Simple Derivation of the First Cumulant for the Rouse Chain

Abstract

Abstract:A simple analytic expression for the first cumulant of the dynamic structure factor of a polymer coil in the Rouse model is derived. The obtained formula is exact within the usual assumption of the continuum distribution of beads along the chain. It reflects the contributions to the scattering of light or neutrons from both the internal motion of the polymer and its diffusion, and is valid in the whole region of the wave-vector change at the scattering.

Langevin theory of anomalous Brownian motion made simple

“…The effects of hydrodynamic memory appeared to be important also for rotational Brownian motion where the memory is revealed in the correlation function of the angular velocity of the spherical particle in the long-time algebraic asymptote ∼ t −5/2 , independently of the size and density of the particle [21]. Analogous results were found later by a number of other authors (see [22]).…”

“…The modern period of the studies of Brownian motion in suspensions can be related to the late 60s and early 70s of the last century, when the famous long-time 'tails' of the molecular velocity autocorrelation function (persistent or long-lived correlations) were discovered. First in computer experiments, and later they have been confirmed theoretically and experimentally (for a review of the literature see [22][23][24][25][26]). In particular, this discovery had put in doubt the commonly accepted conception on the microscopic and macroscopic properties of liquids as being characterized by very different time scales and showed real limits (broader than expected at that time) of hydrodynamic theory.…”

“…The theory that determined the range of applicability of the Langevin theory was published only in 1945 [20]. In this little known paper by Vladimirsky and Terletzky the first hydrodynamic theory of translational Brownian motion has been created (for the discussion of this work and the first theory of hydrodynamic rotational Brownian motion [21] see [22]). In the next sections we formulate and give the solution of the 'hydrodynamic Langevin equation'.…”

“…First of all it concerns the remarkable paper [20]. For the further development of the hydrodynamic theory of Brownian motion we again refer to the paper [22]. Here we only note that the mentioned computer experiments were already explained by Fisher in 1970 [27].…”