The Brownian motion of a particle in a harmonic potential, which is simultaneously exposed either to a linear shear flow or to a plane Poiseuille flow is investigated. In the shear plane of both flows the probability distribution of the particle becomes anisotropic and the dynamics is changed in a characteristic manner compared to a trapped particle in a quiescent fluid. The particle distribution takes either an elliptical or a parachute shape or a superposition of both depending on the mean particle position in the shear plane. Simultaneously, shear-induced cross-correlations between particle fluctuations along orthogonal directions in the shear plane are found. They are asymmetric in time. In Poiseuille flow thermal particle fluctuations perpendicular to the flow direction in the shear plane induce a shift of the particle's mean position away from the potential minimum. Two complementary methods are suggested to measure shear-induced cross-correlations between particle fluctuations along orthogonal directions.

Shear-induced cross-correlations of particle fluctuations perpendicular and along streamlines are investigated experimentally and theoretically. Direct measurements of the Brownian motion of micron-sized beads, held by optical tweezers in a shear-flow cell, show a strong time asymmetry in the cross-correlation, which is caused by the non-normal amplification of fluctuations. Complementary measurements on the single particle probability distribution substantiate this behavior and both results are consistent with a Langevin model. In addition, a shear-induced anticorrelation between orthogonal random displacements of two trapped and hydrodynamically interacting particles is detected, having one or two extrema in time, depending on the positions of the particles.

We investigate the response of two-dimensional pattern-forming systems with a broken up-down symmetry, such as chemical reactions, to spatially resonant forcing and propose related experiments. The nonlinear behavior immediately above threshold is analyzed in terms of amplitude equations suggested for a 1:2 and 1:1 ratio between the wavelength of the spatial periodic forcing and the wavelength of the pattern of the respective system. Both sets of coupled amplitude equations are derived by a perturbative method from the Lengyel-Epstein model describing a chemical reaction showing Turing patterns, which gives us the opportunity to relate the generic response scenarios to a specific pattern-forming system. The nonlinear competition between stripe patterns and distorted hexagons is explored and their range of existence, stability, and coexistence is determined. Whereas without modulations hexagonal patterns are always preferred near onset of pattern formation, single-mode solutions (stripes) are favored close to threshold for modulation amplitudes beyond some critical value. Hence distorted hexagons only occur in a finite range of the control parameter and their interval of existence shrinks to zero with increasing values of the modulation amplitude. Furthermore, depending on the modulation amplitude, the transition between stripes and distorted hexagons is either subcritical or supercritical.

We present a numerical investigation of the Brownian motion and diffusion of a dumbbell in a two-dimensional periodic potential. Its dynamics is described by a Langevin model including the hydrodynamic interaction. With increasing values of the amplitude of the potential we find along the modulated spatial directions a reduction of the diffusion constant and of the impact of the hydrodynamic interaction. For modulation amplitudes of the potential in the range of the thermal energy the dumbbell diffusion exhibits a pronounced local maximum at a wavelength of about 3/2 of the dumbbell extension. This is especially emphasized for stiff springs connecting the two beads. Introduction.-Investigations on the diffusion of different colloidal particles in a homogeneous solvent have a long history [1,2], while studies on the diffusion of small spheres, dimers and polymers in different potentials attract considerable interest only for a short time [3,4,5,6,7,8,9]. Laser-tweezer arrays are a new powerful tool for generating the desired spatially periodic, correlated or unstructured potentials in order to study the effects of inhomogeneous potential landscapes on the motion of colloidal particles [3,4,5,10]. Furthermore recent studies of dumbbells and polymers in random potentials are exciting issues in statistical physics [11,12].

The dynamics of two Brownian particles trapped by two neighboring harmonic potentials in a linear shear flow is investigated. The positional correlation functions in this system are calculated analytically and analyzed as a function of the shear rate and the trap distance. Shear-induced cross-correlations between particle fluctuations along orthogonal directions in the shear plane are found. They are linear in the shear rate, asymmetric in time, and occur for one particle as well as between both particles. Moreover, the shear rate enters as a quadratic correction to the well-known correlations of random displacements along parallel spatial directions. The correlation functions depend on the orientation of the connection vector between the potential minima with respect to the flow direction. As a consequence, the inter-particle cross-correlations between orthogonal fluctuations can have zero, one or two local extrema as a function of time. Possible experiments for detecting these predicted correlations are described.