Entropy of rigid k-mers on a square lattice

Abstract

Using the transfer matrix technique, we estimate the entropy for a gas of rods of sizes equal to k (named k-mers), which cover completely a square lattice. Our calculations were made considering three different constructions, using periodical and helical boundary conditions. One of those constructions, which we call Profile Method, was based on the calculations performed by Dhar and Rajesh [Phys. Rev. E 103, 042130 (2021)] to obtain a lower limit to the entropy of very large chains placed on the square lattice. This method, so far as we know, was never used before to define the transfer matrix, but turned out to be very useful, since it produces matrices with smaller dimensions than those obtained using other approaches. Our results were obtained for chain sizes ranging from k=2 to k=10 and they are compared with results already available in the literature. In the case of dimers ($k=2$) our results are compatible with the exact result, for trimers ($k=3$), recently investigated by Ghosh et al [Phys. Rev. E 75, 011115 (2007)] also our results were compatible, the same happening for the simulational estimates obtained by Pasinetti et al [Physical Review E 104, 054136 (2021)] in the whole range of rod sizes. Our results are consistent with the asymptotic expression for the behavior of the entropy as a function of the size $k$, proposed by Dhar and Rajesh [Phys. Rev. E 103, 042130 (2021)] for very large rods (k>>1).