Self-avoiding loops

Connected sets of missing faces can be represented by their perimeters which form non-intersecting self-avoiding loops. Let us come back to the C (n) model and to the solid and solid model described in Section 4.1. The partition function, in the absence of the chain, is given by eqn (12.4.6)... 

For technical reasons one usually insists that the last step of the SAW also lies in the line y = 0, so that we are studying self-avoiding loops, rather than SAW. The reason for this... 

A classic problem of optimal paths is the travelling Scdesman problem (TSP) given a set of cities and a metric for the intercity distances, the salesman needs to find the shortest closed path (contour) through all the cities. It is a problem of combinatorial optimization and is known to be nondeterministic polynomial complete (NP complete) [46,47]. The optimal contour is necessarily a self-avoiding loop which is a special case of self-avoiding paths discussed in this book. 

In many cases, it is also helpful to have the path repel itself so that the transition pathway is self-avoiding. An acmal dynamic trajectory may oscillate about a minimum energy configuration prior to an activated transition. In the computed restrained, selfavoiding path, there will be no clusters of intermediates isolated in potential energy minima and no loops or redundant segments. The self-avoidance restraint reduces the wasted effort in the search for a characteristic reaction pathway. The constraints and restraints are essential components of the computational protocol. 

The requirement of self-avoidance restricts the admissible path classes and loop classes. A simple subclass of self-avoiding PS models are directed PS models We call a chmn directed if there is a preferred direction such that the order of the chain segments induces the same order on the vertex coordinates (w.r.t. the preferred direction), for each pair of vertices taken from two different chain segments. Such chains are then segment-avoiding. We call a PS model directed if paths, loops and chains are directed. 

This directed model consists of fully directed walks for the paths in the chain. These are clearly self-avoiding, and only take steps in positive directions. The corresponding loops are staircase polygons, which consist of two fully directed walks, which do not intersect or touch, but have a common starting point and end point. Paths are attached to these points. We distinguish the two strands of a loop. 

Since the critical behavior of PS models is essentially determined by the properties of loops, PS [70,71], and later Fisher [72], were led to consider various loop classes (together with straight paths for the double stranded segments). Whereas PS analyzed loop classes derived from random walks, Fisher considered loop classes derived from self-avoiding walks. 

Fisher concluded that the above values of the loop class exponent c intjply a continuous phase transition in d = 2 tuid d = 3. In [89] it is shown that the phase transition condition is satisfied in both d = 2 tuid in d = 3 for the case when the double stranded segments are treated as straight paths. Note that in d = 2, self-avoiding walks as paths will result in no phase transition. 

In summary, as we have shown in brief, and as shown in more detail in [89], a selfavoiding PS model (with unique marking, with self-avoiding bridges and (unrooted) selfavoiding loops) as defined above yields a first order phase transition in both d = 2 and d = 3. 

For the presumably more realistic model of pairs of interacting self-avoiding walks [79,83-85,80-82], the results of [89] suggest an interpretation of excluded volume effects, which complements the common one [83-85]. The self-avoiding PS models discussed above correctly account for excluded volume effects within a loop, but overestimate excluded volume effects between different segments of a chain, due to their directed chain structure. Since this leads to a first order phase transition in d = 2 and d = 3, one can conclude that the relaxation of excluded volume effects between different segments of the chain does not change the nature of the transition. 

Note that the denominator of eq. (X.31) is equal to 1 (from eq. X.30). Then expand the exponential in the numerator. Again, for n = 0, the only graphs which contribute are self-avoiding paths, but here they are not closed loops because eq. (X.31) contains two extra spin factors St Sj. In fact what we have is a sum over all self-avoiding walks linking sites i and j (Fig. X.4). If the walk involves N steps, the resulting contribution to eq. (X.30) is simply... 

A self-consistent field (SCF) is constructed for a polymer chain with excluded volume modeled as a self-avoiding random walk (SAW) of N eps (iV->oo). The SCF requires the introduction of a second exponent 0 in addition to the usual v exponent that dharacterizes the size of a SAW. The SCF equals N times the probability of an interaction of the chain end witb a distant part of the chain. In a-dimensional space scales as (self-consistency of the field yields the relations 0 = (4 - d)/3 and v d + ) =2. It is shown that 0q < < 0-j where 0q is the exponent associated with the probability of a SAW returning to its origin and 0-j is the exponent associated with the probability of a SAW forming a large loop with a long tail. A SCF (4>) is also determined for a semidilute solution of polymer volume fraction The number of binary interactions scales as < > <() - ... 

MOSFET (Ql). R4 is a gate resistor used to limit the current in the gate and to avoid any self-oscillations of the MOSFET because of noise. Components R7 and Cl provide compensation for the control loop. 

The use of self-pressurization based on nuclear heating is desirable to avoid the expense of an auxiliary electric power system, pressure control equipment and the potential for an accident event involving a break of the piping between an external pressurizer and the RPV. Also, the volume inside the RPV serving as an internal pressurizer for an IPWR tends to be larger than that for an external pressurizer with a loop-type PWR and, thus, provides a slower pressure response to transients. Moreover, the integral configuration is more comprehensible to the operator because the water level in the pressurizer dways corresponds directly to the amount of water above the core. 

The move away from transformers introduced several unexpected problems. Installation practices that provide acceptable performance with transformer-isolated equipment may not work for active balanced I/O equipment. Active balanced circuits can be less forgiving of wiring faults and short circuits some output stages will self-destruct if shorted. Avoidance of ground loops is more important with an active balanced system than with transformers. Active balanced input circuits can provide excellent noise rejection under ideal conditions. In the real world, however, their noise rejection can deteriorate rapidly. Transformers remain the best choice for the toughest situations. 

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