Journal of Physics A: Mathematical and Theoretical ( IF 2.1 ) Pub Date : 2021-01-19 , DOI: 10.1088/1751-8121/abd784 A I vila 1 , M I Gonzlez-Flores 2 , W Lebrecht 2
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 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 Nk-mers is related with the general binomial coefficient . 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 , it grows similarly with asymptotic value . Since the similar behavior for large values, we found the limit relationship 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.
中文翻译:
最大化熵的解析方法,用于计算线性链上k -mers的平衡密度
多原子(或k- mers)在线性链上的不可逆吸附与诸如胶体,长分子和蛋白质在固体基质上的吸附等现象有关。此过程将产生阻塞或阻塞的最终状态。在k = 2的情况下,二项式系数计算最终状态的数量。由正则系综中,通过使用斯特灵公式获得的玻尔兹曼吉布斯-香农熵功能,其平衡密度ρ 当量,2是其最大与在值的热力学极限ρ 当量,2≈0.822 991 17.此外,由于在相同的能量下,我们有几种可能的配置,所以我们获得了状态概率密度。最大化熵,收敛为L →∞的高斯分布。
在本文中,我们将这一分析推广到k > 2,以使熵最大化并获得平衡密度ρeq ,k 。我们首先开发一个完整的组合分析,以得到通用的递归公式(GRF),用于计算长度为L且链长为k的链上的所有封闭构型状态,其中k对应于广义的截短的斐波那契序列。分配Nk- mers的配置状态与一般二项式系数有关。由于斯特林近似不能用于GRF,因此我们通过数值计算状态概率密度并近似ρeq ,k 和σ 当量,ķ 对于大ķ聚体以高精度对ķ聚体高达ķ = 1,000,000。我们强调的是ρ 当量,ķ 从减小ķ = 2,...,8达到最小值在ķ = 9,然后与渐近值增大ρ 当量,∞ = 0.9285685。我们与干扰由RSA和在得到的密度相比ķ ≈16,两个曲线相交并且由于遍历性不破裂ρ 果酱,ķ ≈ ρ 当量,ķ 。在的情况下,其渐近值增长相似。由于大值的行为相似,因此对于任何k,我们发现极限关系为L →∞ 。最后,当k →∞时,我们得到了连续阻塞不可逆吸附的高斯分布,或等于不可逆阻塞停车问题。