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Mean Field and the Single Homopolymer

We develop a statistical model for a confined chain molecule based on a monomer grand canonical ensemble. The molecule is subject to an external chemical potential, a backbone interaction, and an attractive interaction between all monomers. Using a Gaussian variable formalism and a mean field approximation, we analytically derive a minimum principle from which we can obtain relevant physical quantities, such as the monomer density, and we explore the limit in which the chain is subject to a tight confinement. Through a numerical implementation of the minimization process we show how we can obtain density profiles in three dimensions for arbitraty potentials, and we test the limits of validity of the theory.