120 semi flexible polymer in two dimensions configurations of model polymer can be described by

1. First, we need to find the probability of a configuration. We know that the probability is pr

Question

12.0 Semi-flexible polymer in two dimensions: Configurations of a model polymer can be described by either a set of vectors {t_i} of length a in two dimensions (for i = 1, N); or alternatively by the angles {φ_i} between successive vectors, as indicated in the figure below. The polymer is at temperature T and subject to an energy H = -K Σ t_i · t_{i+1} = -K a² Σ cos φ_i where K is related to the bending rigidity, such that the probability of any configuration is proportional to exp (-H / kBT). (a) Show that ⟨t_m · t_n⟩ ∝ exp (-|n - m| / ξ) and obtain an expression for the persistence length ℓp = aξ. (You can leave the answer as the ratio of simple integrals) (b) Consider the end-to-end distance R as illustrated in the figure. Obtain an expression for ⟨R²⟩ in the limit of N ≫ 1. Find the probability p(R) in the limit of N ≫ 1.

          12.0 Semi-flexible polymer in two dimensions: Configurations of a model polymer can be described by either a set of vectors {t_i} of length a in two dimensions (for i = 1, N); or alternatively by the angles {φ_i} between successive vectors, as indicated in the figure below.

The polymer is at temperature T and subject to an energy

H = -K Σ t_i · t_{i+1} = -K a² Σ cos φ_i

where K is related to the bending rigidity, such that the probability of any configuration is proportional to exp (-H / kBT).

(a) Show that ⟨t_m · t_n⟩ ∝ exp (-|n - m| / ξ) and obtain an expression for the persistence length ℓp = aξ. (You can leave the answer as the ratio of simple integrals) 
(b) Consider the end-to-end distance R as illustrated in the figure. Obtain an expression for ⟨R²⟩ in the limit of N ≫ 1. Find the probability p(R) in the limit of N ≫ 1.

Added by Jennifer M.

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