(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 24710752, 410898] NotebookOptionsPosition[ 24669299, 409764] NotebookOutlinePosition[ 24672682, 409848] CellTagsIndexPosition[ 24672598, 409843] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[TextData[{ "\n\nThis notebook is intended to run on ", StyleBox["Mathematica", FontSlant->"Italic"], " v.7; it has been tested on v.8, where most commands still run (though the \ commands that draw random walks on Slides 12 and 13 are slow). Also the \ attempt to install the \"Combinatorica\" package on the final slide produces \ the notice that those resources have become built-in resources: there must, \ however, be some mismatching, because some of the graph-theoretic \ constructions (encountered only under the \"Percolation theory\" head), now \ don't produce the results they did in v.7.\n\n\nN. Wheeler\n24 October 2011" }], "Text", CellChangeTimes->{{3.528477844532318*^9, 3.528478195443808*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["Slide 1", "Section", CellChangeTimes->{{3.527732621976136*^9, 3.527732627044087*^9}}], Cell["\<\ Some Miscellaneous Adventures in\ \>", "Subtitle", CellChangeTimes->{{3.527690619524537*^9, 3.527690634485964*^9}, { 3.5276907041759043`*^9, 3.527690705238059*^9}}, TextAlignment->Center, TextJustification->0., FontSize->36], Cell[CellGroupData[{ Cell[TextData[StyleBox["Experimental Mathematical Physics", FontColor->RGBColor[0, 0, 1]]], "Title", CellChangeTimes->{{3.527690647192891*^9, 3.527690656168666*^9}}, TextAlignment->Center, TextJustification->0., FontSize->72], Cell["\<\ ºÚÁϳԹÏÍø Physics Seminar 9 November 2011\ \>", "Subtitle", CellChangeTimes->{{3.527690763902596*^9, 3.527690782694758*^9}, { 3.527690827641246*^9, 3.527690854374992*^9}}, TextAlignment->Center, TextJustification->0., FontSize->36], Cell["\<\ Nicholas Wheeler A. 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sister.\" \ Here she is (1963) with her husband:\ \>", "Text", CellChangeTimes->{{3.524070304386812*^9, 3.52407035263797*^9}, { 3.524071948230966*^9, 3.5240719723839893`*^9}}, FontSize->24], Cell[BoxData[ RowBox[{ GraphicsBox[ TagBox[RasterBox[CompressedData[" 1:eJxsvXdUFdm6vrtP2GeHszvYbbe2Oeecs2JWMGAOiKKSJEqOIiBGBBUUBcWI ijmgYsCsoIAiSBBQQMHc3Xvvc8a54/53n6q3u37ec+8Ykxpz1aq1qDXnM9/v +2aqWT7L2y7/wx/+8G/8neOv2sjHrglZFxW6ISYifkP0zm2b03ZtS9kRn7h5 3e6krbu2b9mbsp2Xeosz69YGJ2yOTtyylrQtPjolaeO+PQm7kzeR58zWTWsS NkeRD/Z3C/B13hQX9uTxrWOHU9bHBG2KC924LmTd2oDYKP/wYM8NsSGhgavW RqxeG+Hv7LTA13PF4f27jh7as3FdaHSUX0S4R0z06pBg17jYoKhI3/jNkVs2 RWzZFLlj27qgALeF822XLJ6xdIn9ksX2Hu5LV/t6RIQHRIT6+Xov91zl6OXp 4OPtGBiwPHatFykuxidhS8iWuMCoUPcQv5WuTrMdF01fsnCa3eQRvbq16trx p+lTR/l4LF3t5RQZ6uXhutjLfYmnm0NIgJv3Kke3lQvmzByvxLvuzgvJzLQb s3i+7SqXRVywYM7kebMmcma54yxekndYYDdiSA9+7P60xPBgj6TE2N3JGxO3 RMVvjCBtXh/GW6QNscEqirjowHVrA9eE+VAmUeG+5NfHBFMIWzZwfSQvSZvX h/OSEqM8t2+NiYsOoty4mAznKXNKfkdCLHk+GxvF1wbog3wP7/JxK8+XkOef 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mathematics in the gymnasium which both attended. At \ college in Berlin he majored in chemical engineering, but attended the \ Wednesday afternoon seminars that featured speakers like Planck, von Laue, \ Heisenberg, Nernst, Pauli, Einstein, and was motivated to take up \ (mathematical) physics as a career. At G\[ODoubleDot]ttingen he worked as an \ assistant to Hilbert, worked later at Princeton, University of Wisconsin, \ Princeton again, and finally at the University of Texas.