PCAO model

It is noteworthy that for investigation of properties of real polymer systems with topological constraints it is not enough to be able to calculate the statistical characteristics of chains in the lattice of obstacles. It is also necessary to be able to compare any concrete physical system with the unique lattice of obstacles, which is a much more complicated problem than the first task. In this way, the model polymer chain in an array of obstacles is an intermediate between the microscopical and phenomenological approaches. The direct investigation of the PCAO-model was fulfilled in Refs. [16-25]. 

The word W consisting of a sequence of letters corresponding to different entanglements (introduced in Sect. 2.21) plays a role of full topological invariant for the PCAO-model. It is closely connected with the concept of the primitive path obtained by means of roughing of the microscopic chain trajectory up to the scale of the lattice cell and by exclusion of all loop fragments not entangled with the obstacles (Fig. 5). 

In a number of papers [16-23,25], the discrete variant of the PCAO-model is considered the chain is modeled by a random walk on the lattice with spacing a and the topological constraints are placed on the dual lattice with period c. 

We have presented only a few basic physical results obtained within the framework of the PCAO-model the mathematical methods for investigating equilibrium and dynamic properties of a test polymer chain of different topology placed in the lattice of obstacles will be reviewed below. 

The model polymer chain in an array of obstacles (PCAO) (see Fig. 4) combines the geometrical clarity of its image with the possibility to investigate the influence of entanglements on equilibrium and dynamic properties of polymers quite precisely. 

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