", SubscriptBox[ StyleBox["p", FontWeight->"Bold"], "1"]}]]}]}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{"=", RowBox[{"\[Integral]", RowBox[{\(\[DifferentialD]x\_2\), RowBox[{"\[Integral]", RowBox[{"\[DifferentialD]", RowBox[{ SubscriptBox[ StyleBox["p", FontWeight->"Bold"], "2"], "[", RowBox[{ RowBox[{ RowBox[{\(f\_1\), "(", SubsuperscriptBox[ StyleBox["p", FontWeight->"Bold"], "1", "'"], ")"}], " ", RowBox[{\(f\_1\), "(", SubsuperscriptBox[ StyleBox["p", FontWeight->"Bold"], "2", "'"], ")"}]}], " ", "-", " ", RowBox[{ RowBox[{\(f\_1\), "(", SubscriptBox[ StyleBox["p", FontWeight->"Bold"], "1"], ")"}], " ", RowBox[{\(f\_1\), "(", SubscriptBox[ StyleBox["p", FontWeight->"Bold"], "2"], ")"}]}]}], "]"}]}]}]}]}]}], TraditionalForm]}], FontSize->18], "\n", StyleBox[" \nIntegrating over momentum allows them to use conservation of \ particle number and momentum to write:\n", FontSize->16], "\n", StyleBox[" ", FontSize->16], StyleBox[" ", FontSize->24], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{\(\[Integral]\_\(-\[Infinity]\)\%\[Infinity]\), RowBox[{ RowBox[{"\[DifferentialD]", StyleBox["p", FontWeight->"Bold"]}], StyleBox[" ", FontWeight->"Bold"], RowBox[{"(", RowBox[{\(\[PartialD]f\_1\/\[PartialD]t\), "+", RowBox[{ FractionBox[ StyleBox["p", FontWeight->"Bold"], "m"], "\[CenterDot]", \(\[PartialD]f\_1\/\[PartialD]t\)}], "-", RowBox[{\(\[PartialD]\(V(x)\)\/\[PartialD]x\), "\[CenterDot]", FractionBox[\(\[PartialD]f\_1\), RowBox[{"\[PartialD]", StyleBox["p", FontWeight->"Bold"]}]]}]}], ")"}]}]}], "=", "0"}], " ", ","}], TraditionalForm]], FontSize->18], StyleBox["\n \n ", FontSize->18], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\[Integral]\_\(-\[Infinity]\)\%\[Infinity]\), RowBox[{ StyleBox["p", FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], RowBox[{"\[DifferentialD]", StyleBox["p", FontWeight->"Bold"]}], StyleBox[" ", FontWeight->"Bold"], RowBox[{"(", RowBox[{\(\[PartialD]f\_1\/\[PartialD]t\), "+", RowBox[{ FractionBox[ StyleBox["p", FontWeight->"Bold"], "m"], "\[CenterDot]", \(\[PartialD]f\_1\/\[PartialD]t\)}], "-", RowBox[{\(\[PartialD]\(V(x)\)\/\[PartialD]x\), "\[CenterDot]", FractionBox[\(\[PartialD]f\_1\), RowBox[{"\[PartialD]", StyleBox["p", FontWeight->"Bold"]}]]}]}], ")"}]}]}], "=", "0"}], TraditionalForm]], FontSize->18], "\n\n", StyleBox["Until now, the above steps are standard. Their original \ contribution comes in examining these balance equations in terms of the \ fourier representations of ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`\(\(f\_1. \)\(.\)\)\)], FontSize->14], StyleBox["They take \n\n", FontSize->14], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(f\_1\), "(", RowBox[{"x", ",", StyleBox["p", FontWeight->"Bold"], StyleBox[",", FontWeight->"Plain"], StyleBox["t", FontWeight->"Plain"]}], StyleBox[")", FontWeight->"Plain"]}], StyleBox[" ", FontWeight->"Plain"], StyleBox["=", FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], RowBox[{ FractionBox[ StyleBox["1", FontWeight->"Plain"], \(\((2\ \[Pi]\ \[Eta])\)\^3\)], RowBox[{\(\[Integral]\_\(-\[Infinity]\)\%\[Infinity]\), RowBox[{ RowBox[{"exp", "(", RowBox[{ RowBox[{"-", " ", StyleBox["i", FontSlant->"Italic"]}], StyleBox[" ", FontSlant->"Plain"], FractionBox[ StyleBox[\(p\[CenterDot]y\), FontWeight->"Bold", FontSlant->"Plain"], "\[Eta]"]}], ")"}], RowBox[{\(f\&^\), "(", RowBox[{ StyleBox["x", FontWeight->"Bold"], StyleBox[",", FontWeight->"Bold"], StyleBox["y", FontWeight->"Bold"], StyleBox[",", FontWeight->"Bold"], StyleBox["t", FontWeight->"Plain"]}], StyleBox[")", FontWeight->"Plain"]}], RowBox[{"\[DifferentialD]", StyleBox["y", FontWeight->"Bold"]}]}]}]}]}], TraditionalForm]], FontSize->14], StyleBox["\n\nThe balance equations then become\n\n ", FontSize->14], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FractionBox[ StyleBox["1", FontWeight->"Plain"], \(\((2\ \[Pi]\ \[Eta])\)\^3\)], " ", RowBox[{"\[Integral]", " ", RowBox[{\(\[Integral]\_\(-\[Infinity]\)\%\[Infinity]\), RowBox[{ RowBox[{"\[DifferentialD]", StyleBox["p", FontWeight->"Bold"]}], RowBox[{"\[DifferentialD]", StyleBox["y", FontWeight->"Bold"]}], StyleBox[" ", FontWeight->"Plain"], RowBox[{ RowBox[{ StyleBox["exp", FontWeight->"Plain"], StyleBox["(", FontWeight->"Plain"], RowBox[{ StyleBox[\(-i\), FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], FractionBox[ StyleBox[\(p\[CenterDot]y\), FontWeight->"Bold"], "\[Eta]"]}], ")"}], "[", RowBox[{\(\[PartialD]f\/\[PartialD]t\), "+", RowBox[{\(\[Eta]\/\(i\ m\)\), RowBox[{\(\[PartialD]\/\[PartialD]x\), "\[CenterDot]", FractionBox[\(\[PartialD]f\&^\), RowBox[{"\[PartialD]", StyleBox["y", FontWeight->"Bold"]}]]}]}]}], "]"}]}]}]}]}], "=", "0"}], TraditionalForm]], FontSize->14], StyleBox["\n \n ", FontSize->14], Cell[BoxData[ FormBox[ RowBox[{ FormBox[ RowBox[{ FractionBox[ StyleBox["1", FontWeight->"Plain"], \(\((2\ \[Pi]\ \[Eta])\)\^3\)], " ", RowBox[{"\[Integral]", " ", RowBox[{\(\[Integral]\_\(-\[Infinity]\)\%\[Infinity]\), RowBox[{ RowBox[{"\[DifferentialD]", StyleBox["p", FontWeight->"Bold"]}], RowBox[{"\[DifferentialD]", StyleBox["y", FontWeight->"Bold"]}], StyleBox[" ", FontWeight->"Plain"], RowBox[{ StyleBox["exp", FontWeight->"Plain"], StyleBox["(", FontWeight->"Plain"], RowBox[{ StyleBox[\(-i\), FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], FractionBox[ StyleBox[\(p\[CenterDot]y\), FontWeight->"Bold"], "\[Eta]"]}], ")"}], " "}]}]}]}], "TraditionalForm"], "[", RowBox[{ RowBox[{\(\[PartialD]\/\[PartialD]t\), RowBox[{"(", RowBox[{\(\[Eta]\/i\), FractionBox[\(\[PartialD]f\&^\), RowBox[{"\[PartialD]", StyleBox["y", FontWeight->"Bold"]}]]}], ")"}]}], "-", RowBox[{\(\[Eta]\^2\/m\), RowBox[{ FractionBox["\[PartialD]", RowBox[{"\[PartialD]", StyleBox["x", FontWeight->"Bold"]}]], "\[CenterDot]", RowBox[{"(", FractionBox[\(\(\[PartialD]\^2 \ f\&^\)\&\[LongLeftRightArrow]\), RowBox[{"\[PartialD]", StyleBox[\(y^2\), FontWeight->"Bold"]}]], StyleBox[")", FontWeight->"Plain"]}]}]}], StyleBox["+", FontWeight->"Plain"], RowBox[{ FractionBox[ StyleBox["\[PartialD]", FontWeight->"Plain"], RowBox[{"\[PartialD]", StyleBox["y", FontWeight->"Bold"]}]], RowBox[{"(", RowBox[{ RowBox[{ StyleBox["y", FontWeight->"Bold"], StyleBox["\[CenterDot]", FontWeight->"Bold"], FractionBox[ StyleBox[\(\[PartialD]V\), FontWeight->"Plain"], RowBox[{"\[PartialD]", StyleBox["x", FontWeight->"Bold"]}]]}], \(f\&^\)}], ")"}]}]}], "]"}], TraditionalForm]], FontSize->14], StyleBox["\n \nThese equations can be manipulated to