Cellular Automata Entropy

P(A)

0.7

rule

110

size

random seed

P(A)  0.7

P(B)  0.3

number of black cells: 214

number of white cells: 206

entropy: -

214P(A)log(P(A))

log(2)

-

206P(B)log(P(B))

log(2)

 184.428

The entropy of a list

Q

is defined by

H(Q)=-∑P(q)

log

2

P(q)

, summing over the elements

q

of

Q

.

P(A)

and

P(B)

are the probabilities of black and white cells, respectively. The initial condition is a finite list of random bits.

The entropy can be used to study the amount of information in the evolution of a cellular automaton; it is lower in ordered systems and higher in chaotic (disordered) systems.