Calculating entropy by Ma's method for a system of $k$-mers on a square lattice

TL;DR

This paper evaluates Ma's coincidence method for estimating the entropy of a system of $k$-mers on a lattice, demonstrating its accuracy in simplified models and proposing a practical approach for large systems.

Contribution

It introduces a novel application of Ma's method to estimate entropy in $k$-mer systems, analyzing its accuracy and defining optimal coincidence schemes.

Findings

Ma's method correlates well with naive estimates in simple models

The choice of coincidence definition affects accuracy

The approach offers a faster alternative for large systems

Abstract

Boltzmann's entropy is an important feature of any dynamic system. Calculating Boltzmann's entropy directly as the logarithm of the total number of microstates for a current macrostate is difficult for large systems. In the case of studying the diffusion of $k$-mers (linear segments of $k$ adjacent cells) on a lattice, the complexity of counting the possible microstates grows exponentially with the number of $k$-mers in the system. The obvious solution to this problem is to obtain only an estimate for the entropy, as this is faster to calculate. The 2D sliding window technique can be used to divide a system of $k$-mers on a lattice into subsystems. We use Ma's "coincidence" method to estimate the total number of possible states for such subsystems. In this study, the accuracy of Ma's method is studied in a simple combinatory model, both experimentally and theoretically. We determine which definition of "coincidence" for this scheme leads to greater accuracy. Where complex interactions between $k$-mers are ignored, Ma's estimate of the number of possible states for a system of $k$-mers correlates well with the estimate obtained using a "naive" method.