(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 33977, 1045] NotebookOptionsPosition[ 30704, 931] NotebookOutlinePosition[ 31097, 948] CellTagsIndexPosition[ 31054, 945] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Self-Avoiding Walks", "Title"], Cell["\<\ A.P. 1601 Columbia University\ \>", "Subsubtitle"], Cell[TextData[{ "With help from ", StyleBox["Computer Simulations with Mathematica", FontSlant->"Plain"], " by R.Gaylord and P.Wellin (Springer-Verlag,1995)" }], "Subsubtitle"], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell[TextData[{ "A random walk consists of random steps without any regard to past history. \ Each step is independent. The behavior of a large number of particles \ undergoing random walk exhibits ", StyleBox["diffusion", FontSlant->"Italic"], "." }], "Text"], Cell["\<\ The self-avoiding walk (SAW) is fundamentally different. As we trace random \ steps, our algorithm checks to make certain the path never contacts itself. \ To implement this algorithm, we need a record of the entire path in order to \ make decisions about the direction of the next step. As the path gets longer \ and longer, the SAW requires more computation. \ \>", "Text"], Cell["\<\ Self-avoiding walks are often used to represent simple models of long polymer \ chains. In two-dimensions, a SAW can represent the precipitation of a polymer \ onto a surface. This notebook presents the \"pivot\" technique for generating \ different self-avoiding walks of fixed length.\ \>", "Text"], Cell[TextData[{ "The behavior of the SAW is ", StyleBox["much different", FontSlant->"Italic"], " than the random walk (which could be easily predicted with \ \[LeftAngleBracket]", StyleBox["r", FontSlant->"Italic"], "\[RightAngleBracket] ~ ", Cell[BoxData[ FormBox[ SqrtBox["n"], TraditionalForm]]], ". " }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["The Pivot Algorithm", "Section"], Cell["\<\ The \"pivot algorithm\" is considered to be a very efficient algorithm for \ generating a large set of \"self-avoiding\" paths. It is based on selecting a \ random position on the path and the applying a random rotation or reflection \ on that part of the path beyond the selected position.\ \>", "Text"], Cell["\<\ We will look at two-dimensional self-avoiding walks, and we need consider \ only three rotations: \t1) rotate by 90 Degrees \t2) rotate by 180 Degrees \t3) rotate by 270 Degrees\ \>", "Text"], Cell[TextData[{ "We will use the method introduced by R.Gaylord and P.Wellin. A walk begins \ at ", StyleBox["{0, 0}", FontWeight->"Bold"], ", and the initial unit step is along the ", StyleBox["x", FontSlant->"Italic"], "-direction. We write:" }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{"initial", " ", "=", " ", RowBox[{"{", RowBox[{"{", RowBox[{"1", ",", "0"}], "}"}], "}"}]}], ";"}]], "Input", CellLabel->"In[1]:="], Cell["For the three rotations, we define:", "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"rot90", " ", "=", " ", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", "1"}], "}"}], ",", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "0"}], "}"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"rot180", " ", "=", " ", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "0"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"0", ",", RowBox[{"-", "1"}]}], "}"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"rot270", " ", "=", " ", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"0", ",", RowBox[{"-", "1"}]}], "}"}], ",", " ", RowBox[{"{", RowBox[{"1", ",", "0"}], "}"}]}], "}"}]}], ";"}]}], "Input", CellLabel->"In[2]:="], Cell[TextData[{ "Then, we are able to use ", 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