Spatial block entropy

The decay of the spatial block entropy, which gives the amount of information contained in a block of N contiguous site values ai,...,aN needed to predict the value (Jn+i is considerably slower than [block-length), and is therefore indicative of very long and complex correlations we will come back to this point later in chapter 4, following our discussion of dynamical system theory. 

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. 

Figure 4.14 shows the first few iteration steps in the evolution of the spatial measure block-entropy of rule R122 for blocks with size B < 5. Although the irreversibility of this rule predictably leads to a decrease of entropy with time, there nonetheless appears to be a relaxation to equilibrium values. Observe also... 

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

There is another notable point. The temperature in the system need not be temporally nor spatially constant. The transfer of a quantity of entropy St from a hot to a cold subarea by use of an auxiliary body that repeatedly undergoes a reversible cyclic process (see Fig. 24.2) delivers useful work to be stored in the system because all the possible paths for energy outflow have been blocked. Energy U remains constant in the process. Transferring the same amount of entropy 5t by... 

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