Thermodynamics

Decreases with time in the spatial entropies and dimensions of equations (2) and (3) signal irreversibility in cellular automaton evolution. Some cellular automaton rules are, however, reversible, so that each and every configuration has a unique predecessor in the evolution, and the spatial entropy and dimension of equations (2) and (3) remain constant with time. Figure 7 shows some examples of the evolution of such rules, constructed by adding a term to equation (1) (ref. [20] and E. Fredkin, personal communication). Again, there are analogues of the four classes of behaviour seen in Fig. 3, distinguished by the range and speed of information propagation.

Conventional thermodynamics gives a general description of systems whose microscopic evolution is reversible; it may, therefore, be applied to reversible cellular automata such as those of Fig. 4. As usual, the `fine-grained' entropy for sets (ensembles) of configurations, computed as in equation (3) with perfect knowledge of each site value, remains constant in time. The `coarse-grained' entropy for configurations is, nevertheless, almost always non-decreasing with time, as required by the second law of thermodynamics. Coarse graining emulates the imprecision of practical measurements, and may be implemented by applying almost any contractive mapping to the configurations (a few iterations of an irreversible cellular automaton rule suffice). For example, coarse-grained entropy might be computed by applying equation (3) to every fifth site value. In an ensemble with low coarse-grained entropy, the values of every fifth site would be highly constrained, but arbitrary values for the intervening sites would be allowed. Then in the evolution of a class 3 or 4 cellular automaton the disorder of the intervening site values would `mix' with the fifth-site values, and the coarse-grained entropy would tend towards its maximum value. Signs of self-organization in such systems must be sought in temporal correlations, often manifest in `fluctuations' or metastable `pockets' of order.

While all fundamental physical laws appear to be reversible, macroscopic systems often behave irreversibly, and are appropriately described by irreversible laws. Thus, for example, although the microscopic molecular dynamics of fluids is reversible, the relevant macroscopic velocity field obeys the irreversible Navier-Stokes equations. Conventional thermodynamics does not apply to such intrinsically irreversible systems; new general principles must be found. Thus, for cellular automata with irreversible evolution rules, coarse-grained entropy typically increases for a short time, but then decreases to follow the fine grained entropy. Measures of the structure generated by self-organization in the large time limit are usually affected very little by coarse graining.

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