Reversible rules

As noted repeatedly in earlier sections, CA rules are generally irreversible a given configuration typically has more than one predecessor. Although it is difficult to write down a reversible rule - for which each configuration has a unique predecessor - from scratch, there is a very simple way of turning an arbitrary (r, k) CA rule, 0, that is first-order in time into a second-order reversible rule, (f-jz-f 

Reversible-, given an arbitrary, even irreversible, rule (j), we can uniquely 

Time-reversal invariant the reverse sequence is obtained by inverting the last two configurations obtained in the forward direction and applying the same, rule, 4 tz- 

Apart from their pedagogical value, reversible rules may be used to explore possible relationships between discrete dynamical systems and the dynamics of real mechanical systems, for which the microscopic laws are known to be time-reversal invariant. What sets such systems apart from continuous idealizations is their exact reversibility, discreteness assures us that computer simulations run for arbitrarily long times will never suffer from roundoff or truncation errors. As Toffoli points out, ...the results that one obtains have thus the force of theorems [toff84a].  

From an analytic point of view, the techniques of conventional thermodynamics, which describes systems whose microscopic dynamics is reversible, may be formally applied to reversible CA as well. We shall, in fact, follow this course in chapter 4. 

---

Figures 3.38 and 3.39 show typical space-time patterns generated by a few r = 1 reversible rules starting from both simple and disordered initial states. Although analogs of the four generic classes of behavior may be discerned, there are important dynamical differences. The most important difference being the absence of attractors, since there can never be a merging of trajectories in a reversible system for finite lattices this means that the state transition graph must consist exclusively of cyclic states. We make a few general observations.

Fig. 3.38 Space-time patterns for a few one-dimensional r = 1 reversible rules starting from simple initial states.

We can now take one of two approaches (1) construct a probabilistic CA along lines with the Metropolis Monte Carlo algorithm outlined above (see section 7.1.3.1), or (2) define a deterministic but reversible rule consistent with the microcanonical prescription. As we shall immediately see, however, neither approach yields the expected results. 

As suggested above, time-reversal invariance is a stronger property than simple invertibility and means that the dynamics is not only invertible, but that the time-reversed evolution proceeds according to the same rule as well. We shall call all invertible CA rules that are also time-reversal invariant, reversible rules. ... 

Amoroso and Patt [amoro72] observed that, as a subset of the set of all possible rules, the set of nontrivial reversible rules appeared to be exceedingly small. 

An important issue concerning reversible CA is their construction. In particular, we are interested in knowing if there is some systematic method by which reversible CA rules can be constructed from scratch. On the one hand - if the desire is to randomly choose a reversible rule out of the set of all possible rules, the outlook for success is very dim. Amoroso and Patt [amoro72], for example, had observed that, within the space of all possible rules, the set of nontrivial reversible rules appeared to be exceedingly small. Sears [sears71] had also shown that the set of invertible CA actually constitutes a vanishingly small subset. On the other hand, if the desire is to simply define some representative samples of reversible rules (or some set of rules that may also possess some additional special features), a number... 

Many second-order reversible rules of the above form allow a pseudo-Hamiltonian prescription. The evolution of such systems may then be defined as any configurational change that conserves an energy function . We discuss this Hamiltonian formulation a bit later in this section. 

We recall from the previous section that a second-order Fredkin-reversible rule can always be redefined as a conventional first-order one, but only at a cost of... 

Fig. 8.2 An example of a Partitioning CA reversible rule, /, mapping (2x 2)-blocks of two-valued states to (2 X 2-blocks / (2 X 2) —> (2 X 2). Note that this rule conserves the total number of I s (indicated by a solid circle) and O s (indicated by an empty square). The system that evolves under this rule is in fact a universal CA (see Billiard Ball Model, later in this section).

The principles of statistical mechanics can be applied to a dynamical systeni provided that it obeys Louiville s Theorem (that is, it preserves volumes in phase space) and that its energy remains constant. The first requirement is easy since all reversible rules 4>r define bijective mappings of the phase space volume... 

The reversibility rule (impact on soil quality by man should be reversible). 

In order to determine whether the ROP of lactones into the corresponding aliphatic polyesters is possible, thermodynamics have to be taken into account. The ROP follows the micro-reversibility rule according to (1) ... 

This reversal rule for transposing products is important, and can be extended to several factors. 

Eq. (12) has commonly been used, e.g. in the analysis of mass and gas transfer in gas-filled systems. From this relationship it may be deduced that varies from 0 to 00 and that = 1 when G = 0.5. Systems for which G 4 0.5 are poorly gas-filled ( low-porous ) and those with G > 0.5 are highly gas-filled . The rule of reciprocals ( reversal rule ) facilitates the analysis of gas-filled structures such as foamed plastics by enabling the use of the so-called complementary gas-filled (porous, cellular) systems. The complementary systems relate to each other as a mold and casting or negative and positive . 

The chief advantage offered by this rule, proposed by Radushkevich is not only that it considerably reduces the number of systems being subject to statistical treatments but also simplifies the latter. For instance, by applying the reversal rule to closed-cell foamed plastics we convert the latter into their reciprocals in which the individual particles (formerly cells) are arranged in such a manner that they have very few contacts, if any. The latter systems formally all exhibit the characteristics of suspensions (or sols) which can at present readily be studied and described using the well-developed statistical apparatus, which is not possible for the original real system (foamed plastic), due to the insurmountable mathematical difficulties involved. 

The surface area expansion process in Figure 3.5 must obey the basic thermodynamic reversibility rules so that the movement from equilibrium to both directions should be so slow that the system can be continually relaxed. For most low-viscosity liquids, their surfaces relax very rapidly, and this reversibility criterion is usually met. However, if the viscosity of the liquid is too high, the equilibrium cannot take place and the thermodynamical equilibrium equations cannot be used in these conditions. For solids, it is impossible to expand a solid surface reversibly under normal experimental conditions because it will break or crack rather than flow under pressure. However, this fact should not confuse us surface tension of solids exists but we cannot apply a reversible area expansion method to solids because it cannot happen. Thus, solid surface tension determination can only be made by indirect methods such as liquid drop contact angle determination, or by applying various assumptions to some mechanical tests (see Chapters 8 and 9). 

The next stage in the development of a systematic asymmetric synthesis strategy addressed the question of whether or not there was any feedback mechanism whereby the seeding of the initial growth solution with product of one chirality (say D, and formed from an earlier solid state reaction) would modify product formation by induced crystallization such that even more D isomer may be formed on further reaction. Green and Heller proposed a cyclic process as shown in Scheme 6.12a. It transpired, however, that in such experiments the product obtained from a crystal grown in the presence of right-handed product repeatedly caused preferential crystallization into a crystal of the left-handed form — the so-called inversion or reversal rule. ... 

---