The Continuous Rouse Model

Equation (3.C.2) or (3.C.3) serves as the boundary condition for the differential Eq. (3.C.1), while the moments of the random forces are given in the continuous limit by 

Equation (3.C.1) represents the Brownian motions of coupled oscillators. Similar to the discrete case (Eqs. (3.31)-(3.33)), the standard method to solve the differential equation of the continuous Rouse model is to find the normal coordinates, each with its own independent motion. Considering the boundary conditions given by Eq. (3.C.3), we may write the Fomier expansion for Rn(t) in terms of the normal coordinates X, as 

Using the orthonormal property of the basis functions (2/A ) / cos q7rn/N), we can express the normal coordinates Xj,(t) as 

Performing the integration dn on the both sides of Eq. (3.C.9) and following a procedure similar to the discrete case, the same diffusion constant (Eq. (3.41)) can be obtained. 

Equation (3.C.10) in combination with Eqs. (3.C.12) and (3.C.13), which describes the motion of the normal coordinate Xp(t), can be represented by the general form of the Langevin equation given by Eq. (3.B.1) in combination with Eqs. (3.B.2) and (3.B.3). Thus, the solution of Eq. (3.B.1) given in Appendix 3.B can be readily used here. Since the random forces of different modes are independent of each other (Eq. (3.C.13)), the motions of the Xp s are also independent of each other. Thus, 

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