show\n\n", FontSize->14], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ UnderscriptBox[ StyleBox["lim", FontSlant->"Italic"], \(y \[Rule] 0\)], "[", RowBox[{\(\[PartialD]f\&^\/\[PartialD]t\), "+", RowBox[{\(\[Eta]\/\(i\ m\)\), RowBox[{ FractionBox["\[PartialD]", RowBox[{"\[PartialD]", StyleBox["x", FontWeight->"Bold"]}]], "\[CenterDot]", FractionBox[\(\[PartialD]f\&^\), RowBox[{"\[PartialD]", StyleBox["y", FontWeight->"Bold"]}]]}]}]}], "]"}], "=", "0"}], TraditionalForm]], FontSize->14], StyleBox["\n\n", FontSize->14], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ UnderscriptBox[ StyleBox["lim", FontSlant->"Italic"], \(y \[Rule] 0\)], RowBox[{"[", RowBox[{ RowBox[{\(\[PartialD]\/\[PartialD]t\), RowBox[{"(", RowBox[{\(\[Eta]\/i\), FractionBox[\(\[PartialD]f\&^\), RowBox[{"\[PartialD]", StyleBox["y", FontWeight->"Bold"]}]]}], ")"}]}], "-", RowBox[{\(\[Eta]\^2\/m\), RowBox[{ FractionBox["\[PartialD]", RowBox[{"\[PartialD]", StyleBox["x", FontWeight->"Bold"]}]], "\[CenterDot]", RowBox[{"(", FractionBox[\(\(\[PartialD]\^2 f\&^\)\&\ \[LongLeftRightArrow]\), RowBox[{"\[PartialD]", StyleBox[\(y^2\), FontWeight->"Bold"]}]], StyleBox[")", FontWeight->"Plain"]}]}]}], StyleBox["+", FontWeight->"Plain"], RowBox[{ FractionBox[ StyleBox[\(\[PartialD]V\), FontWeight->"Plain"], RowBox[{"\[PartialD]", StyleBox["x", FontWeight->"Bold"]}]], \(f\&^\)}], "+", RowBox[{"O", "(", StyleBox["y", FontWeight->"Bold"], StyleBox[")", FontWeight->"Plain"]}]}], "]"}]}], "=", "0"}], TraditionalForm]], FontSize->14], StyleBox["\n\nThe authors highlight the fact that these limits take the \ same functional form that Frohlich found in his derivation of quantum \ hydrodynamics. Using an algebraic change of variables and much manipulation, \ the authors produce:\n\n\[Integral]", FontSize->14], StyleBox[" ", FontSize->14, FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(dx\^''\), RowBox[{ SubscriptBox[\(\(lim\)\(\ \)\), RowBox[{ StyleBox[\(x'\), FontWeight->"Bold"], StyleBox["\[Rule]", FontWeight->"Bold"], StyleBox[\(x''\), FontWeight->"Plain"]}]], "[", RowBox[{\(i\ \[Eta]\ \ \[PartialD]f\&^\/\[PartialD]t\), "+", RowBox[{\(\[Eta]\^2\/\(2\ m\)\), RowBox[{"(", RowBox[{ FractionBox[\(\[PartialD]\^2 f\&^\), RowBox[{"\[PartialD]", SuperscriptBox[ StyleBox["x", FontWeight->"Bold"], \(\('\) \(2\)\)]}]], "-", FractionBox[\(\[PartialD]\^2 f\&^\), RowBox[{"\[PartialD]", SuperscriptBox[ StyleBox["x", FontWeight->"Bold"], \(\(''\) \(2\)\)]}]]}], ")"}]}]}], "]"}]}], " ", "=", " ", "0"}], TraditionalForm]], FontSize->14], StyleBox["\n\n\[Integral]", FontSize->14], StyleBox[" ", FontSize->14, FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(dx\^''\), SubscriptBox[\(\(lim\)\(\ \)\), RowBox[{ StyleBox[\(x'\), FontWeight->"Bold"], StyleBox["\[Rule]", FontWeight->"Bold"], StyleBox[\(x''\), FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"]}]], \(1\/2\), RowBox[{"(", RowBox[{ FractionBox["\[PartialD]", RowBox[{"\[PartialD]", SuperscriptBox[ StyleBox["x", FontWeight->"Bold"], "'"]}]], "-", FractionBox["\[PartialD]", RowBox[{"\[PartialD]", SuperscriptBox[ StyleBox["x", FontWeight->"Bold"], "''"]}]]}], ")"}], " ", "\[Times]", RowBox[{"[", RowBox[{\(\[Eta]\/i\ \ \[PartialD]f\&^\/\[PartialD]t\), "+", RowBox[{\(\[Eta]\^2\/\(2\ m\)\), RowBox[{"(", RowBox[{ FractionBox[\(\[PartialD]\^2 f\&^\), RowBox[{"\[PartialD]", SuperscriptBox[ StyleBox["x", FontWeight->"Bold"], \(\('\) \(2\)\)]}]], "-", FractionBox[\(\[PartialD]\^2 f\&^\), RowBox[{"\[PartialD]", SuperscriptBox[ StyleBox["x", FontWeight->"Bold"], \(\(''\) \(2\)\)]}]]}], ")"}]}], " ", "+", " ", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"V", "(", SuperscriptBox[ StyleBox["x", FontWeight->"Bold"], "'"], ")"}], " ", "-", " ", RowBox[{"V", "(", SuperscriptBox[ StyleBox["x", FontWeight->"Bold"], "''"], ")"}]}], ")"}], \(f\&^\)}]}], "]"}]}], " ", "=", " ", "0"}], TraditionalForm]], FontSize->14], StyleBox["\n\nHere, the authors note two points: one, that the Fourier \ transform has linearized and made separable (up to O(", FontSize->14], StyleBox["y)) ", FontSize->14, FontWeight->"Bold"], StyleBox["the non-linear part of the earlier differential equation, and \ this non-linear term will later become the convective term. The second is \ that these equations are invariant under the exchange ", FontSize->14], StyleBox["x", FontSize->14, FontWeight->"Bold"], StyleBox["\[LeftRightArrow]", FontSize->14], StyleBox["p.\n\n", FontSize->14, FontWeight->"Bold"], StyleBox["Furthermore, the authors state that there are only separable \ solutions for a small subclass of solutions, and require that initial and \ boundary conditions be separable as well. However, they claim this subclass \ is the same (and only) subclass earlier demonstrated to map the \ Sturm-Liouville operator to the Schrodinger equation.\n\nThese separable \ solutions can eventually be manipulated to show:\n\n", FontSize->14], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\[PartialD]\[Rho]\/\[PartialD]t\), "+", RowBox[{ StyleBox["\[Del]", FontWeight->"Bold"], RowBox[{"\[CenterDot]", RowBox[{"(", RowBox[{"\[Rho]", " ", StyleBox["u", FontWeight->"Bold", FontSlant->"Plain"]}], StyleBox[")", FontWeight->"Plain"]}]}]}]}], StyleBox["=", FontWeight->"Plain"], StyleBox["0", FontWeight->"Plain"]}], TraditionalForm]], FontSize->14], StyleBox["\n\n", FontSize->14], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{\(\[PartialD]\/\[PartialD]t\), RowBox[{"(", RowBox[{"m", " ", "\[Rho]", " ", StyleBox["u", FontWeight->"Bold", FontSlant->"Plain"]}], ")"}]}], "+", RowBox[{ StyleBox["\[Del]", FontWeight->"Bold"], RowBox[{ StyleBox["\[CenterDot]", FontWeight->"Bold"], RowBox[{"[", RowBox[{"\[Rho]", " ", RowBox[{"(", RowBox[{ RowBox[{ StyleBox["m", FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"], StyleBox["uu", FontWeight->"Bold"]}], StyleBox[" ", FontWeight->"Bold"], "-", FormBox[ RowBox[{\(\[Eta]\^2\/\(4\ m\)\), FractionBox[\(\(\[PartialD]\^2 ln\[Rho]\)\&\[LongLeftRightArrow]\), RowBox[{ RowBox[{"\[PartialD]", StyleBox["x", FontWeight->"Bold"]}], RowBox[{ StyleBox["\[PartialD]", FontWeight->"Bold"], StyleBox["x", FontWeight->"Plain"]}]}]]}], "TraditionalForm"]}], ")"}]}], "]"}]}]}], " ", "+", " ", RowBox[{"\[Rho]", " ", RowBox[{ StyleBox["\[Del]", FontWeight->"Bold"], StyleBox["V", FontWeight->"Plain"]}]}]}], StyleBox[" ", FontWeight->"Plain"], StyleBox["=", FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], StyleBox["0", FontWeight->"Plain"]}], TraditionalForm]], FontSize->14], StyleBox["\n\nwhere ", FontSize->14], StyleBox["u ", FontSize->14, FontWeight->"Bold"], StyleBox["is defined as the momentum averaged velocity ", FontSize->14], StyleBox["p/", FontSize->14, FontWeight->"Bold"], StyleBox["m. Moreover, this transformation is irreversible, as the authors \ excellently describe: \"This function will give the correct \ probability...when integrated over ", FontSize->14], StyleBox["p", FontSize->14, FontWeight->"Bold"], StyleBox["... but, clearly, it is not equal to the true reduced probability \ density ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`f\_1\)], FontSize->14], StyleBox["... This discrepancy is due to the fact that ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`f\&^\)], FontSize->14], StyleBox[" has been constructed using ", FontSize->14], StyleBox["only", FontSize->14, FontSlant->"Italic"], StyleBox[" the information given by the first two kinetic equations... To \ claim that anti-transforming the ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`f\&^\)], FontSize->14], StyleBox[" we found keeps any resemblance to the full probability ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`\(\(f\_1\)\(\ \)\)\)], FontSize->14], StyleBox["would be similar to claiming that the polynomial built with the \ first two coefficients of a Taylor expansion of a function would be \ equivalent to said function everywhere.\"\n\nThus, a method of evaluating \ averages in the p-conjugate space must be obtained. Using the Hopf-Cole \ transformation ln \[CapitalPsi] = (1/2) ln \[Rho] + ", FontSize->14], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{"\[Eta]", "/", StyleBox["i", FontSlant->"Italic"]}], StyleBox[")", FontSlant->"Italic"]}], StyleBox[" ", FontSlant->"Italic"], StyleBox["S", FontSlant->"Italic"]}], TraditionalForm]], FontSize->14], StyleBox[", where \[CapitalDelta]S = 2\[Pi]", FontSize->14], StyleBox["n", FontSize->14, FontSlant->"Italic"], StyleBox["\[Eta], with n an integer, the authors derive the familiar:\n\n", FontSize->14], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FormBox[ RowBox[{"<", SuperscriptBox[ StyleBox["x", FontWeight->"Bold"], "n"]}], "TraditionalForm"], ">"}], " ", "=", \(\(\[Integral]\(\(\[CapitalPsi]\)\(\ \)\(x\^n\) \(\ \[CapitalPsi]\^*\)\(\ \)\)\)\/\(\[CapitalPsi]\[CapitalPsi]\^*\)\)}], TraditionalForm]], FontSize->24], StyleBox["\n\n", FontSize->14], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"<", SuperscriptBox[ StyleBox["p", FontWeight->"Bold"], "n"], ">"}], " ", "=", " ", FractionBox[ RowBox[{"\[Integral]", RowBox[{\(\[CapitalPsi]\^*\), " ", SuperscriptBox[ RowBox[{"(", RowBox[{\(-i\), " ", "\[Eta]", " ", StyleBox["\[Del]", FontWeight->"Bold"]}], StyleBox[")", FontWeight->"Bold"]}], "n"], "\[CapitalPsi]", " "}]}], \(\[Integral]\(\[CapitalPsi]\[CapitalPsi]\^*\)\)]}], TraditionalForm]], FontSize->24], StyleBox["\nAdditionally, the authors show that an equation of motion in \ p-conjugate space is given by:\n\n", FontSize->14], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\(\(-\[Eta]\^2\)\/\(2 m\)\) \[Del]\^2 \[CapitalPsi]\), " ", "+", " ", RowBox[{ RowBox[{"V", "(", StyleBox["x", FontWeight->"Bold"], StyleBox[")", FontWeight->"Bold"]}], StyleBox["\[CapitalPsi]", FontWeight->"Bold"]}]}], StyleBox[" ", FontWeight->"Bold"], StyleBox["=", FontWeight->"Bold"], StyleBox[" ", FontWeight->"Bold"], RowBox[{ StyleBox["i", FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], StyleBox["\[Eta]", FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], FractionBox[ StyleBox["\[PartialD]", FontWeight->"Plain"], \(\[PartialD]t\)], "\[CapitalPsi]", " "}]}], TraditionalForm]], FontSize->24], StyleBox["\n\nwith the probability as a function of ", FontSize->14], StyleBox["x", FontSize->14, FontWeight->"Bold"], StyleBox[" only given by \[Rho](", FontSize->14], StyleBox["x", FontSize->14, FontWeight->"Bold"], StyleBox[",t) = \[CapitalPsi](", FontSize->14], StyleBox["x,", FontSize->14, FontWeight->"Bold"], StyleBox["t) ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`\(\[CapitalPsi]\^*\)\)], FontSize->14], StyleBox["(", FontSize->14], StyleBox["x,", FontSize->14, FontWeight->"Bold"], StyleBox["t).\n\nAnd, finally, the authors note that as their \ transformation is irreversible, the Fourier transform requires \n\n", FontSize->14], StyleBox["\[CapitalDelta]", FontSize->24], StyleBox["p", FontSize->24, FontSlant->"Italic"], StyleBox["\[CapitalDelta]", FontSize->24], StyleBox["x ", FontSize->24, FontSlant->"Italic"], StyleBox["= \[Eta]/2", FontSize->24], StyleBox["\n\nThus, the familiar postulates of quantum mechanics are \ obtained from purely kinetic considerations.\n\nIn their second paper, the \ authors extend this method to include fluids with rotational movement by \ changing from an assumption of full separability (which does not introduce \ enough degrees of freedom to consider vortical flows), to an assumption of \ the form \n\n", FontSize->14], Cell[BoxData[ \(TraditionalForm\`\(f\&^\)(\(x\^'\), \(x\^''\), t)\ = \ \(\(h\^'\)(\(x\^'\), t)\)\ \(\(h\^''\)(\(x\^''\), t)\)\ + \ \(\(h\^'\)(\(x\^'\), t)\)\ \(\(h\^''\)(\(x\^''\), t)\)\)], FontSize->14], ".\n\nThis provides eight equations, which is sufficient to encompass \ vortical solutions. In this form, an extra term is introduced into the \ earlier equations of motion derived for a fully separable term that is \ proportional to \[Eta]", StyleBox["/2 times <", FontSize->14], StyleBox["\[Sigma]", FontSize->14, FontWeight->"Bold"], StyleBox[">, where ", FontSize->14], StyleBox["\[Sigma] ", FontSize->14, FontWeight->"Bold"], StyleBox["is the vector made from the standard Pauli matrices. This extra, \ intrinsic, degree of freedom, in \[Eta], describes vorticity and plays an \ identical role in the equations of motion as spin does in the Schrodinger \ equation.\n\n\n", FontSize->14], StyleBox["Summary of auxiliary summer reading", FontSize->14, FontWeight->"Bold"], StyleBox["\n\n", FontSize->14, FontWeight->"Bold", FontVariations->{"Underline"->True}], StyleBox["\"", FontSize->14, FontWeight->"Bold"], StyleBox["Limitation on entropy increase imposed by Fisher information,\"", FontSize->14, FontSlant->"Italic"], StyleBox[" B. Nikolov, B. Roy Frieden (Physics Review E, Volume 49, Number \ 6)\n\nThis article discussed, in part, the different basis for Fisher and \ Shannon information metrics. While both information metrics have the \ intuitive inverse dependence on variance, the Fisher information contains \ what the authors call a \"measure-estimation\" criteria related to the \ information on a distribution provided by a single measurement, while \ Shannon information uses the distinguishability of signals in a channel to \ measure entropy and information.