Equilibrium statistics of an infinitely long chain in the severe confined geometry: Rigorous results

We analyze the equlibrium statistics of a long linear homo-polymer chain confined in between two flat geometrical constraints under good solvent condition. The chain is ocupying two dimensional space and geometrical constraints are two impenetrable lines for the two dimensional space. A fully directed self avoiding walk lattice model is used to derive analytical expression of the partition function for the given value of separation in between the impenetrable lines. The exact values of the critical exponents ($\nu_{||}, \nu_{\perp}, \nu $ and $ \gamma_1$) were obtained for different value of separations in between the impenetrable lines. An exact expression of the grand canonical partition function of the confined semiflexible chain is also calculated for the given value of the constraints separation using generating function technique.