The next order of Markov approximation is also known as the mean field theory [14][13]. The mean field theory, like the 0th-order approximation, encodes only the combinatorial information contained in the cellular automaton map from neighborhood blocks to the states of single cells. The mean field theory, like the 0th-order approximation, represents the action of a cellular automaton on general measures by its action on 1-step Markov measures. The mean field theory is superior to the 0th-order approximation in two respects: 1) The initial measure is allowed to be any 1-step Markov measure, and 2) any number of applications of the map from the set of 1-step Markov measures into itself can be considered.
Let #0(B) and #1(B) be the number of 0's and 1's respectively in a block B. In the mean field theory, the probability of a block B is given by
Equation (5) is exact in the case
in which the states of different
cells are completely uncorrelated. Two blocks
B and 
which have the same number of cells in states
0 and 1
will be said to be of the same 1st-order type. Blocks of
the same 1st-order type
are assigned the same probability by equation (5).
Substituting equation (5) into the equation of the form (2) for the evolution of the probability of a 1, we have the mean field equation
Observe that any two blocks B and contribute the
same probability to the sum if 1) they both
lead to a 1 under the rule, and 2) they are of the same 1st-order type.
Let 
be the number of neighborhood
blocks which lead to a 1 under a rule and also contain i 1's (are of the
same 1st-order type).
Equation (6) can now be rewritten as
The coefficients 
may have any integer value in the
range 0- 
inclusive.
This polynomial equation is a model of the evolution of any cellular
automaton which yields the coefficient
values . A fixed point 
in the
range [0,1] of equation (7) is an estimate of
the invariant density of any cellular
automata which yield the coefficient values .
Observe that many different rules of a given radius may have the same
values for the coefficients. Such rules are indistinguishable at
the level of mean field theory. A collection of
rules with the same mean field coefficient values will be referred to as a
1st-order (or mean field) class
determined by the coefficient values .
All rules in a given mean field class also lie in the same 0th-order class.
That is,
the mean field theory supplies a classification of cellular automata which is
a strict refinement of the
0th-order classification. The value 
for the
0th-order class
which contains a mean field class determined by coefficient
values is given by
.