Spatial Measures for Infinite Lattices

Although it is obviously impossible to enumerate all possible configurations for infinite lattices, so long as the values of far separated sites are statistically independent, the average entropy per site can nonetheless be estimated by a limiting procedure. To this end, we first generalize the definitions for the spatial set and spatial measure entropies given above to their respective block-entropy forms. 

Block Entropies The definitions are the same as for finite lattices except that whereas before we could sum over all possible configurations, we now perform the same calculations for all possible sequences of sites within blocks of finite size B  

While Sgf B,t) is determined directly by the number of distinct sequences of length X generated at time t , Smeas B,t) gives the average entropy per site by taking into account whatever correlations exist in blocks of length up to B. 

In practice, reasonable estimates of the limit limc oo 5 (B) may be obtained for relatively small B. 

Dimensions The limiting values of the block entropies as B — oo gives the spatial set and spatial measure dimensions, of the set of configurations  

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