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

Denis V Kokosinskii,Mikhail V Ulyanov

doi:10.1088/1742-6596/1740/1/012015

Abstract

Random walk of rectangular particles on a square lattice leads to a pattern formation when the hard-core interaction between the particles is assumed. To estimate changes in the entropy during this random walk, we propose a modification of Ma’s method. The 2D sliding window technique was used to divide a system under consideration into subsystems. We used 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. 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.

Highlights

Boltzman’s entropy S is defined as S = kB ln(N ), where N is the number of possible microstates, corresponding to the system’s macrostate, and kB is the Boltzmann’s constant
Application of Ma’s method to a system of k-mers To apply Ma’s method, we first need to define a microstate for the system of k-mers on the lattice
The lattice can be subdivided into small non-overlapping squares

Summary

Introduction

Boltzman’s entropy S is defined as S = kB ln(N ), where N is the number of possible microstates, corresponding to the system’s macrostate, and kB is the Boltzmann’s constant. Boltzmann’s entropy is an important characteristic of any evaluating system. Calculating Boltzmann’s entropy directly as the logarithm of the total number of microstates for a current macrostate is difficult for large systems. Dense systems, direct calculation of the Boltamzann’s entropy may require a great deal of computation. The obvious solution to this problem is to obtain only an estimate for the entropy, as this is faster to calculate. Approximations of entropy, that are easier to compute, can help. Different ways to estimate entropy were proposed [1, 2, 4, 5]

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Conclusion