Statistical properties of the 2 D attached Rouse chain

We study various dynamical properties (winding angles, areas) of a set of harmonically bound Brownian particles (monomers), one endpoint of this chain being kept fixed at the origin 0. In particular, we show that, for long times t, the areas {Ai} enclosed by the monomers scale like t1/2, with correlated gaussian distributions. This is at variance with the winding angles {θi} around fixed points that scale like t and are distributed according to independent Cauchy laws. In this paper, we will study the planar motion of a chain of n harmonically bound brownian particles. This model is usually refered in the litterature as the Rouse chain [1] and has shown to be historically very important in polymer science [2, 3]. We will consider such a chain attached at the origin 0 and examine some of its properties from the Brownian motion viewpoint. Representing a given configuration of the chain by a complex n-vector z (the components zi, i = 1, . . . , n, are the complex coordinates of the particles), we consider the set of all the closed trajectories of length t, i.e. z(t) = z(0), and this for all the starting configurations z(0). Practically, we will not weight the starting configurations with any thermodynamical factor. We are aware that this approach is quite different from the one taken in polymer physics [4] where, at t = 0, the chain is supposed to be in equilibrium with the environment at some finite temperature T . 1 Aj and θj being the area enclosed by the j th particle and its winding angle around 0, our goal is to compute the joint probability distributions P ({Ai}) and P ({θi}) for such trajectories. In order to make comparisons, we now recall some of the results concerning the planar Brownian motion. We first quote the area and winding angle distributions, respectively P (A) (Lévy’s law [5]) and P (θ) (Spitzer’s law [6]) for a particle allowed to wander everywhere in the plane: P (A) = π 2t 1 cosh πA t (1) P (θ) = 2 π ln t 1 1 + ( 2θ ln t )2 (2) (the last one holds, in the limit t → ∞, for open curves, the final point being left unspecified). Those two laws were obtained more than 40 years ago and since that time many refinements have been brought. For instance, in [7], the authors pointed out the importance of the small windings occuring when the particle is close to 0. Excluding an arbitrary small zone around 0, they showed that the variance 〈θ2〉 becomes finite in contrast with the Spitzer’s result, eq.(2). On the other hand, for Brownian motion on bounded domains [8, 9], the scaling variables in the limit t → ∞, become, resp., A/ √ t and θ/t with still an infinite variance 〈θ2〉. We close here this brief recall and start our chain study with the following set of coupled Langevin equations : ż1 = k (z2 − 2z1) + η1 żl = k (zl+1 + zl−1 − 2zl) + ηl , 2 ≤ l ≤ n− 1 (3) żn = k (zn−1 − zn) + ηn where k is the spring constant and ηm (≡ ηmx + iηmy) a gaussian white noise: 〈 ηm(t) 〉 = 0 〈 ηm(t) ηm′(t) 〉 = 2 δmm′ δ(t− t) (4) Introducing the complex n-vector η, eq. (3) can be written in a matrix form: ż = − k M z + η (5) 2 where M is the tridiagonal (n× n) matrix: M =   2 −1 0 · · · 0 −1 2 −1 · · · 0 0 −1 2 · · · 0 .. .. .. . . . .. 0 0 0 · · · 1   with an inverse given by: M =   1 1 1 · · · 1 1 2 2 · · · 2 1 2 3 · · · 3 .. .. .. . . . .. 1 2 3 · · · n   The eigenvalues of M are: ωj = 2 ( 1− cos π(2j − 1) 2n+ 1 ) , 1 ≤ j ≤ n (6) With the matrix ω = diag(ωi), we can write: ω = R M R (7) z = R Z (8) where R is an orthogonal matrix and the components of Z are the normal coordinates. Let us call P(z, z0, t) the probability for the chain to go from z0 at t = 0 to z at time t. P satisfies a Fokker-Planck equation [10]: ∂tP = ( ∂z kM z + t ∂z̄ kM z̄ + 2 ∂z̄ ∂z ) P (9) where ∂z (resp. ∂z̄) is a n-vector of components ∂zi (resp. ∂z̄i) and ∂z (resp. ∂z̄) is the transpose of ∂z (resp. ∂z̄). The solution can be written in terms of a path integral (DzDz̄ = ni=1 DziDz̄i): P(z, z0, t) = det ( e ) ∫ z(t)=z z(0)=z0 DzDz̄ exp ( − 2 ∫ t 0 ( ̇̄ z + kMz̄)(ż + kMz) dτ ) (10) ≡ F (z, z0, t).G(z, z0, t) with 3 F (z, z0, t) = det ( e ) e 1 2 (z̄kMz−z̄0kMz0) G(z, z0, t) = ∫ z(t)=z z(0)=z0 DzDz̄ exp ( − 2 ∫ t 0 ( t ̇̄ z ż + k z̄ Mz ) dτ ) = 〈 z|e0 |z0 〉 (11) = det ( S 2π ) exp ( − 2 ( z̄C z + z̄0 C z0 − z̄ S z0 − z̄0 S z )) (12) H0 = −2 ∂z̄ ∂z + 1 2 k z̄M z (13) The matrices S and C appearing in (12) are defined as: S = kM (sinh( t k M )) , C = kM coth( t k M ) (14) In fact, P, eq. (10), can be easily deduced from the gaussian distribution of η (use (5); det(e) is simply the functional Jacobian for the change of variable η → z [11]). (12) is a generalization of the harmonic oscillator propagator [12]. It is obtained by using the normal coordinates. Furthermore, as can be easily checked, P is properly normalized: ∫ dzdz̄ P(z, z0, t) = 1. Remark that an effective measure can be built for a distinguished monomer of the chain [4]: this can be done by integrating the Wiener measure (10) over all the paths of the other monomers. The result is a complicated expression that contains, in particular, a non local part (in time) exhibiting the non-Markovian character of the process for this monomer. Nevertheless, we will show, in the sequel, that, despite this complication, we can compute some joint laws for several monomers (and a fortiori for one monomer). So, let us turn to the computation of the area distribution P ({Ai}) for closed trajectories. Inserting the constraint