Redefinition of site percolation in light of entropy and the second law of thermodynamics

M. S. Rahman, M. K. Hassan

Published 2018-10-01Version 1

In this article, we revisit random site and bond percolation in square lattice focusing primarily on entropy which quantifies the degree of disorder and order parameter that measures the extent of order. Note that being two opposite quantities they can neither be minimum nor be maximum at the same time. This is perfectly consistent with bond percolation where we occupy bonds to connect sites and cluster sizes are measured by the number of sites connected by the occupied bonds. However, the same is not true for site percolation where we occupy site and measure cluster size in terms of the number of contiguous occupied sites. Rather, we find that entropy and order parameter are both zero at occupation probability $p=0$ and it violates the second law of thermodynamics. To overcome this we redefine the site percolation where we occupy sites to connect bonds and we measure cluster size by the number of bonds connected by occupied sites. This resolves the problem without affecting any of the existing known results whatsoever.