\n\nWigner's name is \ attached to a great and diverse number of fundamental developments, of which \ I present a very incomplete list:\n\n \[Bullet] applications of group \ theory to quantum mechanics\n \[Bullet] phase space formulation of quantum \ mechanics (Wigner quasi-probability distribution)\n \[Bullet] connection \ between particle physics and representations of the Lorentz group\n \ \[Bullet] exploration of symmetry principles and the manifestations of broken \ symmetry\n \[Bullet] contributions to pure/applied nuclear physics\n \ \[Bullet] \"Unreasonable effectiveness of mathematics in the natural sciences\ \"\n \[Bullet] the ", StyleBox["Wigner semicircle distribution", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " \n " }], "Text", CellChangeTimes->{{3.524070448975072*^9, 3.52407045977146*^9}, { 3.524070501020075*^9, 3.524070569246356*^9}, {3.524071762182556*^9, 3.524071862940434*^9}, 3.524072040363141*^9, {3.52407225390329*^9, 3.5240723817661667`*^9}, {3.524072429567604*^9, 3.524072525805314*^9}, { 3.5240725700195217`*^9, 3.524072673999584*^9}, {3.5240727284052343`*^9, 3.524072847673642*^9}, {3.5240728948945723`*^9, 3.5240729528896637`*^9}, { 3.5240730026699657`*^9, 3.524073037069508*^9}, {3.524073075218014*^9, 3.524073077768774*^9}, {3.52407316912504*^9, 3.524073173034074*^9}, { 3.5240732208288*^9, 3.524073808244577*^9}, 3.527776221069738*^9, { 3.527776288071278*^9, 3.527776293866544*^9}, {3.527776347480783*^9, 3.527776383526909*^9}, 3.527776579816955*^9, {3.527861625589937*^9, 3.527861630872376*^9}, {3.5279084095377274`*^9, 3.5279084114422607`*^9}}, FontSize->24], Cell["\<\ Wigner's Nobel Prize\[LongDash]which he shared with Maria Goeppert Mayer & J. \ Hans Jensen, who made \"discoveries concerning nuclear shell structure\"\ \[LongDash]recognized his \"contributions to the theory of the atomic nucleus \ and the elementary particles, particularly through the discovery and \ application of fundamental symmetry principles.\" It was a pretty \ contribution to nuclear theory that provided my point of departure. \ \>", "Text", CellChangeTimes->{{3.524080411468164*^9, 3.524080749243332*^9}}, FontSize->24] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Slide 5", "Section", CellChangeTimes->{{3.527691060042124*^9, 3.527691063647613*^9}, 3.527780694903125*^9}], Cell["Here is a table of nuclides", "Text", CellChangeTimes->{{3.5240816068753443`*^9, 3.5240816329272413`*^9}}, FontSize->24], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Hyperlink", "[", RowBox[{ "\"\\"", ",", "\"\\""}], "]"}]], "Input", CellChangeTimes->{{3.527777028092662*^9, 3.527777037168045*^9}, { 3.527777177777707*^9, 3.527777239825469*^9}}], Cell[BoxData[ TagBox[ ButtonBox[ PaneSelectorBox[{False->"\<\"Table of Nuclides\"\>", True-> StyleBox["\<\"Table of Nuclides\"\>", "HyperlinkActive"]}, Dynamic[ CurrentValue["MouseOver"]], BaseStyle->{"Hyperlink"}, BaselinePosition->Baseline, FrameMargins->0, ImageSize->Automatic], BaseStyle->"Hyperlink", ButtonData->{ URL["http://en.wikipedia.org/wiki/Table_of_nuclides_(complete)"], None}, ButtonNote->"http://en.wikipedia.org/wiki/Table_of_nuclides_(complete)"], Annotation[#, "http://en.wikipedia.org/wiki/Table_of_nuclides_(complete)", "Hyperlink"]& ]], "Output", CellChangeTimes->{3.527777305678994*^9}] }, Open ]], Cell[TextData[{ "each of which presents a many-body ", StyleBox["quantum system with ", FontVariations->{"Underline"->True}], StyleBox["unknown Hamiltonian", FontWeight->"Bold", FontVariations->{"Underline"->True}], ". One has interest in the energy levels, and (more particularly) the \ differences between energy levels. In 1954, Wigner was led to ask\n\n", StyleBox["\"What can one say concerning the ", FontColor->RGBColor[0, 0, 1]], StyleBox["statistical", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox[" properties of the spectra of very large hermitian (or real \ symmetric) matrices?