A number of random processes in various fields of science is described by phenomenological equations containing a stochastic force, the best known example being the Langevin equation (LE) for the Brownian motion (BM) of particles. Long ago Vladimirsk
y (1942) proposed a simple method for solving such equations. The method, based on the classical Gibbs statistics, consists in converting the stochastic LE into a deterministic one, and is applicable to linear equations with any kind of memory. When the memory effects are taken into account in the description of the BM, the mean square displacement of the particle at long times can exhibit an anomalous (different from that in the Einstein theory) time dependence. In the present paper we show how some general properties of such anomalous BM can be easily derived using the Vladimirsky approach. The method can be effectively used in solving many of the problems currently considered in the literature. We apply it to the description of the BM when the memory kernel in the Volterra-type integro-differential LE exponentially decreases with the time. The problem of the hydrodynamic BM of a charged particle in an external magnetic field is also solved.
The Rouse-Zimm equation for the position vectors of beads mapping the polymer is generalized by taking into account the viscous aftereffect and the hydrodynamic noise. For the noise, the random fluctuations of the hydrodynamic tensor of stresses are
responsible. The preaveraging of the Oseen tensor for the nonstationary Navier-Stokes equation allowed us to relate the time correlation functions of the Fourier components of the bead position to the correlation functions of the hydrodynamic field created by the noise. The velocity autocorrelation function of the center of inertia of the polymer coil is considered in detail for both the short and long times when it behaves according to the t^(-3/2) law and does not depend on any polymer parameters. The diffusion coefficient of the polymer is close to that from the Zimm theory, with corrections depending on the ratio between the size of the bead and the size of the whole coil.
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.
