Eigen-Sheets

The following spreadsheets are all based on the book Manfred Eigen and Ruthild Winkler 1993, Laws of the Game: How the Principles of Nature Govern Chance, Princeton: Princeton University Press.

Bead Game: Equilibrium (All Black Start).  In this Ehrenfest urn model, there are two urns, a black urn and a white urn, with 256 = 16 x 16 balls between them.  The 256 balls are represented on the spreadsheet by the cells in a 16 x 16 grid.  A black box in a cell means that ball is in the black urn, and a blank cell means that ball in the white urn.  We begin with all the balls in the black urn.  On each iteration, a ball is chosen at random and then switched to the other urn.  There is a natural negative feedback or stabilizing force in the set-up since if most of the balls are in one of the urns (e.g., the black urn initially), then those balls will tend to be selected (since balls are selected at random) and moved to the other urn.  The long-run tendency is thus for the balls to be evenly split with 128 (= 256/2) in each urn. This is a spreadsheet version of the "Equilibrium" Bead Game (Version 2) described on page 36 of Manfred Eigen and Ruthild Winkler 1993, Laws of the Game: How the Principles of Nature Govern Chance, Princeton: Princeton University Press. (Right-click on title to download file.)

Bead Game: Equilibrium (Random Start).  This is like the previous Ehrenfest Urn model except that the initial state is each cell getting randomly assigned a black or white box, i.e., the balls starting randomly distributed between the black and white urns.  This is a spreadsheet version of the "Equilibrium" Bead Game (Version 1) described on page 36 of Manfred Eigen and Ruthild Winkler 1993, Laws of the Game: How the Principles of Nature Govern Chance, Princeton: Princeton University Press. (Right-click on title to download file.)

Bead Game: Once for all. This is a spreadsheet version of the "Once for all" Bead Game described on page 44 of Manfred Eigen and Ruthild Winkler 1993, Laws of the Game: How the Principles of Nature Govern Chance, Princeton: Princeton University Press.  In the 16 x 16 space, each cell starts off randomly as black or white.  In each iteration, a cell is randomly selected, say, a black cell.  Then the program searches the 16 x 16 space top to bottom, left to right, for the first cell with the opposite value and flips it to the value in the chosen cell.  Clearly this is a positive feedback process so after the initial "battle" played out in the initial rows, eventually the black or white cells will dominate. (Right-click on title to download file.)

Bead Game: Selection.  This is a spreadsheet version of the "Selection" Bead Game described on page 52 of Manfred Eigen and Ruthild Winkler 1993, Laws of the Game: How the Principles of Nature Govern Chance, Princeton: Princeton University Press.  Each cell in the 8 x 8 array (B23-I30) is initially assigned a random value 0, 1, 2, or 3 represented by the four suits in a deck of cards.  In each iteration, a cell is randomly selected and then the number or suit in the cell is then put into another randomly selected cell—so there would then tend to be one more cell with that suit (unless that suit was already in the second selected cell).  This is a positive feedback process so eventually one of the suits will be "selected" and will dominate. (Right-click on title to download file.)

Ehrenfest Model with Cooperative Effects.  In the Ehrenfest urn model, a cell is picked randomly and flipped to the opposite color.  In this cooperative Ehrenfest model, the randomly picked cell is flipped with a probability proportional to the number of the eight neighbors that are of opposite color.  The blacks (ones) and whites (zeros) are initially randomly distributed.  The cooperative effect takes time.  Slowly the white cells and black cells will group together.  This is modeled after the description given on pages 71-72 in Eigen, Manfred and Ruthild Winkler 1993, Laws of the Game: How the Principles of Nature Govern Chance, Princeton: Princeton University Press (except that only four neighbors are used there instead of eight). (Right-click on title to download file.)