\"", FontColor->RGBColor[0, 0, 1]], " \n\nHeavy analysis\[LongDash]unsupported by a shred of numerical evidence\ \[LongDash]led Wigner to the famously pretty (and quite unanticipated) result \ reported in \"Characteristic vectors of bordered matrices in infinite \ dimensions,\" Annals of Mathematics ", StyleBox["62", FontWeight->"Bold"], ", 548 - 564 (1955)\[Ellipsis]or which more in a moment.\n" }], "Text", CellChangeTimes->{{3.5240816909381323`*^9, 3.524081847357492*^9}, { 3.524082188263885*^9, 3.524082194549758*^9}, {3.524082271861321*^9, 3.524082339692656*^9}, {3.5240823738760033`*^9, 3.524082396539577*^9}, 3.5241004978469877`*^9, {3.524141479898361*^9, 3.524141482756462*^9}, { 3.5241479277309504`*^9, 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For a simple proof (that assumes symmetry of the positive matrix) see\ \>", "Text", CellChangeTimes->{{3.5279652939775667`*^9, 3.527965294280611*^9}, { 3.528038042242485*^9, 3.528038082970025*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Hyperlink", "[", RowBox[{ "\"\\"", ",", "\"\\""}], "]"}]], "Input", CellChangeTimes->{{3.528038092115431*^9, 3.528038144227306*^9}}], Cell[BoxData[ TagBox[ ButtonBox[ PaneSelectorBox[{False->"\<\"Ninio's Proof\"\>", True-> StyleBox["\<\"Ninio's Proof\"\>", "HyperlinkActive"]}, Dynamic[ CurrentValue["MouseOver"]], BaseStyle->{"Hyperlink"}, BaselinePosition->Baseline, FrameMargins->0, ImageSize->Automatic], BaseStyle->"Hyperlink", ButtonData->{ URL["http://iopscience.iop.org/0305-4470/9/8/017/pdf/0305-4470_9_8_017.\ pdf"], None}, ButtonNote-> "http://iopscience.iop.org/0305-4470/9/8/017/pdf/0305-4470_9_8_017.pdf"], Annotation[#, "http://iopscience.iop.org/0305-4470/9/8/017/pdf/0305-4470_9_8_017.pdf", "Hyperlink"]& ]], "Output", CellChangeTimes->{3.52803815061052*^9}] }, Open ]], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.527960105372134*^9}], Cell[TextData[{ "I happened at this point to be browsing in M. 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Ziegler's ", StyleBox["Proofs from THE BOOK", FontSlant->"Italic"], " (1998) where I came upon a result due to Legendre\[LongDash]presented \ there as a consequence of the inequalities" }], "Text", CellChangeTimes->{{3.527960150808469*^9, 3.527960267081285*^9}, { 3.527961180478609*^9, 3.527961183747078*^9}, {3.52803818768097*^9, 3.5280382547446527`*^9}, {3.52803831708774*^9, 3.528038321790022*^9}, { 3.528038376109315*^9, 3.5280384257705097`*^9}, 3.528038502935121*^9}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"harmonic", " ", "mean"}], " ", "\[LessSlantEqual]", " ", RowBox[{"geometric", " ", "mean"}], " ", "\[LessSlantEqual]", " ", RowBox[{"arithmetic", " ", "mean"}]}], "\[IndentingNewLine]"}]], "DisplayFormula", CellChangeTimes->{{3.5239684604571877`*^9, 3.523968509886692*^9}, 3.528038509503389*^9}], Cell[TextData[{ StyleBox["THEOREM", FontWeight->"Bold"], ": Suppose all roots of the polynomial" }], "Text", CellChangeTimes->{{3.523968558032909*^9, 3.523968583496952*^9}}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["x", "n"], "+", RowBox[{ SubscriptBox["a", RowBox[{"n", "-", "1"}]], SuperscriptBox["x", RowBox[{"n", "-", "1"}]]}], "+", RowBox[{ SubscriptBox["a", RowBox[{"n", "-", "2"}]], SuperscriptBox["x", RowBox[{"n", "-", "2"}]]}]}], " ", "..."