Entropy of rigid $k$-mers on a square lattice

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 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 the usual approach. 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., also our results were compatible, with the same happening for the simulational estimates obtained by Pasinetti et al. in the whole range of rod sizes. Our results are also consistent with the asymptotic expression for the behavior of the entropy as a function of the size $k$, proposed by Dhar and Rajesh for very large rods ($k\gg 1$).

Figure 1

Example of a possible continuation for a state defined by vertical bonds in a strip of width $L=4$, identified by the reference line $R_1$, followed by a state connected to it defined by the reference line $R_2$.

Figure 2

Illustration of one step of the process of filling the strip of width $L=7$ with trimers ($k=3$). The initial profile is the thick black line and its baseline is at the level pointed by the black arrow. The height profile in this case will be (0,0,0,0,1,1,0); notice that there are two steps (5 and 6, from left to right) which are at the same height in both profiles. One possibility is to aggregate one horizontal rod (red on line) and two vertical rods (yellow on line). The new baseline is pointed at by the blue arrow and the new profile will be represented by (0,0,2,2,0,0,0). The contribution of this configuration is $z^3$.

Figure 3

Ratio between the number of states of the transfer matrix considering the usual approach and the profile method. The dashed lines are just a guide for the eye.

Figure 4

Strip of width $L=4$ with helical boundary conditions in the transverse direction. The $L+1$ lattice edges crossed by the dashed line, starting at point A and in the direction indicated by the arrow, define the vector which represents the state at this point. An additional site is incorporated when the transfer matrix is applied, so that the new starting point of the line is B. Trimers are represented by thick lines (red on line).

Figure 5

Panel (a): behavior of the entropy for tetramers ($k=4$) as a function of $L$ separated by sets, where dots, stars, squares, and triangles are related to the remainders $R=0$, 1, 2, and 3, respectively. The dashed lines are fittings of each set according to the relation given by Eq. (10). Panel (b) shows how the number of states of the transfer matrix grows as a function of $L$ for each of the sets. The dashed lines here are used just as a guide for the eyes, indicating the exponential behavior of that number when $L$ is large enough.

Figure 6

Extrapolated entropies of rods filling a square lattice as a function of the size of chains, $k$. The blue dashed and the red dotted lines correspond, respectively, to the lower and upper bounds, obtained in [13] and [22] [Eqs. (14) and (15), respectively]. The dashed-dotted line follows the behavior predicted by Dhar and Rajesh [13] when $k\to \infty$, i.e., $s=lnk/k^2$.

Figure 7

Dimension for the transfer matrix as a function of the width $L$, considering results coming from the periodical (pbc) and helical (hbc) boundary conditions obtained for $k=4$ and $k=5$.

Figure 8

Illustration on how the largest eigenvalues for a transfer matrix $\mathcal{T}$ are distributed with the same modulus on the complex plane (a) and how the translation produced by the parameter $p$ turns the eigenvalue located at the real axis into the only dominant eigenvalue for a transformed matrix ${\mathcal{T}}^{\prime }$. This example corresponds to the situation we get for $k=4$ with hbc, where the largest eigenvalue is fourfold degenerate.

Figure 9

Behavior of the trimer entropies, separating the behavior of remainders $R=0,1$ and $R=2$ (black, blue, and red dots, respectively) as a function of $lnL$. The linear behavior for $R=1$ and $R=2$ shows that $s\left(L\right)$ follows the behavior predicted by Eq. (16), while the logarithmic term is absent for $R=0$, which follows the same dependence observed for the pbc case. The estimates of $s_{\infty }$ were obtained via BST extrapolation as discussed below.