Path integral description for polymers

The path integral technique was first proposed by Feynmann (Feynmann Hibbs, 1965). The purpose of this technique was to deal with questions in quantum mechanics. It has been applied to the study of the statistical mechanics of polymer systems (Kreed, 1972 Doi Edwards, 1986) and liquid crystalline polymers as well (Jahnig, 1981 Warner et al, 1985 Wang Warner, 1986). The path integrals relate the configurations of a polymer chain to the paths of a particle when the particle is undergoing Brownian or diffusive motion. 

The path integral for a jointed-chain polymer, with a step length c, is given by 

If the chain is worm-like, the bending of the chain will cost the energy 

This elastic energy give rise to the Boltzmann factor of Equation 2.100, therefore the path integral becomes the case of a worm-like chain, u = du/ds, 

The unit vector u, enables Equation 2.103 to be rewritten as, by taking 

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