}], " ", "+", SubscriptBox["a", "0"]}]], "DisplayFormula", CellChangeTimes->{{3.523968588997184*^9, 3.523968646475713*^9}, { 3.523975315734182*^9, 3.523975327521276*^9}}], Cell["\<\ are real. 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PlotRange->Automatic, PlotRangeClipping->True, Ticks->{{{ NCache[ Rational[265, 2], 132.5], FormBox[ FractionBox["265", "2"], TraditionalForm]}, {265, FormBox["265", TraditionalForm]}}}]], "Output", CellChangeTimes->{3.524323908096106*^9, 3.52432828751873*^9}], Cell["\<\ we find that the \"brim problem\" has pretty much evaporated, and the trend \ N \[RightArrow] \[Infinity] is clear. \ \>", "Text", CellChangeTimes->{{3.524328315368614*^9, 3.524328380501917*^9}, 3.527780567986795*^9}], Cell[TextData[{ "We have arrived computationally at the result which Wigner obtained \ analytically. The discovery of ", StyleBox["Wigner's semicircle law", "Subsection"], " has given rise to an enormous (and still-expanding) literature (to which \ Freeman Dyson was an important early contributor). The theory of random \ matrices has been found to have important applications to (among other \ fields)\n \[Bullet] the theory of disordered solid state and chaotic \ systems;\n \[Bullet] statistical mechanics\n \[Bullet] physics of \ superconductivity, quantum dots \n \[Bullet] quantum field theory\n \ \[Bullet] signal processing\n \[Bullet] design of web search engines\n \ \[Bullet] number theory\n \[Bullet] the mathematics of graph theory, \ group theory, orthogonal polynomials, integral \[NonBreakingSpace]equations, \ \n non-commutative probability theory, combinatorics, random partitions, \ \[Ellipsis]\n \nSee, for example, M. L. Mehta, ", StyleBox["Random Matrices", FontSlant->"Italic"], " (1991) for a comprehensive review. \n " }], "Text", CellChangeTimes->{{3.5243283944542313`*^9, 3.5243284575972652`*^9}, { 3.524329533677115*^9, 3.524329539258383*^9}, {3.524329641119279*^9, 3.524329718773097*^9}, {3.524329818490102*^9, 3.524330140518375*^9}, { 3.524330175944584*^9, 3.524330448890008*^9}, {3.524331022943345*^9, 3.524331260297484*^9}, 3.528295617528331*^9}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Slide 10", "Section", CellChangeTimes->{{3.527784638715765*^9, 3.527784651752268*^9}}], Cell[BoxData[ RowBox[{"\[IndentingNewLine]", StyleBox[ RowBox[{ "Spectral", " ", "distributions", " ", "of", " ", "large", " ", "supersymmetric", " ", "matrices"}], "Subsection"]}]], "Input", CellChangeTimes->{{3.527784663756143*^9, 3.52778471135432*^9}}], Cell[TextData[{ "Have looked thus far to the ", StyleBox["symmetric parts", FontSize->36], " of large random matrices:" }], "Text", CellChangeTimes->{{3.5277847297982607`*^9, 3.527784752481042*^9}}], Cell[BoxData[ StyleBox[ RowBox[{ RowBox[{"\"\\"", " ", "Matrices"}], "=", FractionBox[ RowBox[{ SuperscriptBox["\[DoubleStruckCapitalM]", "transpose"], "+", "\[DoubleStruckCapitalM]"}], "2"]}], FontColor->RGBColor[0, 0, 1]]], "DisplayFormula"], Cell[TextData[{ "There exists an ", StyleBox["alternative way to generate symmetric matrices", FontSize->36], ":\n" }], "Text", CellChangeTimes->{{3.52778484410044*^9, 3.5277848736109324`*^9}}], Cell[BoxData[ StyleBox[ RowBox[{ RowBox[{"\"\\"", " ", "Matrices"}], "=", RowBox[{ SuperscriptBox["\[DoubleStruckCapitalM]", "transpose"], ".", "\[DoubleStruckCapitalM]"}]}], FontColor->RGBColor[0, 0, 1]]], "DisplayFormula", CellChangeTimes->{{3.524330786565791*^9, 3.524330806047654*^9}, { 3.524330881577231*^9, 3.5243308894198647`*^9}}], Cell[TextData[{ "\nSuch matrices\[LongDash]which are square even when \ \[DoubleStruckCapitalM] is rectangular\[LongDash] are called \"Wishart \ matrices\" (in honor of John Wishart, who studied statistical aspects of such \ matrices\[LongDash]including their spectral distribution\[LongDash]already in \ 1928, nearly 30 years before Wigner entered the random matrix picture) by \ mathematicians: my own \"supersymmetric\" terminology derives from the fact \ that operators of the form ", Cell[BoxData[ RowBox[{ SuperscriptBox["\[DoubleStruckA]", "transpose"], ".", "\[DoubleStruckA]"}]], CellChangeTimes->{{3.5243342284893312`*^9, 3.524334242342342*^9}}], ", which entered physics ", StyleBox["via", FontSlant->"Italic"], " Dirac's ", StyleBox["algebraic formulation of quantum oscillator theory", FontVariations->{"Underline"->True}], ", are fundamental to ", StyleBox["supersymmetric quantum field theory", FontVariations->{"Underline"->True}], ", and\[LongDash]Witten's simplification of that subject\[LongDash]to ", StyleBox["supersymmetric quantum mechanics", FontVariations->{"Underline"->True}], "." }], "Text", CellChangeTimes->{{3.52778490798486*^9, 3.527784915465803*^9}}], Cell["It follows immediately from", "Text", CellChangeTimes->{3.527785020824503*^9}], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["\[DoubleStruckCapitalM]", "transpose"], ".", "\[DoubleStruckCapitalM]"}], " ", "\[Psi]"}], "=", RowBox[{"\[Lambda]", " ", "\[Psi]"}]}]], "Input", CellChangeTimes->{{3.524334675758463*^9, 3.5243346890187063`*^9}}], Cell[TextData[{ "that ", StyleBox["the eigenvalues of supersymmetric matrices are necessarily \ non-negative", FontWeight->"Bold"], ". 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It was in this connection \ that I first encountered the beta distribution, which is actually a \ 2-parameter class of distributions that live naturally on the unit interval. \ It is defined" }], "Text", CellChangeTimes->{3.527785747833496*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"PDF", "[", RowBox[{ RowBox[{"BetaDistribution", "[", RowBox[{"\[Alpha]", ",", "\[Beta]"}], "]"}], ",", "x"}], "]"}]], "Input", CellChangeTimes->{{3.524272401318717*^9, 3.524272402005176*^9}}], Cell[BoxData[ FractionBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "-", "x"}], ")"}], RowBox[{ RowBox[{"-", "1"}], "+", "\[Beta]"}]], " ", SuperscriptBox["x", RowBox[{ RowBox[{"-", "1"}], "+", "\[Alpha]"}]]}], RowBox[{"Beta", "[", RowBox[{"\[Alpha]", ",", "\[Beta]"}], "]"}]]], "Output", CellChangeTimes->{3.528396340325379*^9}] }, Open ]], Cell["\<\ where \[Alpha] and \[Beta] can assume any positive real values, and where \ Beta[\[Alpha], \[Beta]] is the Euler beta function:\ \>", "Text", 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Recent work indicates that\n\n \[Bullet] if ", Cell[BoxData[ StyleBox[ RowBox[{"0", "<", "s", "<", FractionBox["1", "2"]}], "InlineFormula"]], CellChangeTimes->{{3.52529781583202*^9, 3.5252978343048363`*^9}}], " the maximal advance distribution is ", StyleBox["fractal", FontWeight->"Bold"], ";\n \[Bullet] if ", Cell[BoxData[ RowBox[{"s", "=", FractionBox["1", "2"]}]], CellChangeTimes->{{3.52529781583202*^9, 3.5252978343048363`*^9}}], " the distribution is ", StyleBox["uniform", FontWeight->"Bold"], ";\n \[Bullet] if ", Cell[BoxData[ RowBox[{ FractionBox[ StyleBox["1", "InlineFormula"], "2"], StyleBox["<", "InlineFormula"], StyleBox["s", "InlineFormula"], StyleBox["<", "InlineFormula"], StyleBox["1", "InlineFormula"]}]], CellChangeTimes->{{3.52529781583202*^9, 3.5252978343048363`*^9}}], " the distribution is ", StyleBox["continuous", FontWeight->"Bold"], " (but shape generally unknown).