Exact Thermodynamics of a Polymer Confined to a Lattice of Finite Size
We write exact equations for the thermodynamic properties of a linear polymer molecule confined to walk on a lattice of finite size. The dimension of the space in which the lattice resides can be arbitrary. We also calculate polymer density. The boundary can be of arbitrary shape and the attraction of the monomers for the sites can be an arbitrary function of each site. The formalism is even more general in that each monomer can have its own energy of attraction for each lattice site. Multiple occupation of lattice sites is allowed which means that we have not solved the excluded volume problem. For one dimension we recover results obtained previously. The 2-d solution obtained here also solves the problem of an infinite parallelepiped. The method is easily extended by the methods of a previous paper to treat the problem of polymer stars or of branched polymers confined within a finite volume. This exact matrix formalism results in sparse matrices with approximately zM non-zero matrix elements where z is the lattice coordination number and the linear dimension M of the Matrix is equal to the number of lattice sites.