Cellular Automata:Exploring Applications

Erik AguilarAmelia Yzaguirre

Amy Femal

Goal

Explore 3 specific types of cellular automata applications and use MATLAB to model them.

Types of Cellular Automata

•Boolean

•Excitable Media

•Lattice Gas Automata

Excitable MediaBoolean Cellular Automatacan be seen in The Game of Life

Rules for the Game of Life•Any live cell with fewer than 2 live neighbors dies

•Any live cell with more than 3 neighbors dies

•Any live cell with 2 or 3 live neighbors lives on to the next generation

•Any dead cell with exactly 3 live neighbors becomes a live cell

Writing the code

%First we must create an n x n matrix of zeros.n=75; z=zeros(n,n); %Produces random "0" and "1" throughout matrixcells = (rand(n,n))<.12; %Creates an image using the matriximh = image(cat(3,cells,z,z)); %When the image is replaced, the old image is not erased firstset(imh, 'erasemode', 'none')

x=(2:n-1); %creates array x which is used for positiony=(2:n-1); %creates array y which is used for position %creates matrix for summationssum(x,y)=zeros(n-2,n-2);

a=0;while a==0; %equations for the corner cells to check for infected %neighbors sum(1,1)=cells(1,2)+cells(2,1)+cells(2,2); sum(1,n)=cells(2,n)+cells(2,n-1)+cells(1,n-1); sum(n,1)=cells(n,2)+cells(n-1,1)+cells(n-1,2); sum(n,n)=cells(n,n-1)+cells(n-1,n)+cells(n-1,n-1); %equations for the edge rows/columns to check for infected %neighbors sum(1,y)=cells(1,y-1)+cells(1,y+1)+cells(2,y)+cells(2,y+1)+... cells(2,y-1); sum(n,y)=cells(n,y+1)+cells(n,y-1)+cells(n-1,y)+cells(n-1,y+1)+... cells(n-1,y-1); sum(x,1)=cells(x-1,1)+cells(x+1,1)+cells(x,2)+cells(x+1,2)+... cells(x-1,2); sum(x,n)=cells(x-1,n)+cells(x+1,n)+cells(x,n-1)+cells(x+1,n-1)+... cells(x-1,n-1);

%equations for the interior cells to check for infected %neighbors sum(x,y)=cells(x,y+1)+cells(x,y-1)+cells(x+1,y)+cells(x-1,y)+... cells(x+1,y+1)+cells(x+1,y-1)+cells(x-1,y+1)+cells(x-1,y-1); %updating the cells matrix cells = (sum(1:n,1:n)==3)|(sum(1:n,1:n)==2 & cells); %draw the new image set(imh, 'CData', cat(3,cells,z,z)) drawnow %displaying the updated image waitforbuttonpress end

Excitable Mediacan be observed in the BZ reaction

Belousov-Zhabotinsky reaction

What is the BZ reaction?It is a chemical reaction caused by the mixture of

Sulfuric Acid

Sodium Bromate

Malonic Acid

Sodium Bromide

Phenanthroline Ferrous Sulfate

Triton X-100 Surfactant

Rules for the BZ ReactionCells can be in 10 different states. State 0 = resting States 1 – 5 = active States 6 – 9 = refractory

Like LIFE, each cell of the BZ reaction is dependent on its 8 surrounding neighbors. • If 3 or more neighbors are active, cell = 1• A cell in State 1 will change to State 2. • A cell in State 2 will change to State 3 and so on.• A cell in State 9 will change to State 0.

Code for the BZ Reaction

clear all n=200; %size M=zeros(n); %this will give us an n by n matrixgrid=M; %the grid will be made up of the n by n matrixgrid=(rand(n))<.06; sum=M;

bz=image(cat(3,grid,M,M)); x=[2:n-1];y=[2:n-1];axis tight

t = 6; % when t is in active statefor i=1:1000 %duration of loop sum(x,y) = ((grid(x,y-1)>0)&(grid(x,y-1)<t)) + ... ((grid(x,y+1)>0)&(grid(x,y+1)<t)) + ... ((grid(x-1, y)>0)&(grid(x-1, y)<t)) + ... ((grid(x+1,y)>0)&(grid(x+1,y)<t)) + ... ((grid(x-1,y-1)>0)&(grid(x-1,y-1)<t)) + ... ((grid(x-1,y+1)>0)&(grid(x-1,y+1)<t)) + ... ((grid(x+1,y-1)>0)&(grid(x+1,y-1)<t)) + ... ((grid(x+1,y+1)>0)&(grid(x+1,y+1)<t)); %the sum of each cell in active state of the 8 surrounding neighbors grid = ((grid==0) & (sum>=3)) + 2*(grid==1) + 3*(grid==2) + ... 4*(grid==3) + 5*(grid==4) + 6*(grid==5) +... 7*(grid==6) + 8*(grid==7) + 9*(grid==8) +... 0*(grid==9); %when state=1, next state=2... set(bz,'cdata', cat(3,M,grid/10,M) ) %bz is the image, cdata contains %data array drawnow %creates imageend

Lattice Gas AutomataEvolution of Gas Particles

Lattice Gas AutomataRulesCells have 2 states 0 = empty 1 = moving gas particle

Each cell has 3 neighbors for a given time step where a block rule is applied to a 2 x 2 block of cells.