\n " }], "Text", CellChangeTimes->{{3.5278744322138643`*^9, 3.5278744657908983`*^9}, 3.527874530196953*^9, {3.528039834489102*^9, 3.52803986468497*^9}, { 3.528039895107193*^9, 3.528039934556593*^9}, {3.5280399681584873`*^9, 3.528039970146901*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Stretched walks", "Subsection", CellChangeTimes->{{3.527874538096046*^9, 3.5278745421992617`*^9}, { 3.5278745847922573`*^9, 3.5278745857728453`*^9}}], Cell["\<\ Assume s to be slightly greater than unity. Then steps become progressively \ longer (not shorter), and the maximal N-step advance grows without bound as N\ \[LongRightArrow]\[Infinity]. For every finite N there does, however, exist \ a well-defined upper bound. 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CellChangeTimes->{{3.5280633014053097`*^9, 3.528063305105328*^9}}], Cell[CellGroupData[{ Cell["\<\ Principle of Minimal Energy Dissipation (Minimal Entropy Production)\ \>", "Subsection", CellChangeTimes->{{3.5280633290530367`*^9, 3.528063386377962*^9}, { 3.5280670282134953`*^9, 3.5280670303499737`*^9}}], Cell[TextData[{ "It has always seemed to me to be self-evident that current passing through \ a resistive DC network will be so partitioned as to ", StyleBox["minimize the dissipation rate", FontWeight->"Bold"], ". It is a proposition which I have assumed to be well known to electrical \ engineers, and am surprised to discover that as recently as 1999 it was \ presented in the pages of the American Journal of Physics (D. A. Van Baak, \ \"Variational alternatives to Kirchhoff's loop theorem in dc circuits,\" AJP ", StyleBox["67", FontWeight->"Bold"], ", 36 (1999)) as something of a novelty. 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In essence this subject represents the \ marriage of two major branches of theoretical physics\[Ellipsis]", FontSlant->"Italic"], " \"\n\nOf lesser importance is the fact that the method leads to a ", StyleBox["novel interpretation to the concept of \"electric current\"", FontColor->RGBColor[0, 0, 1]], ": ", Cell[BoxData[ FormBox[ SubscriptBox["J", RowBox[{"m", ",", " ", "n"}]], TraditionalForm]], FormatType->"TraditionalForm"], " becomes a measure of how much the \"walking Electron traffic\" m\ \[LongRightArrow]n exceeds that in the opposite direction: \ m\[LongLeftArrow]n. \n" }], "Text", CellChangeTimes->{{3.528216945917424*^9, 3.5282173380439577`*^9}, { 3.528217395736703*^9, 3.528217736387867*^9}, 3.5282178070790367`*^9, { 3.52822965657594*^9, 3.5282296724104767`*^9}, {3.528229709307469*^9, 3.5282299601510067`*^9}, {3.528229995216238*^9, 3.528229999514567*^9}, { 3.528230047231944*^9, 3.528230057678705*^9}, {3.5282300921872063`*^9, 3.5282300974228153`*^9}}] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ Slide 29 \ \>", "Section", CellChangeTimes->{{3.5282178180510597`*^9, 3.5282178301927233`*^9}, 3.5282503493050213`*^9}], Cell[CellGroupData[{ Cell["5. 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