An analytic approach to maximize entropy for computing equilibrium densities of k-mers on linear chains

Abstract

Abstract The irreversible adsorption of polyatomic (or k-mers) on linear chains is related to phenomena such as the adsorption of colloids, long molecules, and proteins on solid substrates. This process generates jammed or blocked final states. In the case of k = 2, the binomial coefficient computes the number of final states. By the canonical ensemble, the Boltzmann–Gibbs–Shannon entropy function is obtained by using Stirling’s approximation, and its equilibrium density ρ eq,2 is its maximum at the thermodynamic limit with value ρ eq,2 ≈ 0.822 991 17. Moreover, since at the same energy we have several possible configurations, we obtain the state probability density. Maximizing the entropy, it converges to a Gaussian distribution N ( ρ e q , 2 , σ e q , 2 2 ) as L → ∞. In this article, we generalize this analysis to k > 2 to maximize the entropy and to get the equilibrium densities ρ eq,k . We first develop a complete combinatorial analysis to get the generalized recurrence formula (GRF) for counting all blocked configuration states on a chain of length L with fixed k, which corresponds to a generalized truncated Fibonacci sequence. The configuration states for allocating N k-mers is related with the general binomial coefficient N + 1 , k m . Since Stirling’s approximation cannot be used for GRF, we numerically compute the state probability density and approximate ρ eq,k and σ eq,k for large k-mers with high precision for k-mers up to k = 1, 000, 000. We highlight that ρ eq,k decreases from k = 2, …, 8 reaching a minimum at k = 9 and then increases with an asymptotic value ρ eq,∞ = 0.9285685. We compared with jamming densities obtained by RSA and at k ≈ 16, both curves intersect and ergodicity is not broken since ρ jam,k ≈ ρ eq,k . In the case of σ e q , k 2 , it grows similarly with asymptotic value σ e q , ∞ 2 = 0.862 9597 . Since the similar behavior for large values, we found the limit relationship σ e q , k 2 = ρ e q , k 2 as L → ∞ for any k. Finally, as k → ∞, we get the Gaussian distribution for the continuous blocked irreversible adsorption or equivalent to the irreversible blocked car parking problem.