 
Summary: MARKOV SUPERPOSITION EXPANSION FOR THE
ENTROPY AND CORRELATION FUNCTIONS IN TWO AND
THREE DIMENSIONS
PHIL ATTARD
Ian Wark Research Institute, University of South Australia, The Levels,
SA 5095 Australia
The r“ole that entropy plays in information theory is described, together with a
pedagogic derivation of Gibb's formula that expresses it in terms of the probabil
ity of states. It is then shown how the entropy of a spin lattice system may be
expanded in terms of successively higher order spin correlation functions. This
expansion is based on a Markov approach and it has previously proven successful
in one dimensional spinlattice systems, which have possible applications to time
series and signal processing. Here the procedure is generalized to higher dimen
sions, where it may be applicable to image analysis and tomography. Tests for
the twodimensional Ising model show that two terms of the expansion suffice for
accurate result at both high and low temperatures. The corresponding Markov
superposition approximation for the higher order correlation functions in terms
of elemental adjacent correlation functions (i.e. between neighbor vertices, bonds,
and squares), is given in both two and threedimensions.
1 Introduction
