Mapping reversible

P. Doruker and W.L. Mattice, A second generation of mapping/reverse mapping ofcoarse-grained and fiilly atomistic models ofpolymermelts,Macro/no/. TheorySimul. 8 463 (1999). 

One of the advantages of the Verlet integrator is that it is time reversible and symplectic[30, 31, 32]. Reversibility means that in the absence of numerical round off error, if the trajectory is run for many time steps, say nAt, and the velocities are then reversed, the trajectory will retrace its path and after nAt more time steps it will land back where it started. An integrator can be viewed as a mapping from one point in phase apace to another. If this mapping is applied to a measurable point set of states at on(> time, it will... 

Wisdom, J. The Origin of the Kirkwood Gaps A Mapping for Asteroidal Motion Near the 3/1 Commensurability. Astron. J. 87 (1982) 577-593 Tuckerman, M., Martyna, G. J., Berne, J. Reversible Multiple Time Scale Molecular Dynamics. J. Chem. Phys. 97 (1992) 1990-2001 Tuckerman, M., Berne, J. Vibrational Relaxation in Simple Fluids Comparison of Theory and Simulation. J. Chem. Phys. 98 (1993) 7301-7318 Humphreys, D. D., Friesner, R. A., Berne, B. J. A Multiple-Time Step Molecular Dynamics Algorithm for Macromolecules. J. Chem. Phys. 98 (1994) 6885-6892... 

The symmetry T p) = T[—p) implies that reversing the order of these three steps and changing the sign of r and p results in exactly the same method. In other words, Verlet is time-reversible. (In practice, the equations are usually reduced to equations for the positions at time-steps and the momenta at halfsteps, only, but for consideration of time-reversibility or symplecticness, the method should be formulated as a mapping of phase space.)... 

FIGURE 14 1 Electro static potential maps of (a) methyl fluoride and (b) methyllithium The electron distribution is reversed in the two compounds Carbon IS electron poor (blue) in methyl fluoride but electron rich (red) in methyllithium... 

Fig. 12. Tryptic map of it-PA (mol wt = 66,000) showing peptides formed from hydrolysis of reduced, alkylated rt-PA. Separation by reversed-phase octadecyl (C g) column using aqueous acetonitrile with an added acidic agent to the mobile phase. Arrows show the difference between A, normal, and B, mutant rt-PA where the glutamic acid residue, D, has replaced the normal arginine residue, C, at position 275.

M. Stromqvist, Peptide mapping using combinations of size-exclusion chromatography, reversed-phase chromatography and capillary electrophoresis , 7. Chromatogr. 667 304-310(1994). 

Chapter 8 describes a number of generalized CA models, including reversible CA, coupled-map lattices, quantum CA, reaction-diffusion models, immunologically motivated CA models, random Boolean networks, sandpile models (in the context of self-organized criticality), structurally dynamic CA (in which the temporal evolution of the value of individual sites of a lattice are dynamically linked to an evolving lattice structure), and simple CA models of combat. 

In the context of CA systems, it turns out that there is a difference between rules that are invertible and rules that are time-reversal invariant. A global CA rule S —> S, mapping a global state ct S to some other global state ct S, is said to be invertible if for all states ct S there exists exactly one predecessor state O S such that (cr) = a. The state transition graphs G for all such rules must therefore consist entirely of cycles. 

Before reviewing some of the more common approaches to constructing reversible CA from scratch, it is first of all important to understand why we should expect any difficulty at all in casually defining them. The difficulty stems mainly from the different cardinalities of the site value set and the set of A -tuples (where J f is the neighborhood size) defining the rule table. That is to say, since a general CA map (j) maps —> Zk, an obvious but important consequence is that (except for the... 

An example of a 2-state partitioning CA rule mapping (2 x 2) blocks to (2 x 2) blocks is shown in figure 8.2. The rule is rotationally symmetric, so that only one instance of the mapping for a block with a given number of rr = 1 sites need be given to completely define the rule. The rule is trivially reversible since each initial state is mapped to a unique final state. Observe also that the number of I s (shown as solid circles in the figure) and O s (shown as clear squares) is conserved, but that this simple conservation law is not a consequence of reversibility. Indeed, we could have just as easily defined a rule that conserved the number of I s and O s as this one but which was not reversible. (We mention here also that, despite its simple appearance, this rule happens to define a universal CA. We will have a chance to discuss reversible computation later on in this section.)... 

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... 

Consider a deteriiiinistic local reversible CA i.o. start with an infinite array of sites, T, arranged in some regular fashion, and a.ssume each site can be any of N states labeled by 0 < cr x) < N. If the number of sites is Af, the Hilbert space spanned by the states <7-(x is N- dimensional. The state at time t + 1, cTf+i(a ) depends only on the values cri x ) that are in the immediate neighborhood of X. Because the cellular automata is reversible, the mapping ai x) crt+i x ) is assumed to have a unique inveuse and the evolution operator U t,t + 1) in this Hilbert space is unitary,... 

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