\n\nThe authors make a persuasive argument \ that Fisher information is more appropriate for deriving physical phenomena \ as measurement error is a more physical description of a system than is \ signal distinguishability.", FontSize->14], StyleBox["\n", FontSize->14, FontWeight->"Bold", FontVariations->{"Underline"->True}], StyleBox["\nFisher information ", FontSize->14], Cell[BoxData[ FormBox[ RowBox[{"I", " ", "=", " ", RowBox[{"\[Integral]", RowBox[{\(\(\[Del]p\[CenterDot]\[Del]p\)\/p\), " ", StyleBox[ RowBox[{"d", StyleBox["r", FontWeight->"Bold"]}]]}]}]}], TraditionalForm]], FontSize->14], StyleBox["\n\nShannon information ", FontSize->14], Cell[BoxData[ FormBox[ RowBox[{"I", "=", RowBox[{"-", RowBox[{"\[Integral]", RowBox[{ StyleBox["p", FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"], \(ln(p)\), " ", StyleBox[ RowBox[{"d", StyleBox["r", FontWeight->"Bold"]}]]}]}]}]}], TraditionalForm]], FontSize->14], StyleBox["\n\nwhere ", FontSize->14], StyleBox["p", FontSize->14, FontSlant->"Italic"], StyleBox[" = ", FontSize->14], StyleBox["p(r|t).\n\n\"Fisher and Jaynesian statistics compared in the \ description of classical fluids\"", FontSize->14, FontSlant->"Italic"], StyleBox[", R E Nettleton (Journal of Physics A, 2002, 295-304)\n\nThis \ paper examines the use of Fisher entropy as a thermodynamic measure and \ compares it to the \"Jaynes-Shannon\" approach, which it references does not \ describe. It demonstrates that the two methods are equivalent at equilibrium, \ for non-equilibrium solutions the Fisher entropy gives results not \ phenomenologically consistent with known non-equilibrium behavior. They also \ show that Fisher entropy solutions are not consistent with \"Onsager-Casimir \ reciprocity\".\n\n\n\"", FontSize->14], StyleBox["Concept of entropy for nonequilibrium states of closed many-body \ systems\",", FontSize->14, FontSlant->"Italic"], StyleBox[" J. L. del Rio-Correa (Physical Review A, Volume 43, Number 12).\n\ \nThis paper also analyzes non-equilibrium states using Jaynesian statisics, \ developing a variational method along the way. Furthermore, it relates the \ behavior of the system to the information available when the system is \ initially prepared, not an arbitrary condition at some later time t. This \ point is illustrated well by Jaynes' own rather emphatic remark, \"If a \ macro-phenomenon is found to be reproducible then it follows that all \ microscopic details that were not under the experimenter's control must be \ irrelevant for understanding and predicting it.\"", FontSize->14] }], "Text", CellDingbat->"\[FilledSquare]"], Cell[BoxData[""], "Input"], Cell[BoxData[""], "Input"] }, FrontEndVersion->"5.0 for Microsoft Windows", ScreenRectangle->{{0, 1400}, {0, 956}}, PageWidth->PaperWidth, WindowSize->{582, 742}, WindowMargins->{{1, Automatic}, {Automatic, 0}}, StyleDefinitions -> "ArticleClassic.nb" ] (******************************************************************* 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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