Peripheral PCA

Consider an isotropic version of the elementary peripheral PCA introduced above, this time also with binary site variables Ji t) = (1/2)(1 — Si t)) 0,1. Even-indexed (or odd-indexed) sites evolve on either even (or odd) time steps according to a set of rules determined by three conditional probabilities, po, p and p2 - 

As before, the conditional probabilities for obtaining Ti(i -P 1) = 0 are given in terms of these three parameters by P(0 01,02) =1 —P(1 01,02)- In the following we set po = 0 to make the zero-state an absorbing state. 

Note that the fact that the transition curve ends at pi = 1/2 and p2 = 1 could be predicted from a simple symmetry property (called duality in [kinzel85b]) namely, that any PCA defined by a set of conditional probabilities P(o 01O2O3) is preserved when each of these probabilities undergoes the duality transformation 

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Fig. 7 4 The triangular space-time lattice generated by the temporal evolution of the peripheral PCA system described in the text.

We have already commented on the equivalence between peripheral PCA and the problem of directed percolation (see footnote on page 343). It is easy to show that the pi and P2 of the isotropic system are given in terms of site- and bond- directed percolation probabilities - ps and pb, respectively - by the expressions pi = PaPb and P2 = PsPfe(2 — p ) see [domany84] and [kinzel85]. 

Except for a few special cases (most notably, /inear rules (see below) and the P2 = I line for the isotropic peripheral PCA discussed above, which happens to be an endpoint of a disorder line of an exactly solved 2-dim Ising model on a triangular lattice and can be solved exactly [kinzel85b]), phase-diagrams for PCA... 

In the case of the peripheral PCA above, for example, this equation appears as... 

EXAMPLE Consider the isotropic peripheral PCA defined in equation 7.69 ... 

EXAMPLES (1) Isotropic Peripheral PGA - consider the isotropic version of the peripheral PCA defined by equations 7.61 and 7.63 i.e. take a-2 — os = 02.3- In this case, the detailed balance condition is satisfied when the 3-spin coupling constant hi23 = 0. From equation 7.96, we see that this condition translates to... 

Thus, the isotropic peripheral PCA effectively describes the parallel 2-step dynamics (following the alternating updating of odd- and even- indexed sites) of the following onedimensional Isiug model ... 

Patient controlled intravenous analgesia (PCIA) is perhaps the simplest and most common method of PCA administration. PCIA is typically administered by inserting a needle into a peripheral vein, and then connecting the needle to a catheter or intravenous... 

PCRA is typically accomplished by inserting a small catheter into the affected site, and then attaching the catheter to some type of pump that enables the patient to self-administer small amounts of medication as needed. For example, a PCA system has been used to deliver local anesthetics such as bupivacaine and ropivacaine into the subacromial space following acromial decompression surgery.3,27 Likewise, PCRA can be used to deliver a local anesthetic to the area around a peripheral nerve (sciatic, popliteal, and so forth), and... 

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