Odd OddOdd Cell Even

Even Even

Code for Lattice Gas Automata

%Cellular Automata model of gas particles in a box with a partition%This will make use of a Margolus neighborhood to create motion of an HPP%(Hardy, Pazzis, Pomeau) lattice gas... Curious if we meet Gibbs' paradox!

clear allclf %clears any frames being used.

%--------------We must first create our grid----------%These variables will be used to define the dimension of the matrix in our%gridnx=52; %must be divisible by 4, since each cell will be divided into groups of fourny=100;z=zeros(nx,ny); %Creates an nx by ny matrix of zeros called zo=ones(nx,ny); %Creates an nx by ny matrix of ones called o

%Initialize each of the matrices to be used latercells = z ; cellsNew = z; ground = z ; diag1 = z; diag2 = z; and12 = z;or12 = z; sums = z; orsum = z;

%create the boxground(1:nx,ny-3)=1 ; % right ground lineground(1:nx,3)=1 ; % left ground lineground(nx/4:nx/2-2,ny/2)=1; % the hole in the middle of the partitionground(nx/2+2:nx,ny/2)=1; %the hole in the middle of the partitionground(nx/4, 1:ny) = 1; %top lineground(3*nx/4, 1:ny) = 1; %bottom line

%We now want to "fill" the left side of the container with "gas particles"r = rand(nx,ny);cells(nx/4+1:3*nx/4-1, 4:ny/2-1) = r(nx/4+1:3*nx/4-1, 4:ny/2-1)<0.3;

%Define the image of the gas particles in the container!imh = image(cat(3,z,cells,ground));set(imh, 'erasemode', 'none')axis equalaxis tight

%This is where we define the motion of the particles

for i=1:1000 p=mod(i,2); %Margolus neighborhood defined %upper left cell update xind = [1+p:2:nx-2+p]; yind = [1+p:2:ny-2+p]; %See if exactly one diagonal is ones %We can only have one of the following to hold: diag1(xind,yind) = (cells(xind,yind)==1) & (cells(xind+1,yind+1)==1) & ... (cells(xind+1,yind)==0) & (cells(xind,yind+1)==0); diag2(xind,yind) = (cells(xind+1,yind)==1) & (cells(xind,yind+1)==1) & ... (cells(xind,yind)==0) & (cells(xind+1,yind+1)==0);

%This gives the diagonals both not occupied by two particles andboth(xind,yind) = (diag1(xind,yind)==0) & (diag2(xind,yind)==0); %This gives one diagonal occupied by two particles orone(xind,yind) = diag1(xind,yind) | diag2(xind,yind); %For a given gas particle, check if it is near the boundary sums(xind,yind) = ground(xind,yind) | ground(xind+1,yind) | ... ground(xind,yind+1) | ground(xind+1,yind+1) ; %Rules: %If (no walls) and (diagonals are both empty) %then there are no particles to swap, so the block stays the same cellsNew(xind,yind) = ... (andboth(xind,yind) & ~sums(xind,yind) & cells(xind+1,yind+1)) + ... (orone(xind,yind) & ~sums(xind,yind) & cells(xind,yind+1)) + ... (sums(xind,yind) & cells(xind,yind));

cellsNew(xind+1,yind) = (andboth(xind,yind) & ~sums(xind,yind) & cells(xind,yind+1)) + (orone(xind,yind) & ~sums(xind,yind) & cells(xind,yind))+ ... (sums(xind,yind) & cells(xind+1,yind)); %If (no walls) and (only one diagonal occupied) %then this is representative of a collision--- treat as though the %particles hit and deflect each other at 90 degrees, i.e. one diagonal %is converted to the other on the time step. cellsNew(xind,yind+1) = ... (andboth(xind,yind) & ~sums(xind,yind) & cells(xind+1,yind)) + ... (orone(xind,yind) & ~sums(xind,yind) & cells(xind+1,yind+1))+ ... (sums(xind,yind) & cells(xind,yind+1)); %If (wall) %then the cell stays the same in the block (causes a reflection) cellsNew(xind+1,yind+1) = ... (andboth(xind,yind) & ~sums(xind,yind) & cells(xind,yind)) + ... (orone(xind,yind) & ~sums(xind,yind) & cells(xind+1,yind))+ ... (sums(xind,yind) & cells(xind+1,yind+1));

cells = cellsNew; set(imh, 'cdata', cat(3,z,cells,ground) ) drawnow waitforbuttonpress;end