**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.)